Dafny Reference Manual
The dafny-lang community
v4.2.0 release snapshot
Abstract: This is the Dafny reference manual; it describes the Dafny programming language and how to use the Dafny verification system. Parts of this manual are more tutorial in nature in order to help the user understand how to do proofs with Dafny.
(Link to current document as html)
- 1. Introduction
- 2. Lexical and Low Level Grammar
- 3. Programs (grammar)
- 4. Modules (grammar)
- 5. Types
- 5.1. Kinds of types
- 5.2. Basic types
- 5.3. Type parameters (grammar)
- 5.4. Generic Instantiation (grammar)
- 5.5. Collection types
- 5.6. Types that stand for other types (grammar)
- 5.7. Newtypes (grammar)
- 5.8. Class types (grammar)
- 5.9. Trait types (grammar)
- 5.10. Array types (grammar)
- 5.11. Iterator types (grammar)
- 5.12. Arrow types (grammar)
- 5.13. Tuple types
- 5.14. Algebraic Datatypes (grammar)
- 5.14.1. Inductive datatypes
- 5.14.2. Coinductive datatypes
- 5.14.3. Coinduction
- 6. Member declarations
- 7. Specifications
- 8. Statements (grammar)
- 8.1. Labeled Statement (grammar)
- 8.2. Block Statement (grammar)
- 8.3. Return Statement (grammar)
- 8.4. Yield Statement (grammar)
- 8.5. Update and Call Statements (grammar)
- 8.6. Update with Failure Statement (
:-
) (grammar)- 8.6.1. Failure compatible types
- 8.6.2. Simple status return with no other outputs
- 8.6.3. Status return with additional outputs
- 8.6.4. Failure-returns with additional data
- 8.6.5. RHS with expression list
- 8.6.6. Failure with initialized declaration.
- 8.6.7. Keyword alternative
- 8.6.8. Key points
- 8.6.9. Failure returns and exceptions
- 8.7. Variable Declaration Statement (grammar)
- 8.8. Guards (grammar)
- 8.9. Binding Guards (grammar)
- 8.10. If Statement (grammar)
- 8.11. Match Statement (grammar)
- 8.12. While Statement (grammar)
- 8.13. For Loops (grammar)
- 8.14. Break and Continue Statements (grammar)
- 8.15. Loop Specifications
- 8.16. Print Statement (grammar)
- 8.17. Assert statement (grammar)
- 8.18. Assume Statement (grammar)
- 8.19. Expect Statement (grammar)
- 8.20. Reveal Statement (grammar)
- 8.21. Forall Statement (grammar)
- 8.22. Modify Statement (grammar)
- 8.23. Calc Statement (grammar)
- 9. Expressions
- 9.1. Lemma-call expressions (grammar)
- 9.2. Equivalence Expressions (grammar)
- 9.3. Implies or Explies Expressions (grammar)
- 9.4. Logical Expressions (grammar)
- 9.5. Relational Expressions (grammar)
- 9.6. Bit Shifts (grammar)
- 9.7. Terms (grammar)
- 9.8. Factors (grammar)
- 9.9. Bit-vector Operations (grammar)
- 9.10. As (Conversion) and Is (type test) Expressions (grammar)
- 9.11. Unary Expressions (grammar)
- 9.12. Primary Expressions (grammar)
- 9.13. Lambda expressions (grammar)
- 9.14. Left-Hand-Side Expressions (grammar)
- 9.15. Right-Hand-Side Expressions (grammar)
- 9.16. Array Allocation (grammar)
- 9.17. Object Allocation (grammar)
- 9.18. Havoc Right-Hand-Side (grammar)
- 9.19. Constant Or Atomic Expressions (grammar)
- 9.20. Literal Expressions (grammar}
- 9.21.
this
Expression (grammar) - 9.22. Old and Old@ Expressions (grammar)
- 9.23. Fresh Expressions (grammar)
- 9.24. Allocated Expressions (grammar)
- 9.25. Unchanged Expressions (grammar)
- 9.26. Cardinality Expressions (grammar)
- 9.27. Parenthesized Expressions (grammar)
- 9.28. Sequence Display Expression (grammar)
- 9.29. Set Display Expression (grammar)
- 9.30. Map Display Expression (grammar)
- 9.31. Endless Expression (grammar)
- 9.31.1. If Expression (grammar)
- 9.31.2. Case and Extended Patterns (grammar)
- 9.31.3. Match Expression (grammar)
- 9.31.4. Quantifier Expression (grammar)
- 9.31.5. Set Comprehension Expressions (grammar)
- 9.31.6. Statements in an Expression (grammar)
- 9.31.7. Let and Let or Fail Expression (grammar)
- 9.31.8. Map Comprehension Expression (grammar)
- 9.32. Name Segment (grammar)
- 9.33. Hash call (grammar)
- 9.34. Suffix (grammar)
- 9.35. Expression Lists (grammar)
- 9.36. Parameter Bindings (grammar)
- 9.37. Compile-Time Constants
- 9.38. List of specification expressions
- 10. Refinement
- 11. Attributes
- 11.1. Attributes on top-level declarations
- 11.2. Attributes on functions and methods
- 11.2.1.
{:abstemious}
- 11.2.2.
{:autoReq}
- 11.2.3.
{:axiom}
- 11.2.4.
{:compile}
- 11.2.5.
{:extern <name>}
- 11.2.6.
{:fuel X}
- 11.2.7.
{:id <string>}
- 11.2.8.
{:induction}
- 11.2.9.
{:only}
- 11.2.10.
{:opaque}
- 11.2.11.
{:print}
- 11.2.12.
{:priority}
- 11.2.13.
{:rlimit}
- 11.2.14.
{:selective_checking}
- 11.2.15.
{:tailrecursion}
- 11.2.16.
{:test}
- 11.2.17.
{:timeLimit N}
- 11.2.18.
{:timeLimitMultiplier X}
- 11.2.19.
{:verify false}
- 11.2.20.
{:vcs_max_cost N}
- 11.2.21.
{:vcs_max_keep_going_splits N}
- 11.2.22.
{:vcs_max_splits N}
- 11.2.23.
{:vcs_split_on_every_assert}
- 11.2.24.
{:synthesize}
- 11.2.25.
{:options OPT0, OPT1, ... }
- 11.2.1.
- 11.3. Attributes on assertions, preconditions and postconditions
- 11.4. Attributes on variable declarations
- 11.5. Attributes on quantifier expressions (forall, exists)
- 11.6. Deprecated attributes
- 11.7. Other undocumented verification attributes
- 12. Advanced Topics
- 13. Dafny User’s Guide
- 13.1. Introduction
- 13.2. Installing Dafny
- 13.3. Dafny Programs and Files
- 13.4. Dafny Code Style
- 13.5. Using Dafny From the Command Line
- 13.5.1. dafny commands
- 13.5.1.1. Options that are not associated with a command
- 13.5.1.2.
dafny resolve
- 13.5.1.3.
dafny verify
- 13.5.1.4.
dafny translate <language>
- 13.5.1.5.
dafny build
- 13.5.1.6.
dafny run
- 13.5.1.7.
dafny server
- 13.5.1.8.
dafny audit
- 13.5.1.9.
dafny format
- 13.5.1.10.
dafny test
- 13.5.1.11.
dafny doc
[Experimental] - 13.5.1.12.
dafny generate-tests
- 13.5.1.13.
dafny find-dead-code
- 13.5.1.14.
dafny measure-complexity
- 13.5.1.15. Plugins
- 13.5.1.16. Legacy operation
- 13.5.2. In-tool help
- 13.5.3. dafny exit codes
- 13.5.4. dafny output
- 13.5.5. Project files
- 13.5.1. dafny commands
- 13.6. Verification
- 13.7. Compilation
- 13.8. Dafny Command Line Options
- 13.8.1. Help and version information
- 13.8.2. Controlling input
- 13.8.3. Controlling plugins
- 13.8.4. Controlling output
- 13.8.5. Controlling language features
- 13.8.6. Controlling warnings
- 13.8.7. Controlling verification
- 13.8.8. Controlling compilation
- 13.8.9. Controlling Boogie
- 13.8.10. Controlling the prover
- 13.8.11. Controlling test generation
- 14. Dafny VSCode extension and the Dafny Language Server
- 15. Plugins to Dafny
- 16. Full list of legacy command-line options {#sec-full-command-line-options}
- 17. Dafny Grammar
- 17.1. Dafny Syntax
- 17.2. Dafny Grammar productions
- 17.2.1. Programs
- 17.2.2. Modules
- 17.2.3. Types
- 17.2.4. Type member declarations
- 17.2.5. Specifications
- 17.2.5.1. Method specifications
- 17.2.5.2. Function specifications
- 17.2.5.3. Lambda function specifications
- 17.2.5.4. Iterator specifications
- 17.2.5.5. Loop specifications
- 17.2.5.6. Requires clauses
- 17.2.5.7. Ensures clauses
- 17.2.5.8. Decreases clauses
- 17.2.5.9. Modifies clauses
- 17.2.5.10. Invariant clauses
- 17.2.5.11. Reads clauses
- 17.2.5.12. Frame expressions
- 17.2.6. Statements
- 17.2.6.1. Labeled statement
- 17.2.6.2. Non-Labeled statement
- 17.2.6.3. Break and continue statements
- 17.2.6.4. Block statement
- 17.2.6.5. Return statement
- 17.2.6.6. Yield statement
- 17.2.6.7. Update and call statement
- 17.2.6.8. Update with failure statement
- 17.2.6.9. Variable declaration statement
- 17.2.6.10. Guards
- 17.2.6.11. Binding guards
- 17.2.6.12. If statement
- 17.2.6.13. While Statement
- 17.2.6.14. For statement
- 17.2.6.15. Match statement
- 17.2.6.16. Assert statement
- 17.2.6.17. Assume statement
- 17.2.6.18. Expect statement
- 17.2.6.19. Print statement
- 17.2.6.20. Reveal statement
- 17.2.6.21. Forall statement
- 17.2.6.22. Modify statement
- 17.2.6.23. Calc statement
- 17.2.7. Expressions
- 17.2.7.1. Top-level expression
- 17.2.7.2. Equivalence expression
- 17.2.7.3. Implies expression
- 17.2.7.4. Logical expression
- 17.2.7.5. Relational expression
- 17.2.7.6. Bit-shift expression
- 17.2.7.7. Term (addition operations)
- 17.2.7.8. Factor (multiplication operations)
- 17.2.7.9. Bit-vector expression
- 17.2.7.10. As/Is expression
- 17.2.7.11. Unary expression
- 17.2.7.12. Primary expression
- 17.2.7.13. Lambda expression
- 17.2.7.14. Left-hand-side expression #g-lhs-expression}
- 17.2.7.15. Right-hand-side expression
- 17.2.7.16. Array allocation right-hand-side expression
- 17.2.7.17. Object allocation right-hand-side expression
- 17.2.7.18. Havoc right-hand-side expression
- 17.2.7.19. Atomic expressions
- 17.2.7.20. Literal expressions
- 17.2.7.21. This expression
- 17.2.7.22. Old and Old@ Expressions
- 17.2.7.23. Fresh Expressions
- 17.2.7.24. Allocated Expressions
- 17.2.7.25. Unchanged Expressions
- 17.2.7.26. Cardinality Expressions
- 17.2.7.27. Parenthesized Expression
- 17.2.7.28. Sequence Display Expression
- 17.2.7.29. Set Display Expression
- 17.2.7.30. Map Display Expression
- 17.2.7.31. Endless Expression
- 17.2.7.32. If expression
- 17.2.7.33. Match Expression
- 17.2.7.34. Case and Extended Patterns
- 17.2.7.35. Quantifier expression
- 17.2.7.36. Set Comprehension Expressions
- 17.2.7.37. Map Comprehension Expression
- 17.2.7.38. Statements in an Expression
- 17.2.7.39. Let and Let or Fail Expression
- 17.2.7.40. Name Segment
- 17.2.7.41. Hash Call
- 17.2.7.42. Suffix
- 17.2.7.43. Augmented Dot Suffix
- 17.2.7.44. Datatype Update Suffix
- 17.2.7.45. Subsequence Suffix
- 17.2.7.46. Subsequence Slices Suffix
- 17.2.7.47. Sequence Update Suffix
- 17.2.7.48. Selection Suffix
- 17.2.7.49. Argument List Suffix
- 17.2.7.50. Expression Lists
- 17.2.7.51. Parameter Bindings
- 17.2.7.52. Quantifier domains
- 17.2.7.53. Basic name and type combinations
- 18. Testing syntax rendering
- 19. References
1. Introduction
Dafny [@Leino:Dafny:LPAR16] is a programming language with built-in specification constructs,
so that verifying a program’s correctness with respect to those specifications
is a natural part of writing software.
The Dafny static program verifier can be used to verify the functional
correctness of programs.
This document is a reference manual for the programming language and a user guide
for the dafny
tool that performs verification and compilation to an
executable form.
The Dafny programming language is designed to support the static verification of programs. It is imperative, sequential, supports generic classes, inheritance and abstraction, methods and functions, dynamic allocation, inductive and coinductive datatypes, and specification constructs. The specifications include pre- and postconditions, frame specifications (read and write sets), and termination metrics. To further support specifications, the language also offers updatable ghost variables, recursive functions, and types like sets and sequences. Specifications and ghost constructs are used only during verification; the compiler omits them from the executable code.
The dafny
verifier is run as part of the compiler. As such, a programmer
interacts with it in much the same way as with the static type
checker—when the tool produces errors, the programmer responds by
changing the program’s type declarations, specifications, and statements.
(This document typically uses “Dafny” to refer to the programming language
and dafny
to refer to the software tool that verifies and compiles programs
in the Dafny language.)
The easiest way to try out the Dafny language is to download the supporting tools and documentation and
run dafny
on your machine as you follow along with the Dafny tutorial.
The dafny
tool can be run from the command line (on Linux, MacOS, Windows or other platforms) or from IDEs
such as emacs and VSCode, which can provide syntax highlighting and code manipulation capabilities.
The verifier is powered by Boogie [@Boogie:Architecture;@Leino:Boogie2-RefMan;@LeinoRuemmer:Boogie2] and Z3 [@deMouraBjorner:Z3:overview].
From verified programs, the dafny
compiler can produce code for a number
of different backends:
the .NET platform via intermediate C# files, Java, Javascript, Go, and C++.
Each language provides a basic Foreign Function Interface (through uses of :extern
)
and a supporting runtime library.
This reference manual for the Dafny verification system is based on the following references: [@Leino:Dafny:LPAR16], [@MSR:dafny:main], [@LEINO:Dafny:Calc], [@LEINO:Dafny:Coinduction], Co-induction Simply.
The main part of the reference manual is in top down order except for an initial section that deals with the lowest level constructs.
The details of using (and contributing to) the dafny tool are described in the User Guide (Section 13).
1.1. Dafny 4.0
The most recent major version of the Dafny language is Dafny 4.0, released in February 2023. It has some backwards incompatibilities with Dafny 3, as decribed in the migration guide.
The user documentation has been expanded with more examples, a FAQ, and an error explanation catalog. There is even a new book, Program Proofs by Dafny designer Rustan Leino.
The IDE now has a framework for showing error explanation information and corresponding quick fixes are being added, with refactoring operations on the horizon.
More details of 4.0 functionality are described in the release notes.
1.2. Dafny Example
To give a flavor of Dafny, here is the solution to a competition problem.
// VSComp 2010, problem 3, find a 0 in a linked list and return
// how many nodes were skipped until the first 0 (or end-of-list)
// was found.
// Rustan Leino, 18 August 2010.
//
// The difficulty in this problem lies in specifying what the
// return value 'r' denotes and in proving that the program
// terminates. Both of these are addressed by declaring a ghost
// field 'List' in each linked-list node, abstractly representing
// the linked-list elements from the node to the end of the linked
// list. The specification can now talk about that sequence of
// elements and can use 'r' as an index into the sequence, and
// termination can be proved from the fact that all sequences in
// Dafny are finite.
//
// We only want to deal with linked lists whose 'List' field is
// properly filled in (which can only happen in an acyclic list,
// for example). To that end, the standard idiom in Dafny is to
// declare a predicate 'Valid()' that is true of an object when
// the data structure representing that object's abstract value
// is properly formed. The definition of 'Valid()' is what one
// intuitively would think of as the ''object invariant'', and
// it is mentioned explicitly in method pre- and postconditions.
//
// As part of this standard idiom, one also declares a ghost
// variable 'Repr' that is maintained as the set of objects that
// make up the representation of the aggregate object--in this
// case, the Node itself and all its successors.
module {:options "--function-syntax:4"} M {
class Node {
ghost var List: seq<int>
ghost var Repr: set<Node>
var head: int
var next: Node? // Node? means a Node value or null
ghost predicate Valid()
reads this, Repr
{
this in Repr &&
1 <= |List| && List[0] == head &&
(next == null ==> |List| == 1) &&
(next != null ==>
next in Repr && next.Repr <= Repr && this !in next.Repr &&
next.Valid() && next.List == List[1..])
}
static method Cons(x: int, tail: Node?) returns (n: Node)
requires tail == null || tail.Valid()
ensures n.Valid()
ensures if tail == null then n.List == [x]
else n.List == [x] + tail.List
{
n := new Node;
n.head, n.next := x, tail;
if (tail == null) {
n.List := [x];
n.Repr := {n};
} else {
n.List := [x] + tail.List;
n.Repr := {n} + tail.Repr;
}
}
}
method Search(ll: Node?) returns (r: int)
requires ll == null || ll.Valid()
ensures ll == null ==> r == 0
ensures ll != null ==>
0 <= r && r <= |ll.List| &&
(r < |ll.List| ==>
ll.List[r] == 0 && 0 !in ll.List[..r]) &&
(r == |ll.List| ==> 0 !in ll.List)
{
if (ll == null) {
r := 0;
} else {
var jj,i := ll,0;
while (jj != null && jj.head != 0)
invariant jj != null ==>
jj.Valid() &&
i + |jj.List| == |ll.List| &&
ll.List[i..] == jj.List
invariant jj == null ==> i == |ll.List|
invariant 0 !in ll.List[..i]
decreases |ll.List| - i
{
jj := jj.next;
i := i + 1;
}
r := i;
}
}
method Main()
{
var list: Node? := null;
list := list.Cons(0, list);
list := list.Cons(5, list);
list := list.Cons(0, list);
list := list.Cons(8, list);
var r := Search(list);
print "Search returns ", r, "\n";
assert r == 1;
}
}
2. Lexical and Low Level Grammar
As with most languages, Dafny syntax is defined in two levels. First the stream of input characters is broken up into tokens. Then these tokens are parsed using the Dafny grammar.
The Dafny grammar is designed as an attributed grammar, which is a conventional BNF-style set of productions, but in which the productions can have arguments. The arguments control some alternatives within the productions, such as whether an alternative is allowed or not in a specific context. These arguments allow for a more compact and understandable grammar.
The precise, technical details of the grammar are presented together in Section 17. The expository parts of this manual present the language structure less formally. Throughout this document there are embedded hyperlinks to relevant grammar sections, marked as grammar.
2.1. Dafny Input
Dafny source code files are readable text encoded in UTF-8.
All program text other than the contents of comments, character, string and verbatim string literals
consists of printable and white-space ASCII characters,
that is, ASCII characters in the range !
to ~
, plus space, tab,
carriage return and newline (ASCII 9, 10, 13, 32) characters.
(In some past versions of Dafny, non-ASCII, unicode representations of some mathematical symbols were
permitted in Dafny source text; these are no longer recognized.)
String and character literals and comments may contain any unicode character, either directly or as an escape sequence.
2.2. Tokens and whitespace
The characters used in a Dafny program fall into four groups:
- White space characters: space, tab, carriage return and newline
- alphanumerics: letters, digits, underscore (
_
), apostrophe ('
), and question mark (?
) - punctuation:
(){}[],.`;
- operator characters (the other printable characters)
Except for string and character literals, each Dafny token consists of a sequence of consecutive characters from just one of these groups, excluding white-space. White-space is ignored except that it separates tokens and except in the bodies of character and string literals.
A sequence of alphanumeric characters (with no preceding or following additional
alphanumeric characters) is a single token. This is true even if the token
is syntactically or semantically invalid and the sequence could be separated into
more than one valid token. For example, assert56
is one identifier token,
not a keyword assert
followed by a number; ifb!=0
begins with the token
ifb
and not with the keyword if
and token b
; 0xFFFFZZ
is an illegal
token, not a valid hex number 0xFFFF
followed by an identifier ZZ
.
White-space must be used to separate two such tokens in a program.
Somewhat differently, operator tokens need not be separated.
Only specific sequences of operator characters are recognized and these
are somewhat context-sensitive. For example, in seq<set<int>>
, the grammar
knows that >>
is two individual >
tokens terminating the nested
type parameter lists; the right shift operator >>
would never be valid here. Similarly, the
sequence ==>
is always one token; even if it were invalid in its context,
separating it into ==
and >
would always still be invalid.
In summary, except for required white space between alphanumeric tokens, adding or removing white space between tokens can never result in changing the meaning of a Dafny program. For most of this document, we consider Dafny programs as sequences of tokens.
2.3. Character Classes
This section defines character classes used later in the token definitions. In this section
- a backslash is used to start an escape sequence (so for example
'\n'
denotes the single linefeed character) - double quotes enclose the set of characters constituting a character class
- enclosing single
quotes are used when there is just one character in the class
(perhaps expressed with a
\
escape character) +
indicates the union of two character classes-
is the set-difference between the two classesANY
designates all unicode characters.
name | description |
---|---|
letter | ASCII upper or lower case letter; no unicode characters |
digit | base-ten digit (“0123456789”) |
posDigit | digits, excluding 0 (“123456789”) |
posDigitFrom2 | digits excluding 0 and 1 (“23456789”) |
hexdigit | a normal hex digit (“0123456789abcdefABCDEF”) |
special | `?_” |
cr | carriage return character (ASCII 10) |
lf | line feed character (ASCII 13) |
tab | tab character (ASCII 9) |
space | space character (ASCII 32) |
nondigitIdChar | characters allowed in an identifier, except digits (letter + special) |
idchar | characters allowed in an identifier (nondigitIdChar + digits) |
nonidchar | characters not in identifiers (ANY - idchar) |
charChar | characters allowed in a character constant (ANY - ‘'’ - ‘\’ - cr - lf) |
stringChar | characters allowed in a string constant (ANY - ‘”’ - ‘\’ - cr - lf) |
verbatimStringChar | characters allowed in a verbatim string constant (ANY - ‘”’) |
The special characters are the characters in addition to alphanumeric characters that are allowed to appear in a Dafny identifier. These are
'
because mathematicians like to put primes on identifiers and some ML programmers like to start names of type parameters with a'
,_
because computer scientists expect to be able to have underscores in identifiers, and?
because it is useful to have?
at the end of names of predicates, e.g.,Cons?
.
A nonidchar
is any character except those that can be used in an identifier.
Here the scanner generator will interpret ANY
as any unicode character.
However, nonidchar
is used only to mark the end of the !in
token;
in this context any character other than whitespace or printable ASCII
will trigger a subsequent scanning or parsing error.
2.4. Comments
Comments are in two forms.
- They may go from
/*
to*/
. - They may go from
//
to the end of the line.
A comment is identified as a token during the tokenization of
input text and is then discarded for the purpose of interpreting the
Dafny program. (It is retained to enable auto-formatting
and provide accurate source locations for error messages.)
Thus comments are token separators: a/*x*/b
becomes two tokens
a
and b
.
Comments may be nested, but note that the nesting of multi-line comments is behavior that is different from most programming languages. In Dafny,
method m() {
/* comment
/* nested comment
*/
rest of outer comment
*/
}
is permitted; this feature is convenient for commenting out blocks of program statements that already have multi-line comments within them. Other than looking for end-of-comment delimiters, the contents of a comment are not interpreted. Comments may contain any characters.
Note that the nesting is not fool-proof. In
method m() {
/* var i: int;
// */ line comment
var j: int;
*/
}
and
method m() {
/* var i: int;
var s: string := "a*/b";
var j: int;
*/
}
the */
inside the line comment and the string are seen as the end of the outer
comment, leaving trailing text that will provoke parsing errors.
2.5. Documentation comments
Like many other languages, Dafny permits documentation comments in a program file. Such comments contain natural language descriptions of program elements and may be used by IDEs and documentation generation tools to present information to users.
In Dafny programs.
- Documentation comments (a) either begin with
/**
or (b) begin with//
or /*` in specific locations - Doc-comments may be associated with any declaration, including type definitions, export declarations, and datatype constructors.
- They may be placed before or after the declaration.
- If before, it must be a
/**
comment and may not have any blank or white-space lines between the comment and the declaration. - If after, any comments are placed after the signature (with no intervening lines), but before any
specifications or left-brace that starts a body, and may be
//
or/**
or/*
comments. - If doc-comments are in both places, only the comments after the declaration are used.
- If before, it must be a
- Doc-comments after the declaration are preferred.
- If the first of a series of single-line or multi-line comments is interpreted as a doc-string, then any subsequent comments are appended to it, so long as there are no intervening lines, whether blank, all white-space or containing program text.
- The extraction of the doc-string from a multiline comment follow these rules
- On the first line, an optional
*
right after/*
and an optional space are removed, if present - On other lines, the indentation space (with possibly one star in it) is removed, as if the content was supposed to align with A if the comment started with
/** A
for example.
- On the first line, an optional
- The documentation string is interpreted as plain text, but it is possible to provide a user-written plugin that provides other interpretations. VSCode as used by Dafny interprets any markdown syntax in the doc-string.
Here are examples:
const c0 := 8
/** docstring about c0 */
/** docstring about c1 */
const c1 := 8
/** first line of docstring */
const c2 := 8
/** second line of docstring */
const c3 := 8
// docstring about c3
// on two lines
const c4 := 8
// just a comment
// just a comment
const c5 := 8
Datatype constructors may also have comments:
datatype T = // Docstring for T
| A(x: int,
y: int) // Docstring for A
| B() /* Docstring for B */ |
C() // Docstring for C
/** Docstring for T0*/
datatype T0 =
| /** Docstring for A */
A(x: int,
y: int)
| /** Docstring for B */
B()
| /** Docstring for C */
C()
As can export
declarations:
module M {
const A: int
const B: int
const C: int
const D: int
export
// This is the eponymous export set intended for most clients
provides A, B, C
export Friends extends M
// This export set is for clients who need to know more of the
// details of the module's definitions.
reveals A
provides D
}
2.6. Tokens (grammar)
The Dafny tokens are defined in this section.
2.6.1. Reserved Words
Dafny has a set of reserved words that may not be used as identifiers of user-defined entities. These are listed here.
In particular note that
array
,array2
,array3
, etc. are reserved words, denoting array types of given rank. However,array1
andarray0
are ordinary identifiers.array?
,array2?
,array3?
, etc. are reserved words, denoting possibly-null array types of given rank, but notarray1?
orarray0?
.bv0
,bv1
,bv2
, etc. are reserved words that denote the types of bitvectors of given length. The sequence of digits after ‘array’ or ‘bv’ may not have leading zeros: for example,bv02
is an ordinary identifier.
2.6.2. Identifiers
In general, an ident
token (an identifier) is a sequence of idchar
characters where
the first character is a nondigitIdChar
. However tokens that fit this pattern
are not identifiers if they look like a character literal
or a reserved word (including array or bit-vector type tokens).
Also, ident
tokens that begin with an _
are not permitted as user identifiers.
2.6.3. Digits
A digits
token is a sequence of decimal digits (digit
), possibly interspersed with
underscores for readability (but not beginning or ending with an underscore).
Example: 1_234_567
.
A hexdigits
token denotes a hexadecimal constant, and is a sequence of hexadecimal digits (hexdigit
)
prefaced by 0x
and
possibly interspersed with underscores for readability (but not beginning or ending with an underscore).
Example: 0xffff_ffff
.
A decimaldigits
token is a decimal fraction constant, possibly interspersed with underscores for readability (but not beginning or ending with an underscore).
It has digits both before and after a single period (.
) character. There is no syntax for floating point numbers with exponents.
Example: 123_456.789_123
.
2.6.4. Escaped Character
The escapedChar
token is a multi-character sequence that denotes a non-printable or non-ASCII character.
Such tokens begin with a backslash characcter (\
) and denote
a single- or double-quote character, backslash,
null, new line, carriage return, tab, or a
Unicode character with given hexadecimal representation.
Which Unicode escape form is allowed depends on the value of the --unicode-char
option.
If --unicode-char:false
is stipulated,
\uXXXX
escapes can be used to specify any UTF-16 code unit.
If --unicode-char:true
is stipulated,
\U{X..X}
escapes can be used to specify any Unicode scalar value.
There must be at least one hex digit in between the braces, and at most six.
Surrogate code points are not allowed.
The hex digits may be interspersed with underscores for readability
(but not beginning or ending with an underscore), as in \U{1_F680}
.
The braces are part of the required character sequence.
Note that although Unicode letters are not allowed in Dafny identifiers, Dafny does support Unicode in its character, string, and verbatim strings constants and in its comments.
2.6.5. Character Constant Token
The charToken
token denotes a character constant.
It is either a charChar
or an escapedChar
enclosed in single quotes.
2.6.6. String Constant Token
A stringToken
denotes a string constant.
It consists of a sequence of stringChar
and escapedChar
characters enclosed in
double quotes.
A verbatimStringToken
token also denotes a string constant.
It is a sequence of any verbatimStringChar
characters (which includes newline characters),
enclosed between @"
and "
, except that two
successive double quotes represent one quote character inside
the string. This is the mechanism for escaping a double quote character,
which is the only character needing escaping in a verbatim string.
Within a verbatim string constant, a backslash character represents itself
and is not the first character of an escapedChar
.
2.6.7. Ellipsis
The ellipsisToken
is the character sequence ...
and is typically used to designate something missing that will
later be inserted through refinement or is already present in a parent declaration.
2.7. Low Level Grammar Productions
2.7.1. Identifier Variations
2.7.1.1. Identifier
A basic ordinary identifier is just an ident
token.
It may be followed by a sequence of suffixes to denote compound entities.
Each suffix is a dot (.
) and another token, which may be
- another
ident
token - a
digits
token - the
requires
reserved word - the
reads
reserved word
Note that
- Digits can be used to name fields of classes and destructors of
datatypes. For example, the built-in tuple datatypes have destructors
named 0, 1, 2, etc. Note that as a field or destructor name, a digit sequence
is treated as a string, not a number: internal
underscores matter, so
10
is different from1_0
and from010
. m.requires
is used to denote the precondition for methodm
.m.reads
is used to denote the things that methodm
may read.
2.7.1.2. No-underscore-identifier
A NoUSIdent
is an identifier except that identifiers with a leading
underscore are not allowed. The names of user-defined entities are
required to be NoUSIdent
s or, in some contexts, a digits
.
We introduce more mnemonic names
for these below (e.g. ClassName
).
A no-underscore-identifier is required for the following:
- module name
- class or trait name
- datatype name
- newtype name
- synonym (and subset) type name
- iterator name
- type variable name
- attribute name
A variation, a no-underscore-identifier or a digits
, is allowed for
- datatype member name
- method or function or constructor name
- label name
- export id
- suffix that is a typename or constructor
All user-declared names do not start with underscores, but there are internally generated names that a user program might use that begin with an underscore or are just an underscore.
2.7.1.3. Wild identifier
A wild identifier is a no-underscore-identifier except that the singleton
_
is allowed. The _
is replaced conceptually by a unique
identifier distinct from all other identifiers in the program.
A _
is used when an identifier is needed, but its content is discarded.
Such identifiers are not used in expressions.
Wild identifiers may be used in these contexts:
- formal parameters of a lambda expression
- the local formal parameter of a quantifier
- the local formal parameter of a subset type or newtype declaration
- a variable declaration
- a case pattern formal parameter
- binding guard parameter
- for loop parameter
- LHS of update statements
2.7.2. Qualified Names
A qualified name starts with the name of a top-level entity and then is followed by
zero or more DotSuffix
s which denote a component. Examples:
Module.MyType1
MyTuple.1
MyMethod.requires
A.B.C.D
The identifiers and dots are separate tokens and so may optionally be separated by whitespace.
2.7.3. Identifier-Type Combinations
Identifiers are typically declared in combination with a type, as in
var i: int
However, Dafny infers types in many circumstances, and in those, the type can be omitted. The type is required for field declarations and formal parameters of methods, functions and constructors (because there is no initializer). It may be omitted (if the type can be inferred) for local variable declarations, pattern matching variables, quantifiers,
Similarly, there are circumstances in which the identifier name is not needed, because it is not used. This is allowed in defining algebraic datatypes.
In some other situations a wild identifier can be used, as described above.
2.7.4. Quantifier Domains (grammar)
Several Dafny constructs bind one or more variables to a range of possible values.
For example, the quantifier forall x: nat | x <= 5 :: x * x <= 25
has the meaning
“for all integers x between 0 and 5 inclusive, the square of x is at most 25”.
Similarly, the set comprehension set x: nat | x <= 5 :: f(x)
can be read as
“the set containing the result of applying f to x, for each integer x from 0 to 5 inclusive”.
The common syntax that specifies the bound variables and what values they take on
is known as the quantifier domain; in the previous examples this is x: nat | x <= 5
,
which binds the variable x
to the values 0
, 1
, 2
, 3
, 4
, and 5
.
Here are some more examples.
x: byte
(where a value of typebyte
is an int-based numberx
in the range0 <= x < 256
)x: nat | x <= 5
x <- integerSet
x: nat <- integerSet
x: nat <- integerSet | x % 2 == 0
x: nat, y: nat | x < 2 && y < 2
x: nat | x < 2, y: nat | y < x
i | 0 <= i < |s|, y <- s[i] | i < y
A quantifier domain declares one or more quantified variables, separated by commas. Each variable declaration can be nothing more than a variable name, but it may also include any of three optional elements:
-
The optional syntax
: T
declares the type of the quantified variable. If not provided, it will be inferred from context. -
The optional syntax
<- C
attaches a collection expressionC
as a quantified variable domain. Here a collection is any value of a type that supports thein
operator, namely sets, multisets, maps, and sequences. The domain restricts the bindings to the elements of the collection:x <- C
impliesx in C
. The example above can also be expressed asvar c := [0, 1, 2, 3, 4, 5]; forall x <- c :: x * x <= 25
. -
The optional syntax
| E
attaches a boolean expressionE
as a quantified variable range, which restricts the bindings to values that satisfy this expression. In the example abovex <= 5
is the range attached to thex
variable declaration.
Note that a variable’s domain expression may reference any variable declared before it,
and a variable’s range expression may reference the attached variable (and usually does) and any variable declared before it.
For example, in the quantifier domain i | 0 <= i < |s|, y <- s[i] | i < y
, the expression s[i]
is well-formed
because the range attached to i
ensures i
is a valid index in the sequence s
.
Allowing per-variable ranges is not fully backwards compatible, and so it is not yet allowed by default;
the --quantifier-syntax:4
option needs to be provided to enable this feature (See Section 13.8.5).
2.7.5. Numeric Literals (grammar)
Integer and bitvector literals may be expressed in either decimal or hexadecimal (digits
or hexdigits
).
Real number literals are written as decimal fractions (decimaldigits
).
3. Programs (grammar)
At the top level, a Dafny program (stored as files with extension .dfy
)
is a set of declarations. The declarations introduce (module-level)
constants, methods, functions, lemmas, types (classes, traits, inductive and
coinductive datatypes, newtypes, type synonyms, abstract types, and
iterators) and modules, where the order of introduction is irrelevant.
Some types, notably classes, also may contain a set of declarations, introducing fields, methods,
and functions.
When asked to compile a program, Dafny looks for the existence of a
Main()
method. If a legal Main()
method is found, the compiler will emit
an executable appropriate to the target language; otherwise it will emit
a library or individual files.
The conditions for a legal Main()
method are described in the User Guide
(Section 3.4).
If there is more than one Main()
, Dafny will emit an error message.
An invocation of Dafny may specify a number of source files.
Each Dafny file follows the grammar of the Dafny
non-terminal.
A file consists of
- a sequence of optional include directives, followed by
- top level declarations, followed by
- the end of the file.
3.1. Include Directives (grammar)
Examples:
include "MyProgram.dfy"
include @"/home/me/MyFile.dfy"
Include directives have the form "include" stringToken
where
the string token is either a normal string token or a
verbatim string token. The stringToken
is interpreted as the name of
a file that will be included in the Dafny source. These included
files also obey the Dafny
grammar. Dafny parses and processes the
transitive closure of the original source files and all the included files,
but will not invoke the verifier on the included files unless they have been listed
explicitly on the command line or the --verify-included-files
option is
specified.
The file name may be a path using the customary /
, .
, and ..
punctuation.
The interpretation of the name (e.g., case-sensitivity) will depend on the
underlying operating system. A path not beginning with /
is looked up in
the underlying file system relative to the
location of the file in which the include directive is stated.
Paths beginning with a device
designator (e.g., C:
) are only permitted on Windows systems.
Better style advocates using relative paths in include directives so that
groups of files may be moved as a whole to a new location.
Paths of files on the command-line or named in --library
options are
relative the the current working directory.
3.2. Top Level Declarations (grammar)
Examples:
abstract module M { }
trait R { }
class C { }
datatype D = A | B
newtype pos = i: int | i >= 0
type T = i: int | 0 <= i < 100
method m() {}
function f(): int
const c: bool
Top-level declarations may appear either at the top level of a Dafny file, or within a (sub)module declaration. A top-level declaration is one of various kinds of declarations described later. Top-level declarations are implicitly members of a default (unnamed) top-level module.
Declarations within a module or at the top-level all begin with reserved keywords and do not end with semicolons.
These declarations are one of these kinds:
- methods and functions, encapsulating computations or actions
- const declarations, which are names (of a given type) initialized to an unchanging value; declarations of variables and mutable fields are not allowed at the module level
- type declarations of various kinds (Section 5 and the following sections)
Methods, functions and const declarations are placed in an implicit class declaration
that is in the top-level implicit module. These declarations are all implicitly
static
(and may not be declared explicitly static).
3.3. Declaration Modifiers (grammar)
Examples:
abstract module M {
class C {
static method m() {}
}
}
ghost opaque const c : int
Top level declarations may be preceded by zero or more declaration modifiers. Not all of these are allowed in all contexts.
The abstract
modifier may only be used for module declarations.
An abstract module can leave some entities underspecified.
Abstract modules are not compiled.
The ghost
modifier is used to mark entities as being used for
specification only, not for compilation to code.
The opaque
modifier may be used on const declarations and functions.
The static
modifier is used for class members that
are associated with the class as a whole rather than with
an instance of the class. This modifier may not be used with
declarations that are implicitly static, as are members of the
top-level, unnamed implicit class.
The following table shows modifiers that are available for each of the kinds of declaration. In the table we use already-ghost (already-non-ghost) to denote that the item is not allowed to have the ghost modifier because it is already implicitly ghost (non-ghost).
Declaration | allowed modifiers |
---|---|
module | abstract |
class | - |
trait | - |
datatype or codatatype | - |
field (const) | ghost opaque |
newtype | - |
synonym types | - |
iterators | - |
method | ghost static |
lemma | already-ghost static |
least lemma | already-ghost static |
greatest lemma | already-ghost static |
constructor | ghost |
function | ghost static opaque (Dafny 4) |
function method | already-non-ghost static opaque (Dafny 3) |
function (non-method) | already-ghost static opaque (Dafny 3) |
predicate | ghost static opaque (Dafny 4) |
predicate method | already-non-ghost static opaque (Dafny 3) |
predicate (non-method) | already-ghost static opaque (Dafny 3) |
least predicate | already-ghost static opaque |
greatest predicate | already-ghost static opaque |
3.4. Executable programs
Dafny programs have an important emphasis on verification, but the programs may also be executable.
To be executable, the program must have exactly one Main
method and that
method must be a legal main entry point.
- The program is searched for a method with the attribute
{:main}
. If exactly one is found, that method is used as the entry point; if more than one method has the{:main}
attribute, an error message is issued. - Otherwise, the program is searched for a method with the name
Main
. If more than one is found an error message is issued.
Any abstract modules are not searched for candidate entry points, but otherwise the entry point may be in any module or type. In addition, an entry-point candidate must satisfy the following conditions:
- The method has no type parameters and either has no parameters or one non-ghost parameter of type
seq<string>
. - The method has no non-ghost out-parameters.
- The method is not a ghost method.
- The method has no requires or modifies clauses, unless it is marked
{:main}
. - If the method is an instance (that is, non-static) method and the enclosing type is a class, then that class must not declare any constructor. In this case, the runtime system will allocate an object of the enclosing class and will invoke the entry-point method on it.
- If the method is an instance (that is, non-static) method and the enclosing type is not a class, then the enclosing type must, when instantiated with auto-initializing type parameters, be an auto-initializing type. In this case, the runtime system will invoke the entry-point method on a value of the enclosing type.
Note, however, that the following are allowed:
- The method is allowed to have
ensures
clauses - The method is allowed to have
decreases
clauses, including adecreases *
. (IfMain()
has adecreases *
, then its execution may go on forever, but in the absence of adecreases *
onMain()
,dafny
will have verified that the entire execution will eventually terminate.)
If no legal candidate entry point is identified, dafny
will still produce executable output files, but
they will need to be linked with some other code in the target language that
provides a main
entry point.
If the Main
method takes an argument (of type seq<string>
), the value of that input argument is the sequence
of command-line arguments, with the first entry of the sequence (at index 0) being a system-determined name for the
executable being run.
The exit code of the program, when executed, is not yet specified.
4. Modules (grammar)
Examples:
module N { }
import A
export A reveals f
Structuring a program by breaking it into parts is an important part of creating large programs. In Dafny, this is accomplished via modules. Modules provide a way to group together related types, classes, methods, functions, and other modules, as well as to control the scope of declarations. Modules may import each other for code reuse, and it is possible to abstract over modules to separate an implementation from an interface.
Module declarations are of three types:
- a module definition
- a module import
- a module export definition
Module definitions and imports each declare a submodule of its enclosing module, which may be the implicit, undeclared, top-level module.
4.1. Declaring New Modules (grammar)
Examples:
module P { const i: int }
abstract module A.Q { method m() {} }
module M { module N { } }
A module definition
- has an optional modifier (only
abstract
is allowed) - followed by the keyword “module”
- followed by a name (a sequence of dot-separated identifiers)
- followed by a body enclosed in curly braces
A module body consists of any declarations that are allowed at the top level: classes, datatypes, types, methods, functions, etc.
module Mod {
class C {
var f: int
method m()
}
datatype Option = A(int) | B(int)
type T
method m()
function f(): int
}
You can also put a module inside another, in a nested fashion:
module Mod {
module Helpers {
class C {
method doIt()
var f: int
}
}
}
Then you can refer to the members of the Helpers
module within the
Mod
module by prefixing them with “Helpers.”. For example:
module Mod {
module Helpers {
class C {
constructor () { f := 0; }
method doIt()
var f: int
}
}
method m() {
var x := new Helpers.C();
x.doIt();
x.f := 4;
}
}
Methods and functions defined at the module level are available like classes, with just the module name prefixing them. They are also available in the methods and functions of the classes in the same module.
module Mod {
module Helpers {
function addOne(n: nat): nat {
n + 1
}
}
method m() {
var x := 5;
x := Helpers.addOne(x); // x is now 6
}
}
Note that everything declared at the top-level (in all the files constituting the program) is implicitly part of a single implicit unnamed global module.
4.2. Declaring nested modules standalone
As described in the previous section, module declarations can be nested. It is also permitted to declare a nested module outside of its “containing” module. So instead of
module A {
module B {
}
}
one can write
module A {
}
module A.B {
}
The second module is completely separate; for example, it can be in
a different file.
This feature provides flexibility in writing and maintenance;
for example, it can reduce the size of module A
by extracting module A.B
into a separate body of text.
However, it can also lead to confusion, and program authors need to take care.
It may not be apparent to a reader of module A
that module A.B
exists;
the existence of A.B
might cause names to be resolved differently and
the semantics of the program might be (silently) different if A.B
is
present or absent.
4.3. Importing Modules (grammar)
Examples:
import A
import opened B
import A = B
import A : B
import A.B
import A`E
import X = A.B`{E,F}
Sometimes you want to refer to
things from an existing module, such as a library. In this case, you
can import one module into another. This is done via the import
keyword, which has two forms with different meanings.
The simplest form is the concrete import, which has
the form import A = B
. This declaration creates a reference to the
module B
(which must already exist), and binds it to the new local name
A
. This form can also be used to create a reference to a nested
module, as in import A = B.C
. The other form, using a :
, is
described in Section 4.6.
As modules in the same scope must have different names, this ability
to bind a module to a new name allows disambiguating separately developed
external modules that have the same name.
Note that the new name is only bound in the scope containing
the import declaration; it does not create a global alias. For
example, if Helpers
was defined outside of Mod
, then we could import
it:
module Helpers {
function addOne(n: nat): nat {
n + 1
}
}
module Mod {
import A = Helpers
method m() {
assert A.addOne(5) == 6;
}
}
Note that inside m()
, we have to use A
instead of Helpers
, as we bound
it to a different name. The name Helpers
is not available inside m()
(or anywhere else inside Mod
),
as only names that have been bound inside Mod
are available. In order
to use the members from another module, that other module either has to be declared
there with module
or imported with import
. (As described below, the
resolution of the ModuleQualifiedName
that follows the =
in the import
statement or the refines
in a module declaration uses slightly
different rules.)
We don’t have to give Helpers
a new name, though, if we don’t want
to. We can write import Helpers = Helpers
to import the module under
its own name; Dafny
even provides the shorthand import Helpers
for this behavior. You
can’t bind two modules with the same name at the same time, so
sometimes you have to use the = version to ensure the names do not
clash. When importing nested modules, import B.C
means import C = B.C
;
the implicit name is always the last name segment of the module designation.
The first identifier in the dot-separated sequence of identifers that constitute the qualified name of the module being imported is resolved as (in order)
- a submodule of the importing module,
- or a sibling module of the importing module,
- or a sibling module of some containing module, traversing outward. There is no way to refer to a containing module, only sibling modules (and their submodules).
Import statements may occur at the top-level of a program (that is, in the implicit top-level module of the program) as well. There they serve as a way to give a new name, perhaps a shorthand name, to a module. For example,
module MyModule { } // declare MyModule
import MyModule // error: cannot add a module named MyModule
// because there already is one
import M = MyModule // OK. M and MyModule are equivalent
4.4. Opening Modules
Sometimes, prefixing the members of the module you imported with its
name is tedious and ugly, even if you select a short name when
importing it. In this case, you can import the module as opened
,
which causes all of its members to be available without adding the
module name. The opened
keyword, if present, must immediately follow import
.
For example, we could write the previous example as:
module Helpers {
function addOne(n: nat): nat {
n + 1
}
}
module Mod {
import opened Helpers
method m() {
assert addOne(5) == 6;
}
}
When opening modules, the newly bound members have lower priority
than local definitions. This means if you define
a local function called addOne
, the function from Helpers
will no
longer be available under that name. When modules are opened, the
original name binding is still present however, so you can always use
the name that was bound to get to anything that is hidden.
module Helpers {
function addOne(n: nat): nat {
n + 1
}
}
module Mod {
import opened H = Helpers
function addOne(n: nat): nat {
n - 1
}
method m() {
assert addOne(5) == 6; // this is now false,
// as this is the function just defined
assert H.addOne(5) == 6; // this is still true
}
}
If you open two modules that both declare members with the same name,
then neither member can be referred to without a module prefix, as it
would be ambiguous which one was meant. Just opening the two modules
is not an error, however, as long as you don’t attempt to use members
with common names. However, if the ambiguous references actually
refer to the same declaration, then they are permitted.
The opened
keyword may be used with any kind of
import
declaration, including the module abstraction form.
An import opened
may occur at the top-level as well. For example,
module MyModule { } // declares MyModule
import opened MyModule // does not declare a new module, but does
// make all names in MyModule available in
// the current scope, without needing
// qualification
import opened M = MyModule // names in MyModule are available in
// the current scope without qualification
// or qualified with either M (because of this
// import) or MyModule (because of the original
// module definition)
The Dafny style guidelines suggest using opened imports sparingly. They are best used when the names being imported have obvious and unambiguous meanings and when using qualified names would be verbose enough to impede understanding.
There is a special case in which the behavior described above is altered.
If a module M
declares a type M
and M
is import opened
without renaming inside
another module X
, then the rules above would have, within X
,
M
mean the module and M.M
mean the type. This is verbose. So in this
somewhat common case, the type M
is effectively made a local declaration within X
so that it has precedence over the module name. Now M
refers to the type.
If one needs to refer to the module, it will have to be renamed as part of
the import opened
statement.
This special-case behavior does give rise to a source of ambiguity. Consider the example
module Option {
const a := 1
datatype Option = A|B { static const a := 2 }
}
module X {
import opened Option
method M() { print Option.a; }
}
Option.a
now means the a
in the datatype instead of the a
in the module.
To avoid confusion in such cases, it is an ambiguity error if a name
that is declared in both the datatype and the module is used
when there is an import open
of
the module (without renaming).
4.5. Export Sets and Access Control (grammar)
Examples:
export E extends F reveals f,g provides g,h
export E reveals *
export reveals f
export E
export
export E ... reveals f
In some programming languages, keywords such as public
, private
, and protected
are used to control access to (that is, visibility of) declared program entities.
In Dafny, modules and export sets provide that capability.
Modules combine declarations into logically related groups.
Export sets then permit selectively exposing subsets of a module’s declarations;
another module can import the export set appropriate to its needs.
A user can define as many export sets as are needed to provide different
kinds of access to the module’s declarations.
Each export set designates a list of names, which must be
names that are declared in the module (or in a refinement parent).
By default (in the absence of any export set declarations)
all the names declared in a module are available outside the
module using the import
mechanism.
An export set enables a module to disallow the
use of some declarations outside the module.
Export sets have names; those names are used in import
statements to
designate which export set of a module is being imported.
If a module M
has export sets
E1
and E2
, we can write import A = M`E1
to create a module alias
A
that contains only the
names in E1
. Or we can write import A = M`{E1,E2}
to import the union
of names in E1
and E2
as module alias A
.
As before, import M`E1
is an
abbreviation of import M = M`E1
.
If no export set is given in an import statement, the default export set of the module is used.
There are various
defaults that apply differently in different cases.
The following description is with respect to an example module M
:
M
has no export sets declared. Then another module may simply import Z = M
to obtain access to all of M’s declarations.
M
has one or more named export sets (e.g., E
, F
). Then another module can
write import Z = M`E
or import Z = M`{E,F}
to obtain access to the
names that are listed in export set E
or to the union of those in export sets
E
and F
, respectively. If no export set has the same name as the module,
then an export set designator must be used: in that case you cannot write
simply import Z = M
.
M
has an unnamed export set, along with other export sets (e.g., named E
). The unnamed
export set is the default export set and implicitly has the same name as
the module. Because there is a default export set, another module may write
either import Z = M
or import Z = M`M
to import the names in that
default export set. You can also still use the other export sets with the
explicit designator: import Z = M`E
M
declares an export set with the same name as the module. This is equivalent
to declaring an export set without a name. import M
and import M`M
perform the same function in either case; the export set with or without
the name of the module is the default export set for the module.
Note that names of module aliases (declared by import statements) are
just like other names in a module; they can be included or omitted from
export sets.
Names brought into a module by refinement are treated the same as
locally declared names and can be listed in export set declarations.
However, names brought into a module by import opened
(either into a module
or a refinement parent of a module) may
not be further exported. For example,
module A {
const a := 10
const z := 10
}
module B {
import opened Z = A // includes a, declares Z
const b := Z.a // OK
}
module C {
import opened B // includes b, Z, but not a
method m() {
//assert b == a; // error: a is not known
//assert b == B.a; // error: B.a is not valid
//assert b == A.a; // error: A is not known
assert b == Z.a; // OK: module Z is known and includes a
}
}
However, in the above example,
- if
A
has one export setexport Y reveals a
then the import in moduleB
is invalid becauseA
has no default export set; - if
A
has one export setexport Y reveals a
andB
hasimport Z = A`Y
thenB
’s import is OK. So is the use ofZ.a
in the assert becauseB
declaresZ
andC
brings inZ
through theimport opened
andZ
containsa
by virtue of its declaration. (The aliasZ
is not able to have export sets; all of its names are visible.) - if
A
has one export setexport provides z
thenA
does have a default export set, so the import inB
is OK, but neither the use ofa
inB
nor asZ.a
in C would be valid, becausea
is not inZ
.
The default export set is important in the resolution of qualified names, as described in Section 4.8.
There are a few unusual cases to be noted:
- an export set can be completely empty, as in
export Nothing
- an eponymous export set can be completely empty, as in
export
, which by default has the same name as the enclosing module; this is a way to make the module completely private - an export set declaration followed by an extreme predicate declaration looks like this:
export least predicate P() { true }
In this case, theleast
(orgreatest
) is the identifier naming the export set. Consequently,export least predicate P[nat]() { true }
is illegal because[nat]
cannot be part of a non-extreme predicate. So, it is not possible to declare an eponymous, empty export set by omitting the export id immediately prior to a declaration of an extreme predicate, because theleast
orgreatest
token is parsed as the export set identifier. The workaround for this situation is to either put the name of the module in explicitly as the export ID (not leaving it to the default) or reorder the declarations. - To avoid confusion, the code
module M { export least predicate P() { true } }
provokes a warning telling the user that the
least
goes with theexport
.
4.5.1. Provided and revealed names
Names can be exported from modules in two ways, designated by provides
and reveals
in the export set declaration.
When a name is exported as provided, then inside a module that has imported the name only the name is known, not the details of the name’s declaration.
For example, in the following code the constant a
is exported as provided.
module A {
export provides a
const a := 10
const b := 20
}
module B {
import A
method m() {
assert A.a == 10; // a is known, but not its value
// assert A.b == 20; // b is not known through A`A
}
}
Since a
is imported into module B
through the default export set A`A
,
it can be referenced in the assert statement. The constant b
is not
exported, so it is not available. But the assert about a
is not provable
because the value of a
is not known in module B
.
In contrast, if a
is exported as revealed, as shown in the next example,
its value is known and the assertion can be proved.
module A {
export reveals a
const a := 10
const b := 20
}
module B {
import A
method m() {
assert A.a == 10; // a and its value are known
// assert A.b == 20; // b is not known through A`A
}
}
The following table shows which parts of a declaration are exported by an
export set that provides
or reveals
the declaration.
declaration | what is exported | what is exported
| with provides | with reveals
---------------------|---------------------|---------------------
const x: X := E | const x: X | const x: X := E
---------------------|---------------------|---------------------
var x: X | var x: X | not allowed
---------------------|---------------------|---------------------
function F(x: X): Y | function F(x: X): Y | function F(x: X): Y
specification... | specification... | specification...
{ | | {
Body | | Body
} | | }
---------------------|---------------------|---------------------
method M(x: X) | method M(x: X) | not allowed
returns (y: Y) | returns (y: Y) |
specification... | specification... |
{ | |
Body; | |
} | |
---------------------|---------------------|---------------------
type Opaque | type Opaque | type Opaque
{ | |
// members... | |
} | |
---------------------|---------------------|---------------------
type Synonym = T | type Synonym | type Synonym = T
---------------------|---------------------|---------------------
type S = x: X | type S | type S = x: X
| P witness E | | | P witness E
---------------------|---------------------|---------------------
newtype N = x: X | type N | newtype N = x: X
| P witness E | | | P witness E
{ | |
// members... | |
} | |
---------------------|---------------------|---------------------
datatype D = | type D | datatype D =
Ctor0(x0: X0) | | Ctor0(x0: X0)
| Ctor1(x1: X1) | | | Ctor1(x1: X1)
| ... | | | ...
{ | |
// members... | |
} | |
---------------------|---------------------|---------------------
class Cl | type Cl | class Cl
extends T0, ... | | extends T0, ...
{ | | {
constructor () | | constructor ()
spec... | | spec...
{ | |
Body; | |
} | |
// members... | |
} | | }
---------------------|---------------------|---------------------
trait Tr | type Tr | trait Tr
extends T0, ... | | extends T0, ...
{ | |
// members... | |
} | |
---------------------|---------------------|---------------------
iterator Iter(x: X) | type Iter | iterator Iter(x: X)
yields (y: Y) | | yields (y: Y)
specification... | | specification...
{ | |
Body; | |
} | |
---------------------|---------------------|---------------------
module SubModule | module SubModule | not allowed
... | ... |
{ | { |
export SubModule | export SubModule |
... | ... |
export A ... | |
// decls... | // decls... |
} | } |
---------------------|---------------------|---------------------
import L = MS | import L = MS | not allowed
---------------------|---------------------|---------------------
Variations of functions (e.g., predicate
, twostate function
) are
handled like function
above, and variations of methods (e.g.,
lemma
and twostate lemma
) are treated like method
above. Since
the whole signature is exported, a function or method is exported to
be of the same kind, even through provides
. For example, an exported
twostate lemma
is exported as a twostate lemma
(and thus is known
by importers to have two implicit heap parameters), and an exported
least predicate P
is exported as a least predicate P
(and thus
importers can use both P
and its prefix predicate P#
).
If C
is a class
, trait
, or iterator
, then provides C
exports
the non-null reference type C
as an abstract type. This does not reveal
that C
is a reference type, nor does it export the nullable type C?
.
In most cases, exporting a class
, trait
, datatype
, codatatype
, or
abstract type does not automatically export its members. Instead, any member
to be exported must be listed explicitly. For example, consider the type
declaration
trait Tr {
function F(x: int): int { 10 }
function G(x: int): int { 12 }
function H(x: int): int { 14 }
}
An export set that contains only reveals Tr
has the effect of exporting
trait Tr {
}
and an export set that contains only provides Tr, Tr.F reveals Tr.H
has
the effect of exporting
type Tr {
function F(x: int): int
function H(x: int): int { 14 }
}
There is no syntax (for example, Tr.*
) to export all members of a type.
Some members are exported automatically when the type is revealed. Specifically:
- Revealing a
datatype
orcodatatype
automatically exports the type’s discriminators and destructors. - Revealing an
iterator
automatically exports the iterator’s members. - Revealing a class automatically exports the class’s anonymous constructor, if any.
For a class
, a constructor
member can be exported only if the class is revealed.
For a class
or trait
, a var
member can be exported only if the class or trait is revealed
(but a const
member can be exported even if the enclosing class or trait is only provided).
When exporting a sub-module, only the sub-module’s eponymous export set is exported.
There is no way for a parent module to export any other export set of a sub-module, unless
it is done via an import
declaration of the parent module.
The effect of declaring an import as opened
is confined to the importing module. That
is, the ability of use such imported names as unqualified is not passed on to further
imports, as the following example illustrates:
module Library {
const xyz := 16
}
module M {
export
provides Lib
provides xyz // error: 'xyz' is not declared locally
import opened Lib = Library
const k0 := Lib.xyz
const k1 := xyz
}
module Client {
import opened M
const a0 := M.Lib.xyz
const a1 := Lib.xyz
const a2 := M.xyz // error: M does not have a declaration 'xyz'
const a3 := xyz // error: unresolved identifier 'xyz'
}
As highlighted in this example, module M
can use xyz
as if it were a local
name (see declaration k1
), but the unqualified name xyz
is not made available
to importers of M
(see declarations a2
and a3
), nor is it possible for
M
to export the name xyz
.
A few other notes:
- A
provides
list can mention*
, which means that all local names (except export set names) in the module are exported asprovides
. - A
reveals
list can mention*
, which means that all local names (except export set names) in the module are exported asreveals
, if the declaration is allowed to appear in areveals
clause, or asprovides
, if the declaration is not allowed to appear in areveals
clause. - If no export sets are declared, then the implicit
export set is
export reveals *
. - A refinement module acquires all the export sets from its refinement parent.
- Names acquired by a module from its refinement parent are also subject to export lists. (These are local names just like those declared directly.)
4.5.2. Extends list
An export set declaration may include an extends list, which is a list of one or more export set names from the same module containing the declaration (including export set names obtained from a refinement parent). The effect is to include in the declaration the union of all the names in the export sets in the extends list, along with any other names explicitly included in the declaration. So for example in
module M {
const a := 10
const b := 10
const c := 10
export A reveals a
export B reveals b
export C extends A, B
reveals c
}
export set C
will contain the names a
, b
, and c
.
4.6. Module Abstraction
Sometimes, using a specific implementation is unnecessary; instead,
all that is needed is a module that implements some interface. In
that case, you can use an abstract module import. In Dafny, this is
written import A : B
. This means bind the name A
as before, but
instead of getting the exact module B
, you get any module which
adheres to B
. Typically, the module B
may have abstract type
definitions, classes with bodiless methods, or otherwise be unsuitable
to use directly. Because of the way refinement is defined, any
refinement of B
can be used safely. For example, suppose we start with
these declarations:
abstract module Interface {
function addSome(n: nat): nat
ensures addSome(n) > n
}
abstract module Mod {
import A : Interface
method m() {
assert 6 <= A.addSome(5);
}
}
We can be more precise if we know that addSome
actually adds
exactly one. The following module has this behavior. Further, the
postcondition is stronger, so this is actually a refinement of the
Interface module.
module Implementation {
function addSome(n: nat): nat
ensures addSome(n) == n + 1
{
n + 1
}
}
We can then substitute Implementation
for A
in a new module, by
declaring a refinement of Mod
which defines A
to be Implementation
.
abstract module Interface {
function addSome(n: nat): nat
ensures addSome(n) > n
}
abstract module Mod {
import A : Interface
method m() {
assert 6 <= A.addSome(5);
}
}
module Implementation {
function addSome(n: nat): nat
ensures addSome(n) == n + 1
{
n + 1
}
}
module Mod2 refines Mod {
import A = Implementation
...
}
When you refine an abstract import into a concrete one Dafny checks that the concrete module is a refinement of the abstract one. This means that the methods must have compatible signatures, all the classes and datatypes with their constructors and fields in the abstract one must be present in the concrete one, the specifications must be compatible, etc.
A module that includes an abstract import must be declared abstract
.
4.7. Module Ordering and Dependencies
Dafny isn’t particular about the textual order in which modules are declared, but they must follow some rules to be well formed. In particular, there must be a way to order the modules in a program such that each only refers to things defined before it in the ordering. That doesn’t mean the modules have to be given textually in that order in the source text. Dafny will figure out that order for you, assuming you haven’t made any circular references. For example, this is pretty clearly meaningless:
import A = B
import B = A // error: circular
You can have import statements at the toplevel and you can import modules defined at the same level:
import A = B
method m() {
A.whatever();
}
module B { method whatever() {} }
In this case, everything is well defined because we can put B
first,
followed by the A
import, and then finally m()
. If there is no
permitted ordering, then Dafny will give an error, complaining about a cyclic
dependency.
Note that when rearranging modules and imports, they have to be kept in the same containing module, which disallows some pathological module structures. Also, the imports and submodules are always considered to be before their containing module, even at the toplevel. This means that the following is not well formed:
method doIt() { }
module M {
method m() {
doIt(); // error: M precedes doIt
}
}
because the module M
must come before any other kind of members, such
as methods. To define global functions like this, you can put them in
a module (called Globals
, say) and open it into any module that needs
its functionality. Finally, if you import via a path, such as import A
= B.C
, then this creates a dependency of A
on B
, and B
itself
depends on its own nested module B.C
.
4.8. Name Resolution
When Dafny sees something like A<T>.B<U>.C<V>
, how does it know what each part
refers to? The process Dafny uses to determine what identifier
sequences like this refer to is name resolution. Though the rules may
seem complex, usually they do what you would expect. Dafny first looks
up the initial identifier. Depending on what the first identifier
refers to, the rest of the identifier is looked up in the appropriate
context.
In terms of the grammar, sequences like the above are represented as
a NameSegment
followed by 0 or more Suffix
es.
The form shown above contains three instances of
AugmentedDotSuffix_
.
The resolution is different depending on whether it is in a module context, an expression context or a type context.
4.8.1. Modules and name spaces
A module is a collection of declarations, each of which has a name. These names are held in two namespaces.
- The names of export sets
- The names of all other declarations, including submodules and aliased modules
In addition names can be classified as local or imported.
- Local declarations of a module are the declarations
that are explicit in the module and the
local declarations of the refinement parent. This includes, for
example, the
N
ofimport N =
in the refinement parent, recursively. - Imported names of a module are those brought in by
import opened
plus the imported names in the refinement parent.
Within each namespace, the local names are unique. Thus a module may not reuse a name that a refinement parent has declared (unless it is a refining declaration, which replaces both declarations, as described in Section 10).
Local names take precedence over imported names. If a name is used more than once among imported names (coming from different imports), then it is ambiguous to use the name without qualification.
4.8.2. Module Id Context Name Resolution
A qualified name may be used to refer to a module in an import statement or a refines clause of a module declaration.
Such a qualified name is resolved as follows, with respect to its syntactic
location within a module Z
:
-
The leading identifier of the qualified name is resolved as a local or imported module name of
Z
, if there is one with a matching name. The target of arefines
clause does not consider local names, that is, inmodule Z refines A.B.C
, any contents ofZ
are not considered in findingA
. -
Otherwise, it is resolved as a local or imported module name of the most enclosing module of
Z
, iterating outward to each successive enclosing module until a match is found or the default toplevel module is reached without a match. No consideration of export sets, default or otherwise, is used in this step. However, if at any stage a matching name is found that is not a module declaration, the resolution fails. See the examples below.
3a. Once the leading identifier is resolved as say module M
, the next
identifier in the quallified name
is resolved as a local or imported module name within M
.
The resolution is restricted to the default export set of M
.
3b. If the resolved module name is a module alias (from an import
statement)
then the target of the alias is resolved as a new qualified name
with respect to its syntactic context (independent of any resolutions or
modules so far). Since Z
depends on M
, any such alias target will
already have been resolved, because modules are resolved in order of
dependency.
- Step 3 is iterated for each identifier in the qualified module id, resulting in a module that is the final resolution of the complete qualified id.
Ordinarily a module must be imported in order for its constituent
declarations to be visible inside a given module M
. However, for the
resolution of qualified names this is not the case.
This example shows that the resolution of the refinement parent does not use any local names:
module A {
const a := 10
}
module B refines A { // the top-level A, not the submodule A
module A { const a := 30 }
method m() { assert a == 10; } // true
}
In the example, the A
in refines A
refers to the global A
, not the submodule A
.
4.8.3. Expression Context Name Resolution
The leading identifier is resolved using the first following rule that succeeds.
-
Local variables, parameters and bound variables. These are things like
x
,y
, andi
invar x;, ... returns (y: int)
, andforall i :: ....
The declaration chosen is the match from the innermost matching scope. -
If in a class, try to match a member of the class. If the member that is found is not static an implicit
this
is inserted. This works for fields, functions, and methods of the current class (if in a static context, then only static methods and functions are allowed). You can refer to fields of the current class either asthis.f
orf
, assuming of course thatf
is not hidden by one of the above. You can always prefixthis
if needed, which cannot be hidden. (Note that a field whose name is a string of digits must always have some prefix.) -
If there is no
Suffix
, then look for a datatype constructor, if unambiguous. Any datatypes that don’t need qualification (so the datatype name itself doesn’t need a prefix) and also have a uniquely named constructor can be referred to just by name. So ifdatatype List = Cons(List) | Nil
is the only datatype that declaresCons
andNil
constructors, then you can writeCons(Cons(Nil))
. If the constructor name is not unique, then you need to prefix it with the name of the datatype (for exampleList.Cons(List.Nil)))
. This is done per constructor, not per datatype. -
Look for a member of the enclosing module.
-
Module-level (static) functions and methods
In each module, names from opened modules are also potential matches, but only after names declared in the module. If an ambiguous name is found or a name of the wrong kind (e.g. a module instead of an expression identifier), an error is generated, rather than continuing down the list.
After the first identifier, the rules are basically the same, except in the new context. For example, if the first identifier is a module, then the next identifier looks into that module. Opened modules only apply within the module it is opened into. When looking up into another module, only things explicitly declared in that module are considered.
To resolve expression E.id
:
First resolve expression E and any type arguments.
- If
E
resolved to a moduleM
:- If
E.id<T>
is not followed by any further suffixes, look for unambiguous datatype constructor. - Member of module M: a sub-module (including submodules of imports), class, datatype, etc.
- Static function or method.
- If
- If
E
denotes a type:- Look up id as a member of that type
- If
E
denotes an expression:- Let T be the type of E. Look up id in T.
4.8.4. Type Context Name Resolution
In a type context the priority of identifier resolution is:
-
Type parameters.
-
Member of enclosing module (type name or the name of a module).
To resolve expression E.id
:
- If
E
resolved to a moduleM
:- Member of module M: a sub-module (including submodules of imports), class, datatype, etc.
- If
E
denotes a type:- Then the validity and meaning of
id
depends on the type and must be a user-declared or pre-defined member of the type.
- Then the validity and meaning of
5. Types
A Dafny type is a (possibly-empty) set of values or heap data-structures, together with allowed operations on those values. Types are classified as mutable reference types or immutable value types, depending on whether their values are stored in the heap or are (mathematical) values independent of the heap.
Dafny supports the following kinds of types, all described in later sections of this manual:
- builtin scalar types,
- builtin collection types,
- reference types (classes, traits, iterators),
- tuple types (including as a special case a parenthesized type),
- inductive and coinductive datatypes,
- function (arrow) types, and
- types, such as subset types, derived from other types.
5.1. Kinds of types
5.1.1. Value Types
The value types are those whose values do not lie in the program heap. These are:
- The basic scalar types:
bool
,char
,int
,real
,ORDINAL
, bitvector types - The built-in collection types:
set
,iset
,multiset
,seq
,string
,map
,imap
- Tuple Types
- Inductive and coinductive types
- Function (arrow) types
- Subset and newtypes that are based on value types
Data items having value types are passed by value. Since they are not considered to occupy memory, framing expressions do not reference them.
The nat
type is a pre-defined subset type of int
.
Dafny does not include types themselves as values, nor is there a type of types.
5.1.2. Reference Types
Dafny offers a host of reference types. These represent references to objects allocated dynamically in the program heap. To access the members of an object, a reference to (that is, a pointer to or object identity of) the object is dereferenced.
The reference types are class types, traits and array types.
Dafny supports both reference types that contain the special null
value
(nullable types) and reference types that do not (non-null types).
5.1.3. Named Types (grammar)
A Named Type is used to specify a user-defined type by a (possibly module- or class-qualified) name. Named types are introduced by class, trait, inductive, coinductive, synonym and opaque type declarations. They are also used to refer to type variables. A Named Type is denoted by a dot-separated sequence of name segments (Section 9.32).
A name segment (for a type) is a type name optionally followed by a generic instantiation, which supplies type parameters to a generic type, if needed.
The following sections describe each of these kinds of types in more detail.
5.2. Basic types
Dafny offers these basic types: bool
for booleans, char
for
characters, int
and nat
for integers, real
for reals,
ORDINAL
, and bit-vector types.
5.2.1. Booleans (grammar)
There are two boolean values and each has a corresponding literal in
the language: false
and true
.
Type bool
supports the following operations:
operator | precedence | description |
---|---|---|
<==> |
1 | equivalence (if and only if) |
==> |
2 | implication (implies) |
<== |
2 | reverse implication (follows from) |
&& |
3 | conjunction (and) |
|| |
3 | disjunction (or) |
== |
4 | equality |
!= |
4 | disequality |
! |
10 | negation (not) |
Negation is unary; the others are binary. The table shows the operators in groups of increasing binding power, with equality binding stronger than conjunction and disjunction, and weaker than negation. Within each group, different operators do not associate, so parentheses need to be used. For example,
A && B || C // error
would be ambiguous and instead has to be written as either
(A && B) || C
or
A && (B || C)
depending on the intended meaning.
5.2.1.1. Equivalence Operator
The expressions A <==> B
and A == B
give the same value, but note
that <==>
is associative whereas ==
is chaining and they have
different precedence. So,
A <==> B <==> C
is the same as
A <==> (B <==> C)
and
(A <==> B) <==> C
whereas
A == B == C
is simply a shorthand for
A == B && B == C
Also,
A <==> B == C <==> D
is
A <==> (B == C) <==> D
5.2.1.2. Conjunction and Disjunction
Conjunction and disjunction are associative. These operators are
short circuiting (from left to right), meaning that their second
argument is evaluated only if the evaluation of the first operand does
not determine the value of the expression. Logically speaking, the
expression A && B
is defined when A
is defined and either A
evaluates to false
or B
is defined. When A && B
is defined, its
meaning is the same as the ordinary, symmetric mathematical
conjunction &
. The same holds for ||
and |
.
5.2.1.3. Implication and Reverse Implication
Implication is right associative and is short-circuiting from left
to right. Reverse implication B <== A
is exactly the same as
A ==> B
, but gives the ability to write the operands in the opposite
order. Consequently, reverse implication is left associative and is
short-circuiting from right to left. To illustrate the
associativity rules, each of the following four lines expresses the
same property, for any A
, B
, and C
of type bool
:
A ==> B ==> C
A ==> (B ==> C) // parentheses redundant, ==> is right associative
C <== B <== A
(C <== B) <== A // parentheses redundant, <== is left associative
To illustrate the short-circuiting rules, note that the expression
a.Length
is defined for an array a
only if a
is not null
(see
Section 5.1.2), which means the following two
expressions are well-formed:
a != null ==> 0 <= a.Length
0 <= a.Length <== a != null
The contrapositives of these two expressions would be:
a.Length < 0 ==> a == null // not well-formed
a == null <== a.Length < 0 // not well-formed
but these expressions are not well-formed, since well-formedness
requires the left (and right, respectively) operand, a.Length < 0
,
to be well-formed by itself.
Implication A ==> B
is equivalent to the disjunction !A || B
, but
is sometimes (especially in specifications) clearer to read. Since,
||
is short-circuiting from left to right, note that
a == null || 0 <= a.Length
is well-formed, whereas
0 <= a.Length || a == null // not well-formed
is not.
In addition, booleans support logical quantifiers (forall and exists), described in Section 9.31.4.
5.2.2. Numeric Types (grammar)
Dafny supports numeric types of two kinds, integer-based, which
includes the basic type int
of all integers, and real-based, which
includes the basic type real
of all real numbers. User-defined
numeric types based on int
and real
, either subset types or newtypes,
are described in Section 5.6.3 and Section 5.7.
There is one built-in subset type,
nat
, representing the non-negative subrange of int
.
The language includes a literal for each integer, like
0
, 13
, and 1985
. Integers can also be written in hexadecimal
using the prefix “0x
”, as in 0x0
, 0xD
, and 0x7c1
(always with
a lower case x
, but the hexadecimal digits themselves are case
insensitive). Leading zeros are allowed. To form negative literals,
use the unary minus operator, as in -12
, but not -(12)
.
There are also literals for some of the reals. These are
written as a decimal point with a nonempty sequence of decimal digits
on both sides, optionally prefixed by a -
character.
For example, 1.0
, 1609.344
, -12.5
, and 0.5772156649
.
Real literals using exponents are not supported in Dafny. For now, you’d have to write your own function for that, e.g.
// realExp(2.37, 100) computes 2.37e100
function realExp(r: real, e: int): real decreases if e > 0 then e else -e {
if e == 0 then r
else if e < 0 then realExp(r/10.0, e+1)
else realExp(r*10.0, e-1)
}
For integers (in both decimal and hexadecimal form) and reals, any two digits in a literal may be separated by an underscore in order to improve human readability of the literals. For example:
const c1 := 1_000_000 // easier to read than 1000000
const c2 := 0_12_345_6789 // strange but legal formatting of 123456789
const c3 := 0x8000_0000 // same as 0x80000000 -- hex digits are
// often placed in groups of 4
const c4 := 0.000_000_000_1 // same as 0.0000000001 -- 1 Angstrom
In addition to equality and disequality, numeric types support the following relational operations, which have the same precedence as equality:
operator | description |
---|---|
< |
less than |
<= |
at most |
>= |
at least |
> |
greater than |
Like equality and disequality, these operators are chaining, as long as they are chained in the “same direction”. That is,
A <= B < C == D <= E
is simply a shorthand for
A <= B && B < C && C == D && D <= E
whereas
A < B > C
is not allowed.
There are also operators on each numeric type:
operator | precedence | description |
---|---|---|
+ |
6 | addition (plus) |
- |
6 | subtraction (minus) |
* |
7 | multiplication (times) |
/ |
7 | division (divided by) |
% |
7 | modulus (mod) – int only |
- |
10 | negation (unary minus) |
The binary operators are left associative, and they associate with
each other in the two groups.
The groups are listed in order of
increasing binding power, with equality binding less strongly than any of these operators.
There is no implicit conversion between int
and real
: use as int
or
as real
conversions to write an explicit conversion (cf. Section 9.10).
Modulus is supported only for integer-based numeric types. Integer
division and modulus are the Euclidean division and modulus. This
means that modulus always returns a non-negative value, regardless of the
signs of the two operands. More precisely, for any integer a
and
non-zero integer b
,
a == a / b * b + a % b
0 <= a % b < B
where B
denotes the absolute value of b
.
Real-based numeric types have a member Floor
that returns the
floor of the real value (as an int value), that is, the largest integer not exceeding
the real value. For example, the following properties hold, for any
r
and r'
of type real
:
method m(r: real, r': real) {
assert 3.14.Floor == 3;
assert (-2.5).Floor == -3;
assert -2.5.Floor == -2; // This is -(2.5.Floor)
assert r.Floor as real <= r;
assert r <= r' ==> r.Floor <= r'.Floor;
}
Note in the third line that member access (like .Floor
) binds
stronger than unary minus. The fourth line uses the conversion
function as real
from int
to real
, as described in
Section 9.10.
5.2.3. Bit-vector Types (grammar)
Dafny includes a family of bit-vector types, each type having a specific,
constant length, the number of bits in its values.
Each such type is
distinct and is designated by the prefix bv
followed (without white space) by
a positive integer without leading zeros or zero, stating the number of bits. For example,
bv1
, bv8
, and bv32
are legal bit-vector type names.
The type bv0
is also legal; it is a bit-vector type with no bits and just one value, 0x0
.
Constant literals of bit-vector types are given by integer literals converted automatically to the designated type, either by an implicit or explicit conversion operation or by initialization in a declaration. Dafny checks that the constant literal is in the correct range. For example,
const i: bv1 := 1
const j: bv8 := 195
const k: bv2 := 5 // error - out of range
const m := (194 as bv8) | (7 as bv8)
Bit-vector values can be converted to and from int
and other bit-vector types, as long as
the values are in range for the target type. Bit-vector values are always considered unsigned.
Bit-vector operations include bit-wise operators and arithmetic operators (as well as equality, disequality, and comparisons). The arithmetic operations truncate the high-order bits from the results; that is, they perform unsigned arithmetic modulo 2^{number of bits}, like 2’s-complement machine arithmetic.
operator | precedence | description |
---|---|---|
<< |
5 | bit-limited bit-shift left |
>> |
5 | unsigned bit-shift right |
+ |
6 | bit-limited addition |
- |
6 | bit-limited subtraction |
* |
7 | bit-limited multiplication |
& |
8 | bit-wise and |
| |
8 | bit-wise or |
^ |
8 | bit-wise exclusive-or |
- |
10 | bit-limited negation (unary minus) |
! |
10 | bit-wise complement |
.RotateLeft(n) | 11 | rotates bits left by n bit positions |
.RotateRight(n) | 11 | rotates bits right by n bit positions |
The groups of operators lower in the table above bind more tightly.1
All operators bind more tightly than equality, disequality, and comparisons.
All binary operators are left-associative, but the
bit-wise &
, |
, and ^
do not associate together (parentheses are required to disambiguate).
The +
, |
, ^
, and &
operators are commutative.
The right-hand operand of bit-shift operations is an int
value,
must be non-negative, and
no more than the number of bits in the type.
There is no signed right shift as all bit-vector values correspond to
non-negative integers.
Bit-vector negation returns an unsigned value in the correct range for the type.
It has the properties x + (-x) == 0
and (!x) + 1 == -x
, for a bitvector value x
of at least one bit.
The argument of the RotateLeft
and RotateRight
operations is a
non-negative int
that is no larger than the bit-width of the value being rotated.
RotateLeft
moves bits to higher bit positions (e.g., (2 as bv4).RotateLeft(1) == (4 as bv4)
and (8 as bv4).RotateLeft(1) == (1 as bv4)
);
RotateRight
moves bits to lower bit positions, so b.RotateLeft(n).RotateRight(n) == b
.
Here are examples of the various operations (all the assertions are true except where indicated):
const i: bv4 := 9
const j: bv4 := 3
method m() {
assert (i & j) == (1 as bv4);
assert (i | j) == (11 as bv4);
assert (i ^ j) == (10 as bv4);
assert !i == (6 as bv4);
assert -i == (7 as bv4);
assert (i + i) == (2 as bv4);
assert (j - i) == (10 as bv4);
assert (i * j) == (11 as bv4);
assert (i as int) / (j as int) == 3;
assert (j << 1) == (6 as bv4);
assert (i << 1) == (2 as bv4);
assert (i >> 1) == (4 as bv4);
assert i == 9; // auto conversion of literal to bv4
assert i * 4 == j + 8 + 9; // arithmetic is modulo 16
assert i + j >> 1 == (i + j) >> 1; // + - bind tigher than << >>
assert i + j ^ 2 == i + (j^2);
assert i * j & 1 == i * (j&1); // & | ^ bind tighter than + - *
}
The following are incorrectly formed:
const i: bv4 := 9
const j: bv4 := 3
method m() {
assert i & 4 | j == 0 ; // parentheses required
}
const k: bv4 := 9
method p() {
assert k as bv5 == 9 as bv6; // error: mismatched types
}
These produce assertion errors:
const i: bv4 := 9
method m() {
assert i as bv3 == 1; // error: i is out of range for bv3
}
const j: bv4 := 9
method n() {
assert j == 25; // error: 25 is out of range for bv4
}
Bit-vector constants (like all constants) can be initialized using expressions, but pay attention to how type inference applies to such expressions. For example,
const a: bv3 := -1
is legal because Dafny interprets -1
as a bv3
expression, because a
has type bv3
.
Consequently the -
is bv3
negation and the 1
is a bv3
literal; the value of the expression -1
is
the bv3
value 7
, which is then the value of a
.
On the other hand,
const b: bv3 := 6 & 11
is illegal because, again, the &
is bv3
bit-wise-and and the numbers must be valid bv3
literals.
But 11
is not a valid bv3
literal.
5.2.4. Ordinal type (grammar)
Values of type ORDINAL
behave like nat
s in many ways, with one important difference:
there are ORDINAL
values that are larger than any nat
. The smallest of these non-nat ordinals is
represented as $\omega$ in mathematics, though there is no literal expression in Dafny that represents this value.
The natural numbers are ordinals.
Any ordinal has a successor ordinal (equivalent to adding 1
).
Some ordinals are limit ordinals, meaning they are not a successor of any other ordinal;
the natural number 0
and $\omega$ are limit ordinals.
The offset of an ordinal is the number of successor operations it takes to reach it from a limit ordinal.
The Dafny type ORDINAL
has these member functions:
o.IsLimit
– true ifo
is a limit ordinal (including0
)o.IsSucc
– true ifo
is a successor to something, soo.IsSucc <==> !o.IsLimit
o.IsNat
– true ifo
represents anat
value, so forn
anat
,(n as ORDINAL).IsNat
is true and ifo.IsNat
is true then(o as nat)
is well-definedo.Offset
– is thenat
value giving the offset of the ordinal
In addition,
- non-negative numeric literals may be considered
ORDINAL
literals, soo + 1
is allowed ORDINAL
s may be compared, using== != < <= > >=
- two
ORDINAL
s may be added and the result is>=
either one of them; addition is associative but not commutative *
,/
and%
are not defined forORDINAL
s- two
ORDINAL
s may be subtracted if the RHS satisfies.IsNat
and the offset of the LHS is not smaller than the offset of the RHS
In Dafny, ORDINAL
s are used primarily in conjunction with extreme functions and lemmas.
5.2.5. Characters (grammar)
Dafny supports a type char
of characters.
Its exact meaning is controlled by the command-line switch --unicode-char:true|false
.
If --unicode-char
is disabled, then char
represents any UTF-16 code unit.
This includes surrogate code points.
If --unicode-char
is enabled, then char
represents any Unicode scalar value.
This excludes surrogate code points.
Character literals are enclosed in single quotes, as in 'D'
.
To write a single quote as a
character literal, it is necessary to use an escape sequence.
Escape sequences can also be used to write other characters. The
supported escape sequences are the following:
escape sequence | meaning |
---|---|
\' |
the character ' |
\" |
the character " |
\\ |
the character \ |
\0 |
the null character, same as \u0000 or \U{0} |
\n |
line feed |
\r |
carriage return |
\t |
horizontal tab |
\u xxxx |
UTF-16 code unit whose hexadecimal code is xxxx, where each x is a hexadecimal digit |
\U{ x..x} |
Unicode scalar value whose hexadecimal code is x..x, where each x is a hexadecimal digit |
The escape sequence for a double quote is redundant, because
'"'
and '\"'
denote the same
character—both forms are provided in order to support the same
escape sequences in string literals (Section 5.5.3.5).
In the form \u
xxxx, which is only allowed if --unicode-char
is disabled,
the u
is always lower case, but the four
hexadecimal digits are case insensitive.
In the form \U{
x..x}
,
which is only allowed if --unicode-char
is enabled,
the U
is always upper case,
but the hexadecimal digits are case insensitive, and there must
be at least one and at most six digits.
Surrogate code points are not allowed.
The hex digits may be interspersed with underscores for readability
(but not beginning or ending with an underscore), as in \U{1_F680}
.
Character values are ordered and can be compared using the standard relational operators:
operator | description |
---|---|
< |
less than |
<= |
at most |
>= |
at least |
> |
greater than |
Sequences of characters represent strings, as described in Section 5.5.3.5.
Character values can be converted to and from int
values using the
as int
and as char
conversion operations. The result is what would
be expected in other programming languages, namely, the int
value of a
char
is the ASCII or Unicode numeric value.
The only other operations on characters are obtaining a character
by indexing into a string, and the implicit conversion to string
when used as a parameter of a print
statement.
5.3. Type parameters (grammar)
Examples:
type G1<T>
type G2<T(0)>
type G3<+T(==),-U>
Many of the types, functions, and methods in Dafny can be parameterized by types. These type parameters are declared inside angle brackets and can stand for any type.
Dafny has some inference support that makes certain signatures less cluttered (described in Section 12.2).
5.3.1. Declaring restrictions on type parameters
It is sometimes necessary to restrict type parameters so that they can only be instantiated by certain families of types, that is, by types that have certain properties. These properties are known as type characteristics. The following subsections describe the type characteristics that Dafny supports.
In some cases, type inference will infer that a type-parameter
must be restricted in a particular way, in which case Dafny
will add the appropriate suffix, such as (==)
, automatically.
If more than one restriction is needed, they are either
listed comma-separated,
inside the parentheses or as multiple parenthesized elements:
T(==,0)
or T(==)(0)
.
When an actual type is substituted for a type parameter in a generic type instantiation,
the actual type must have the declared or inferred type characteristics of the type parameter.
These characteristics might also be inferred for the actual type. For example, a numeric-based
subset or newtype automatically has the ==
relationship of its base type. Similarly,
type synonyms have the characteristics of the type they represent.
An abstract type has no known characteristics. If it is intended to be defined only as types that have certain characteristics, then those characteristics must be declared. For example,
class A<T(00)> {}
type Q
const a: A<Q>
will give an error because it is not known whether the type Q
is non-empty (00
).
Instead, one needs to write
class A<T(00)> {}
type Q(00)
const a: A?<Q> := null
5.3.1.1. Equality-supporting type parameters: T(==)
Designating a type parameter with the (==)
suffix indicates that
the parameter may only be replaced in non-ghost contexts
with types that are known to
support run-time equality comparisons (==
and !=
).
All types support equality in ghost contexts,
as if, for some types, the equality function is ghost.
For example,
method Compare<T(==)>(a: T, b: T) returns (eq: bool)
{
if a == b { eq := true; } else { eq := false; }
}
is a method whose type parameter is restricted to equality-supporting types when used in a non-ghost context. Again, note that all types support equality in ghost contexts; the difference is only for non-ghost (that is, compiled) code. Coinductive datatypes, arrow types, and inductive datatypes with ghost parameters are examples of types that are not equality supporting.
5.3.1.2. Auto-initializable types: T(0)
At every access of a variable x
of a type T
, Dafny ensures that
x
holds a legal value of type T
.
If no explicit initialization is given, then an arbitrary value is
assumed by the verifier and supplied by the compiler,
that is, the variable is auto-initialized, but to an arbitrary value.
For example,
class Example<A(0), X> {
var n: nat
var i: int
var a: A
var x: X
constructor () {
new; // error: field 'x' has not been given a value`
assert n >= 0; // true, regardless of the value of 'n'
assert i >= 0; // possibly false, since an arbitrary 'int' may be negative
// 'a' does not require an explicit initialization, since 'A' is auto-init
}
}
In the example above, the class fields do not need to be explicitly initialized in the constructor because they are auto-initialized to an arbitrary value.
Local variables and out-parameters are however, subject to definite assignment
rules. The following example requires --relax-definite-assignment
,
which is not the default.
method m() {
var n: nat; // Auto-initialized to an arbitrary value of type `nat`
assert n >= 0; // true, regardless of the value of n
var i: int;
assert i >= 0; // possibly false, arbitrary ints may be negative
}
With the default behavior of definite assignment, n
and i
need to be initialized
to an explicit value of their type or to an arbitrary value using, for example,
var n: nat := *;
.
For some types (known as auto-init types), the compiler can choose an
initial value, but for others it does not.
Variables and fields whose type the compiler does not auto-initialize
are subject to definite-assignment rules. These ensure that the program
explicitly assigns a value to a variable before it is used.
For more details see Section 12.6 and the --relax-definite-assignment
command-line option.
More detail on auto-initializing is in this document.
Dafny supports auto-init as a type characteristic.
To restrict a type parameter to auto-init types, mark it with the
(0)
suffix. For example,
method AutoInitExamples<A(0), X>() returns (a: A, x: X)
{
// 'a' does not require an explicit initialization, since A is auto-init
// error: out-parameter 'x' has not been given a value
}
In this example, an error is reported because out-parameter x
has not
been assigned—since nothing is known about type X
, variables of
type X
are subject to definite-assignment rules. In contrast, since
type parameter A
is declared to be restricted to auto-init types,
the program does not need to explicitly assign any value to the
out-parameter a
.
5.3.1.3. Nonempty types: T(00)
Auto-init types are important in compiled contexts. In ghost contexts, it
may still be important to know that a type is nonempty. Dafny supports
a type characteristic for nonempty types, written with the suffix (00)
.
For example, with --relax-definite-assignment
, the following example happens:
method NonemptyExamples<B(00), X>() returns (b: B, ghost g: B, ghost h: X)
{
// error: non-ghost out-parameter 'b' has not been given a value
// ghost out-parameter 'g' is fine, since its type is nonempty
// error: 'h' has not been given a value
}
Because of B
’s nonempty type characteristic, ghost parameter g
does not
need to be explicitly assigned. However, Dafny reports an error for the
non-ghost b
, since B
is not an auto-init type, and reports an error
for h
, since the type X
could be empty.
Note that every auto-init type is nonempty.
In the default definite-assignment mode (that is, without --relax-definite-assignment
)
there will be errors for all three formal parameters in the example just given.
For more details see Section 12.6.
5.3.1.4. Non-heap based: T(!new)
Dafny makes a distinction between types whose values are on the heap,
i.e. references, like
classes and arrays, and those that are strictly value-based, like basic
types and datatypes.
The practical implication is that references depend on allocation state
(e.g., are affected by the old
operation) whereas non-reference values
are not.
Thus it can be relevant to know whether the values of a type parameter
are heap-based or not. This is indicated by the mode suffix (!new)
.
A type parameter characterized by (!new)
is recursively independent
of the allocation state. For example, a datatype is not a reference, but for
a parameterized data type such as
datatype Result<T> = Failure(error: string) | Success(value: T)
the instantiation Result<int>
satisfies (!new)
, whereas
Result<array<int>>
does not.
Note that this characteristic of a type parameter is operative for both
verification and compilation.
Also, abstract types at the topmost scope are always implicitly (!new)
.
Here are some examples:
datatype Result<T> = Failure(error: string) | Success(v: T)
datatype ResultN<T(!new)> = Failure(error: string) | Success(v: T)
class C {}
method m() {
var x1: Result<int>;
var x2: ResultN<int>;
var x3: Result<C>;
var x4: ResultN<C>; // error
var x5: Result<array<int>>;
var x6: ResultN<array<int>>; // error
}
5.3.2. Type parameter variance
Type parameters have several different variance and cardinality properties.
These properties of type parameters are designated in a generic type definition.
For instance, in type A<+T> = ...
, the +
indicates that the T
position
is co-variant. These properties are indicated by the following notation:
notation | variance | cardinality-preserving |
---|---|---|
(nothing) | non-variant | yes |
+ |
co-variant | yes |
- |
contra-variant | not necessarily |
* |
co-variant | not necessarily |
! |
non-variant | not necessarily |
- co-variance (
A<+T>
orA<*T>
) means that ifU
is a subtype ofV
thenA<U>
is a subtype ofA<V>
- contra-variance (
A<-T>
) means that ifU
is a subtype ofV
thenA<V>
is a subtype ofA<U>
- non-variance (
A<T>
orA<!T>
) means that ifU
is a different type thanV
then there is no subtyping relationship betweenA<U>
andA<V>
Cardinality preserving
means that the cardinality of the type being defined never exceeds the cardinality of any of its type parameters.
For example type T<X> = X -> bool
is illegal and returns the error message formal type parameter 'X' is not used according to its variance specification (it is used left of an arrow) (perhaps try declaring 'X' as '-X' or '!X')
The type X -> bool
has strictly more values than the type X
.
This affects certain uses of the type, so Dafny requires the declaration of T
to explicitly say so.
Marking the type parameter X
with -
or !
announces that the cardinality of T<X>
may by larger than that of X
.
If you use -
, you’re also declaring T
to be contravariant in its type argument, and if you use !
, you’re declaring that T
is non-variant in its type argument.
To fix it, we use the variance !
:
type T<!X> = X -> bool
This states that T
does not preserve the cardinality of X
, meaning there could be strictly more values of type T<E>
than values of type E
for any E
.
A more detailed explanation of these topics is here.
5.4. Generic Instantiation (grammar)
A generic instantiation consists of a comma-separated list of 1 or more Types,
enclosed in angle brackets (<
>
),
providing actual types to be used in place of the type parameters of the
declaration of the generic type.
If there is no instantion for a generic type, type inference will try
to fill these in (cf. Section 12.2).
5.5. Collection types
Dafny offers several built-in collection types.
5.5.1. Sets (grammar)
For any type T
, each value of type set<T>
is a finite set of
T
values.
Set membership is determined by equality in the type T
,
so set<T>
can be used in a non-ghost context only if T
is
equality supporting.
For any type T
, each value of type iset<T>
is a potentially infinite
set of T
values.
A set can be formed using a set display expression, which is a possibly empty, unordered, duplicate-insensitive list of expressions enclosed in curly braces. To illustrate,
{} {2, 7, 5, 3} {4+2, 1+5, a*b}
are three examples of set displays. There is also a set comprehension expression (with a binder, like in logical quantifications), described in Section 9.31.5.
In addition to equality and disequality, set types support the following relational operations:
operator | precedence | description |
---|---|---|
< |
4 | proper subset |
<= |
4 | subset |
>= |
4 | superset |
> |
4 | proper superset |
Like the arithmetic relational operators, these operators are chaining.
Sets support the following binary operators, listed in order of increasing binding power:
operator | precedence | description |
---|---|---|
!! |
4 | disjointness |
+ |
6 | set union |
- |
6 | set difference |
* |
7 | set intersection |
The associativity rules of +
, -
, and *
are like those of the
arithmetic operators with the same names. The expression A !! B
,
whose binding power is the same as equality (but which neither
associates nor chains with equality), says that sets A
and B
have
no elements in common, that is, it is equivalent to
A * B == {}
However, the disjointness operator is chaining though in a slightly different way than other chaining operators:
A !! B !! C !! D
means that A
, B
, C
and D
are all mutually disjoint, that is
A * B == {} && (A + B) * C == {} && (A + B + C) * D == {}
In addition, for any set s
of type set<T>
or iset<T>
and any
expression e
of type T
, sets support the following operations:
expression | precedence | result type | description |
---|---|---|---|
e in s |
4 | bool |
set membership |
e !in s |
3 | bool |
set non-membership |
|s| |
11 | nat |
set cardinality (not for iset ) |
The expression e !in s
is a syntactic shorthand for !(e in s)
.
(No white space is permitted between !
and in
, making !in
effectively
the one example of a mixed-character-class token in Dafny.)
5.5.2. Multisets (grammar)
A multiset is similar to a set, but keeps track of the multiplicity
of each element, not just its presence or absence. For any type T
,
each value of type multiset<T>
is a map from T
values to natural
numbers denoting each element’s multiplicity. Multisets in Dafny
are finite, that is, they contain a finite number of each of a finite
set of elements. Stated differently, a multiset maps only a finite
number of elements to non-zero (finite) multiplicities.
Like sets, multiset membership is determined by equality in the type
T
, so multiset<T>
can be used in a non-ghost context only if T
is equality supporting.
A multiset can be formed using a multiset display expression, which
is a possibly empty, unordered list of expressions enclosed in curly
braces after the keyword multiset
. To illustrate,
multiset{} multiset{0, 1, 1, 2, 3, 5} multiset{4+2, 1+5, a*b}
are three examples of multiset displays. There is no multiset comprehension expression.
In addition to equality and disequality, multiset types support the following relational operations:
operator | precedence | description |
---|---|---|
< |
4 | proper multiset subset |
<= |
4 | multiset subset |
>= |
4 | multiset superset |
> |
4 | proper multiset superset |
Like the arithmetic relational operators, these operators are chaining.
Multisets support the following binary operators, listed in order of increasing binding power:
operator | precedence | description |
---|---|---|
!! |
4 | multiset disjointness |
+ |
6 | multiset sum |
- |
6 | multiset difference |
* |
7 | multiset intersection |
The associativity rules of +
, -
, and *
are like those of the
arithmetic operators with the same names. The +
operator
adds the multiplicity of corresponding elements, the -
operator
subtracts them (but 0 is the minimum multiplicity),
and the *
has multiplicity that is the minimum of the
multiplicity of the operands. There is no operator for multiset
union, which would compute the maximum of the multiplicities of the operands.
The expression A !! B
says that multisets A
and B
have no elements in common, that is,
it is equivalent to
A * B == multiset{}
Like the analogous set operator, !!
is chaining and means mutual disjointness.
In addition, for any multiset s
of type multiset<T>
,
expression e
of type T
, and non-negative integer-based numeric
n
, multisets support the following operations:
expression | precedence | result type | description |
---|---|---|---|
e in s |
4 | bool |
multiset membership |
e !in s |
4 | bool |
multiset non-membership |
|s| |
11 | nat |
multiset cardinality |
s[e] |
11 | nat |
multiplicity of e in s |
s[e := n] |
11 | multiset<T> |
multiset update (change of multiplicity) |
The expression e in s
returns true
if and only if s[e] != 0
.
The expression e !in s
is a syntactic shorthand for !(e in s)
.
The expression s[e := n]
denotes a multiset like
s
, but where the multiplicity of element e
is n
. Note that
the multiset update s[e := 0]
results in a multiset like s
but
without any occurrences of e
(whether or not s
has occurrences of
e
in the first place). As another example, note that
s - multiset{e}
is equivalent to:
if e in s then s[e := s[e] - 1] else s
5.5.3. Sequences (grammar)
For any type T
, a value of type seq<T>
denotes a sequence of T
elements, that is, a mapping from a finite downward-closed set of natural
numbers (called indices) to T
values.
5.5.3.1. Sequence Displays
A sequence can be formed using a sequence display expression, which is a possibly empty, ordered list of expressions enclosed in square brackets. To illustrate,
[] [3, 1, 4, 1, 5, 9, 3] [4+2, 1+5, a*b]
are three examples of sequence displays.
There is also a sequence comprehension expression (Section 9.28):
seq(5, i => i*i)
is equivalent to [0, 1, 4, 9, 16]
.
5.5.3.2. Sequence Relational Operators
In addition to equality and disequality, sequence types support the following relational operations:
operator | precedence | description |
---|---|---|
< | 4 | proper prefix |
<= | 4 | prefix |
Like the arithmetic relational operators, these operators are
chaining. Note the absence of >
and >=
.
5.5.3.3. Sequence Concatenation
Sequences support the following binary operator:
operator | precedence | description |
---|---|---|
+ |
6 | concatenation |
Operator +
is associative, like the arithmetic operator with the
same name.
5.5.3.4. Other Sequence Expressions
In addition, for any sequence s
of type seq<T>
, expression e
of type T
, integer-based numeric i
satisfying 0 <= i < |s|
, and
integer-based numerics lo
and hi
satisfying
0 <= lo <= hi <= |s|
, sequences support the following operations:
expression | precedence | result type | description |
---|---|---|---|
e in s |
4 | bool |
sequence membership |
e !in s |
4 | bool |
sequence non-membership |
|s| |
11 | nat |
sequence length |
s[i] |
11 | T |
sequence selection |
s[i := e] |
11 | seq<T> |
sequence update |
s[lo..hi] |
11 | seq<T> |
subsequence |
s[lo..] |
11 | seq<T> |
drop |
s[..hi] |
11 | seq<T> |
take |
s[ slices] |
11 | seq<seq<T>> |
slice |
multiset(s) |
11 | multiset<T> |
sequence conversion to a multiset<T> |
Expression s[i := e]
returns a sequence like s
, except that the
element at index i
is e
. The expression e in s
says there
exists an index i
such that s[i] == e
. It is allowed in non-ghost
contexts only if the element type T
is
equality supporting.
The expression e !in s
is a syntactic shorthand for !(e in s)
.
Expression s[lo..hi]
yields a sequence formed by taking the first
hi
elements and then dropping the first lo
elements. The
resulting sequence thus has length hi - lo
. Note that s[0..|s|]
equals s
. If the upper bound is omitted, it
defaults to |s|
, so s[lo..]
yields the sequence formed by dropping
the first lo
elements of s
. If the lower bound is omitted, it
defaults to 0
, so s[..hi]
yields the sequence formed by taking the
first hi
elements of s
.
In the sequence slice operation, slices is a nonempty list of
length designators separated and optionally terminated by a colon, and
there is at least one colon. Each length designator is a non-negative
integer-based numeric; the sum of the length designators is no greater than |s|
. If there
are k colons, the operation produces k + 1 consecutive subsequences
from s
, with the length of each indicated by the corresponding length
designator, and returns these as a sequence of
sequences.
If slices is terminated by a
colon, then the length of the last slice extends until the end of s
,
that is, its length is |s|
minus the sum of the given length
designators. For example, the following equalities hold, for any
sequence s
of length at least 10
:
method m(s: seq<int>) {
var t := [3.14, 2.7, 1.41, 1985.44, 100.0, 37.2][1:0:3];
assert |t| == 3 && t[0] == [3.14] && t[1] == [];
assert t[2] == [2.7, 1.41, 1985.44];
var u := [true, false, false, true][1:1:];
assert |u| == 3 && u[0][0] && !u[1][0] && u[2] == [false, true];
assume |s| > 10;
assert s[10:][0] == s[..10];
assert s[10:][1] == s[10..];
}
The operation multiset(s)
yields the multiset of elements of
sequence s
. It is allowed in non-ghost contexts only if the element
type T
is equality supporting.
5.5.3.5. Strings (grammar)
A special case of a sequence type is seq<char>
, for which Dafny
provides a synonym: string
. Strings are like other sequences, but
provide additional syntax for sequence display expressions, namely
string literals. There are two forms of the syntax for string
literals: the standard form and the verbatim form.
String literals of the standard form are enclosed in double quotes, as
in "Dafny"
. To include a double quote in such a string literal,
it is necessary to use an escape sequence. Escape sequences can also
be used to include other characters. The supported escape sequences
are the same as those for character literals (Section 5.2.5).
For example, the Dafny expression "say \"yes\""
represents the
string 'say "yes"'
.
The escape sequence for a single quote is redundant, because
"\'"
and "\'"
denote the same
string—both forms are provided in order to support the same
escape sequences as do character literals.
String literals of the verbatim form are bracketed by
@"
and "
, as in @"Dafny"
. To include
a double quote in such a string literal, it is necessary to use the
escape sequence ""
, that is, to write the character
twice. In the verbatim form, there are no other escape sequences.
Even characters like newline can be written inside the string literal
(hence spanning more than one line in the program text).
For example, the following three expressions denote the same string:
"C:\\tmp.txt"
@"C:\tmp.txt"
['C', ':', '\\', 't', 'm', 'p', '.', 't', 'x', 't']
Since strings are sequences, the relational operators <
and <=
are defined on them. Note, however, that these operators
still denote proper prefix and prefix, respectively, not some kind of
alphabetic comparison as might be desirable, for example, when
sorting strings.
5.5.4. Finite and Infinite Maps (grammar)
For any types T
and U
, a value of type map<T,U>
denotes a
(finite) map
from T
to U
. In other words, it is a look-up table indexed by
T
. The domain of the map is a finite set of T
values that have
associated U
values. Since the keys in the domain are compared
using equality in the type T
, type map<T,U>
can be used in a
non-ghost context only if T
is
equality supporting.
Similarly, for any types T
and U
, a value of type imap<T,U>
denotes a (possibly) infinite map. In most regards, imap<T,U>
is
like map<T,U>
, but a map of type imap<T,U>
is allowed to have an
infinite domain.
A map can be formed using a map display expression (see Section 9.30),
which is a possibly empty, ordered list of maplets, each maplet having the
form t := u
where t
is an expression of type T
and u
is an
expression of type U
, enclosed in square brackets after the keyword
map
. To illustrate,
map[]
map[20 := true, 3 := false, 20 := false]
map[a+b := c+d]
are three examples of map displays. By using the keyword imap
instead of map
, the map produced will be of type imap<T,U>
instead of map<T,U>
. Note that an infinite map (imap
) is allowed
to have a finite domain, whereas a finite map (map
) is not allowed
to have an infinite domain.
If the same key occurs more than
once in a map display expression, only the last occurrence appears in the resulting
map.2 There is also a map comprehension expression,
explained in Section 9.31.8.
For any map fm
of type map<T,U>
,
any map m
of type map<T,U>
or imap<T,U>
,
any expression t
of type T
,
any expression u
of type U
, and any d
in the domain of m
(that
is, satisfying d in m
), maps support the following operations:
expression | precedence | result type | description |
---|---|---|---|
t in m |
4 | bool |
map domain membership |
t !in m |
4 | bool |
map domain non-membership |
|fm| |
11 | nat |
map cardinality |
m[d] |
11 | U |
map selection |
m[t := u] |
11 | map<T,U> |
map update |
m.Keys |
11 | (i)set<T> |
the domain of m |
m.Values |
11 | (i)set<U> |
the range of m |
m.Items |
11 | (i)set<(T,U)> |
set of pairs (t,u) in m |
|fm|
denotes the number of mappings in fm
, that is, the
cardinality of the domain of fm
. Note that the cardinality operator
is not supported for infinite maps.
Expression m[d]
returns the U
value that m
associates with d
.
Expression m[t := u]
is a map like m
, except that the
element at key t
is u
. The expression t in m
says t
is in the
domain of m
and t !in m
is a syntactic shorthand for
!(t in m)
.3
The expressions m.Keys
, m.Values
, and m.Items
return, as sets,
the domain, the range, and the 2-tuples holding the key-value
associations in the map. Note that m.Values
will have a different
cardinality than m.Keys
and m.Items
if different keys are
associated with the same value. If m
is an imap
, then these
expressions return iset
values.
Here is a small example, where a map cache
of type map<int,real>
is used to cache computed values of Joule-Thomson coefficients for
some fixed gas at a given temperature:
if K in cache { // check if temperature is in domain of cache
coeff := cache[K]; // read result in cache
} else {
coeff := ComputeJTCoefficient(K); // do expensive computation
cache := cache[K := coeff]; // update the cache
}
Dafny also overloads the +
and -
binary operators for maps.
The +
operator merges two maps or imaps of the same type, as if each
(key,value) pair of the RHS is added in turn to the LHS (i)map.
In this use, +
is not commutative; if a key exists in both
(i)maps, it is the value from the RHS (i)map that is present in the result.
The -
operator implements a map difference operator. Here the LHS
is a map<K,V>
or imap<K,V>
and the RHS is a set<K>
(but not an iset
); the operation removes
from the LHS all the (key,value) pairs whose key is a member of the RHS set.
To avoid cuasing circular reasoning chains or providing too much informatino that might complicate Dafny’s prover finding proofs, not all properties of maps are known by the prover by default. For example, the following does not prove:
method mmm<K(==),V(==)>(m: map<K,V>, k: K, v: V) {
var mm := m[k := v];
assert v in mm.Values;
}
Rather, one must provide an intermediate step, which is not entirely obvious:
method mmm<K(==),V(==)>(m: map<K,V>, k: K, v: V) {
var mm := m[k := v];
assert k in mm.Keys;
assert v in mm.Values;
}
5.5.5. Iterating over collections
Collections are very commonly used in programming and one frequently needs to iterate over the elements of a collection. Dafny does not have built-in iterator methods, but the idioms by which to do so are straightforward. The subsections below give some introductory examples; more detail can be found in this power user note.
5.5.5.1. Sequences and arrays
Sequences and arrays are indexable and have a length. So the idiom to iterate over the contents is well-known. For an array:
method m(a: array<int>) {
var i := 0;
var sum := 0;
while i < a.Length {
sum := sum + a[i];
i := i + 1;
}
}
For a sequence, the only difference is the length operator:
method m(s: seq<int>) {
var i := 0;
var sum := 0;
while i < |s| {
sum := sum + s[i];
i := i + 1;
}
}
The forall
statement (Section 8.21) can also be used
with arrays where parallel assignment is needed:
method m(s: array<int>) {
var rev := new int[s.Length];
forall i | 0 <= i < s.Length {
rev[i] := s[s.Length-i-1];
}
}
See Section 5.10.2 on how to convert an array to a sequence.
5.5.5.2. Sets
There is no intrinsic order to the elements of a set. Nevertheless, we can extract an arbitrary element of a nonempty set, performing an iteration as follows:
method m(s: set<int>) {
var ss := s;
while ss != {}
decreases |ss|
{
var i: int :| i in ss;
ss := ss - {i};
print i, "\n";
}
}
Because iset
s may be infinite, Dafny does not permit iteration over an iset
.
5.5.5.3. Maps
Iterating over the contents of a map
uses the component sets: Keys
, Values
, and Items
. The iteration loop follows the same patterns as for sets:
method m<T(==),U(==)> (m: map<T,U>) {
var items := m.Items;
while items != {}
decreases |items|
{
var item :| item in items;
items := items - { item };
print item.0, " ", item.1, "\n";
}
}
There are no mechanisms currently defined in Dafny for iterating over imap
s.
5.6. Types that stand for other types (grammar)
It is sometimes useful to know a type by several names or to treat a type abstractly. There are several mechanisms in Dafny to do this:
- (Section 5.6.1) A typical synonym type, in which a type name is a synonym for another type
- (Section 5.6.2) An abstract type, in which a new type name is declared as an uninterpreted type
- (Section 5.6.3) A subset type, in which a new type name is given to a subset of the values of a given type
- ([Section 0.0){#sec-newtypes)) A newtype, in which a subset type is declared, but with restrictions on converting to and from its base type
5.6.1. Type synonyms (grammar)
type T = int
type SS<T> = set<set<T>>
A type synonym declaration:
type Y<T> = G
declares Y<T>
to be a synonym for the type G
.
If the = G
is omitted then the declaration just declares a name as an uninterpreted
opaque type, as described in Section 5.6.2. Such types may be
given a definition elsewhere in the Dafny program.
Here, T
is a
nonempty list of type parameters (each of which optionally
has a type characteristics suffix), which can be used as free type
variables in G
. If the synonym has no type parameters, the “<T>
”
is dropped. In all cases, a type synonym is just a synonym. That is,
there is never a difference, other than possibly in error messages
produced, between Y<T>
and G
.
For example, the names of the following type synonyms may improve the readability of a program:
type Replacements<T> = map<T,T>
type Vertex = int
The new type name itself may have type characteristics declared, and may need to if there is no definition. If there is a definition, the type characteristics are typically inferred from the definition. The syntax is like this:
type Z(==)<T(0)>
As already described in Section 5.5.3.5, string
is a built-in
type synonym for seq<char>
, as if it would have been declared as
follows:
type string_(==,0,!new) = seq<char>
If the implicit declaration did not include the type characteristics, they would be inferred in any case.
Note that although a type synonym can be declared and used in place of a type name, that does not affect the names of datatype or class constructors. For example, consider
datatype Pair<T> = Pair(first: T, second: T)
type IntPair = Pair<int>
const p: IntPair := Pair(1,2) // OK
const q: IntPair := IntPair(3,4) // Error
In the declaration of q
, IntPair
is the name of a type, not the name of a function or datatype constructor.
5.6.2. Abstract types (grammar)
Examples:
type T
type Q { function toString(t: T): string }
An abstract type is a special case of a type synonym that is underspecified. Such a type is declared simply by:
type Y<T>
Its definition can be stated in a
refining module. The name Y
can be immediately followed by
a type characteristics suffix (Section 5.3.1).
Because there is no defining RHS, the type characteristics cannot be inferred and so
must be stated. If, in some refining module, a definition of the type is given, the
type characteristics must match those of the new definition.
For example, the declarations
type T
function F(t: T): T
can be used to model an uninterpreted function F
on some
arbitrary type T
. As another example,
type Monad<T>
can be used abstractly to represent an arbitrary parameterized monad.
Even as an abstract type, the type may be given members such as constants, methods or functions. For example,
abstract module P {
type T {
function ToString(): string
}
}
module X refines P {
newtype T = i | 0 <= i < 10 {
function ToString(): string { "" }
}
}
The abstract type P.T
has a declared member ToString
, which can be called wherever P.T
may be used.
In the refining module X
, T
is declared to be a newtype
, in which ToString
now has a body.
It would be an error to refine P.T
as a simple type synonym or subset type in X
, say type T = int
, because
type synonyms may not have members.
5.6.3. Subset types (grammar)
Examples:
type Pos = i: int | i > 0 witness 1
type PosReal = r | r > 0.0 witness 1.0
type Empty = n: nat | n < 0 witness *
type Big = n: nat | n > 1000 ghost witness 10000
A subset type is a restricted use of an existing type, called the base type of the subset type. A subset type is like a combined use of the base type and a predicate on the base type.
An assignment from a subset type to its base type is always allowed. An assignment in the other direction, from the base type to a subset type, is allowed provided the value assigned does indeed satisfy the predicate of the subset type. This condition is checked by the verifier, not by the type checker. Similarly, assignments from one subset type to another (both with the same base type) are also permitted, as long as it can be established that the value being assigned satisfies the predicate defining the receiving subset type. (Note, in contrast, assignments between a newtype and its base type are never allowed, even if the value assigned is a value of the target type. For such assignments, an explicit conversion must be used, see Section 9.10.)
The declaration of a subset type permits an optional witness
clause, to declare that there is
a value that satisfies the subset type’s predicate; that is, the witness clause establishes that the defined
type is not empty. The compiler may, but is not obligated to, use this value when auto-initializing a
newly declared variable of the subset type.
Dafny builds in three families of subset types, as described next.
5.6.3.1. Type nat
The built-in type nat
, which represents the non-negative integers
(that is, the natural numbers), is a subset type:
type nat = n: int | 0 <= n
A simple example that
puts subset type nat
to good use is the standard Fibonacci
function:
function Fib(n: nat): nat
{
if n < 2 then n else Fib(n-2) + Fib(n-1)
}
An equivalent, but clumsy, formulation of this function (modulo the
wording of any error messages produced at call sites) would be to use
type int
and to write the restricting predicate in pre- and
postconditions:
function Fib(n: int): int
requires 0 <= n // the function argument must be non-negative
ensures 0 <= Fib(n) // the function result is non-negative
{
if n < 2 then n else Fib(n - 2) + Fib(n - 1)
}
5.6.3.2. Non-null types
Every class, trait, and iterator declaration C
gives rise to two types.
One type has the name C?
(that is, the name of the class, trait,
or iterator declaration with a ?
character appended to the end).
The values of C?
are the references to C
objects, and also
the value null
.
In other words, C?
is the type of possibly null references
(aka, nullable references) to C
objects.
The other type has the name C
(that is, the same name as the
class, trait, or iterator declaration).
Its values are the references to C
objects, and does not contain
the value null
.
In other words, C
is the type of non-null references to C
objects.
The type C
is a subset type of C?
:
type C = c: C? | c != null
(It may be natural to think of the type C?
as the union of
type C
and the value null
, but, technically, Dafny defines
C
as a subset type with base type C?
.)
From being a subset type, we get that C
is a subtype of C?
.
Moreover, if a class or trait C
extends a trait B
, then
type C
is a subtype of B
and type C?
is a subtype of B?
.
Every possibly-null reference type is a subtype of the
built-in possibly-null trait type object?
, and
every non-null reference type is a subtype of the
built-in non-null trait type object
. (And, from the fact
that object
is a subset type of object?
, we also have that
object
is a subtype of object?
.)
Arrays are references and array types also come in these two flavors.
For example,
array?
and array2?
are possibly-null (1- and 2-dimensional) array types, and
array
and array2
are their respective non-null types.
Note that ?
is not an operator. Instead, it is simply the last
character of the name of these various possibly-null types.
5.6.3.3. Arrow types: ->
, -->
, and ~>
For more information about arrow types (function types), see Section 5.12. This section is a preview to point out the subset-type relationships among the kinds of function types.
The built-in type
->
stands for total functions,-->
stands for partial functions (that is, functions with possiblerequires
clauses), and~>
stands for all functions.
More precisely, type constructors
exist for any arity (() -> X
, A -> X
, (A, B) -> X
, (A, B, C) -> X
,
etc.).
For a list of types TT
and a type U
, the values of the arrow type (TT) ~> U
are functions from TT
to U
. This includes functions that may read the
heap and functions that are not defined on all inputs. It is not common
to need this generality (and working with such general functions is
difficult). Therefore, Dafny defines two subset types that are more common
(and much easier to work with).
The type (TT) --> U
denotes the subset of (TT) ~> U
where the functions
do not read the (mutable parts of the) heap.
Values of type (TT) --> U
are called partial functions,
and the subset type (TT) --> U
is called the partial arrow type.
(As a mnemonic to help you remember that this is the partial arrow, you may
think of the little gap between the two hyphens in -->
as showing a broken
arrow.)
The built-in partial arrow type is defined as follows (here shown for arrows with arity 1):
type A --> B = f: A ~> B | forall a :: f.reads(a) == {}
(except that what is shown here left of the =
is not legal Dafny syntax).
That is, the partial arrow type is defined as those functions f
whose reads frame is empty for all inputs.
More precisely, taking variance into account, the partial arrow type
is defined as
type -A --> +B = f: A ~> B | forall a :: f.reads(a) == {}
The type (TT) -> U
is, in turn, a subset type of (TT) --> U
, adding the
restriction that the functions must not impose any precondition. That is,
values of type (TT) -> U
are total functions, and the subset type
(TT) -> U
is called the total arrow type.
The built-in total arrow type is defined as follows (here shown for arrows with arity 1):
type -A -> +B = f: A --> B | forall a :: f.requires(a)
That is, the total arrow type is defined as those partial functions f
whose precondition evaluates to true
for all inputs.
Among these types, the most commonly used are the total arrow types.
They are also the easiest to work with. Because they are common, they
have the simplest syntax (->
).
Note, informally, we tend to speak of all three of these types as arrow types,
even though, technically, the ~>
types are the arrow types and the
-->
and ->
types are subset types thereof. The one place where you may need to
remember that -->
and ->
are subset types is in some error messages.
For example, if you try to assign a partial function to a variable whose
type is a total arrow type and the verifier is not able to prove that the
partial function really is total, then you’ll get an error saying that the subset-type
constraint may not be satisfied.
For more information about arrow types, see Section 5.12.
5.6.3.4. Witness clauses
The declaration of a subset type permits an optional witness
clause.
Types in Dafny are generally expected to be non-empty, in part because
variables of any type are expected to have some value when they are used.
In many cases, Dafny can determine that a newly declared type has
some value.
For example, in the absence of a witness clause,
a numeric type that includes 0 is known by Dafny
to be non-empty.
However, Dafny cannot always make this determination.
If it cannot, a witness
clause is required. The value given in
the witness
clause must be a valid value for the type and assures Dafny
that the type is non-empty. (The variation witness *
is described below.)
For example,
type OddInt = x: int | x % 2 == 1
will give an error message, but
type OddInt = x: int | x % 2 == 1 witness 73
does not. Here is another example:
type NonEmptySeq = x: seq<int> | |x| > 0 witness [0]
If the witness is only available in ghost code, you can declare the witness
as a ghost witness
. In this case, the Dafny verifier knows that the type
is non-empty, but it will not be able to auto-initialize a variable of that
type in compiled code.
There is even room to do the following:
type BaseType
predicate RHS(x: BaseType)
type MySubset = x: BaseType | RHS(x) ghost witness MySubsetWitness()
function MySubsetWitness(): BaseType
ensures RHS(MySubsetWitness())
Here the type is given a ghost witness: the result of the expression
MySubsetWitness()
, which is a call of a (ghost) function.
Now that function has a postcondition saying that the returned value
is indeed a candidate value for the declared type, so the verifier is
satisfied regarding the non-emptiness of the type. However, the function
has no body, so there is still no proof that there is indeed such a witness.
You can either supply a, perhaps complicated, body to generate a viable
candidate or you can be very sure, without proof, that there is indeed such a value.
If you are wrong, you have introduced an unsoundness into your program.
In addition though, types are allowed to be empty or possibly empty.
This is indicated by the clause witness *
, which tells the verifier not to check for a satisfying witness.
A declaration like this produces an empty type:
type ReallyEmpty = x: int | false witness *
The type can be used in code like
method M(x: ReallyEmpty) returns (seven: int)
ensures seven == 7
{
seven := 10;
}
which does verify. But the method can never be called because there is no value that can be supplied as the argument. Even this code
method P() returns (seven: int)
ensures seven == 7
{
var x: ReallyEmpty;
seven := 10;
}
does not complain about x
unless x
is actually used, in which case it must have a value.
The postcondition in P
does not verify, but not because of the empty type.
5.7. Newtypes (grammar)
Examples:
newtype I = int
newtype D = i: int | 0 <= i < 10
newtype uint8 = i | 0 <= i < 256
A newtype is like a type synonym or subset type except that it declares a wholly new type
name that is distinct from its base type. It also accepts an optional witness
clause.
A new type can be declared with the newtype declaration, for example:
newtype N = x: M | Q
where M
is a type and Q
is a boolean expression that can
use x
as a free variable. If M
is an integer-based numeric type,
then so is N
; if M
is real-based, then so is N
. If the type M
can be inferred from Q
, the “: M
” can be omitted. If Q
is just
true
, then the declaration can be given simply as:
newtype N = M
Type M
is known as the base type of N
. At present, Dafny only supports
int
and real
as base types of newtypes.
A newtype is a type that supports the same operations as its
base type. The newtype is distinct from and incompatible with other
types; in particular, it is not assignable to its base type
without an explicit conversion. An important difference between the
operations on a newtype and the operations on its base type is that
the newtype operations are defined only if the result satisfies the
predicate Q
, and likewise for the literals of the
newtype.
For example, suppose lo
and hi
are integer-based numerics that
satisfy 0 <= lo <= hi
and consider the following code fragment:
var mid := (lo + hi) / 2;
If lo
and hi
have type int
, then the code fragment is legal; in
particular, it never overflows, since int
has no upper bound. In
contrast, if lo
and hi
are variables of a newtype int32
declared
as follows:
newtype int32 = x | -0x8000_0000 <= x < 0x8000_0000
then the code fragment is erroneous, since the result of the addition
may fail to satisfy the predicate in the definition of int32
. The
code fragment can be rewritten as
var mid := lo + (hi - lo) / 2;
in which case it is legal for both int
and int32
.
An additional point with respect to arithmetic overflow is that for (signed)
int32
values hi
and lo
constrained only by lo <= hi
, the difference hi - lo
can also overflow the bounds of the int32
type. So you could also write:
var mid := lo + (hi/2 - lo/2);
Since a newtype is incompatible with its base type and since all
results of the newtype’s operations are members of the newtype, a
compiler for Dafny is free to specialize the run-time representation
of the newtype. For example, by scrutinizing the definition of
int32
above, a compiler may decide to store int32
values using
signed 32-bit integers in the target hardware.
The incompatibility of a newtype and its basetype is intentional, as newtypes are meant to be used as distinct types from the basetype. If numeric types are desired that mix more readily with the basetype, the subset types described in Section 5.6.3 may be more appropriate.
Note that the bound variable x
in Q
has type M
, not N
.
Consequently, it may not be possible to state Q
about the N
value. For example, consider the following type of 8-bit 2’s
complement integers:
newtype int8 = x: int | -128 <= x < 128
and consider a variable c
of type int8
. The expression
-128 <= c < 128
is not well-defined, because the comparisons require each operand to
have type int8
, which means the literal 128
is checked to be of
type int8
, which it is not. A proper way to write this expression
is to use a conversion operation, described in Section 5.7.1, on c
to
convert it to the base type:
-128 <= c as int < 128
If possible, Dafny compilers will represent values of the newtype using
a native type for the sake of efficiency. This action can
be inhibited or a specific native data type selected by
using the {:nativeType}
attribute, as explained in
Section 11.1.2.
Furthermore, for the compiler to be able to make an appropriate choice of representation, the constants in the defining expression as shown above must be known constants at compile-time. They need not be numeric literals; combinations of basic operations and symbolic constants are also allowed as described in Section 9.37.
5.7.1. Conversion operations
For every type N
, there is a conversion operation with the
name as N
, described more fully in Section 9.10.
It is a partial function defined when the
given value, which can be of any type, is a member of the type
converted to. When the conversion is from a real-based numeric type
to an integer-based numeric type, the operation requires that the
real-based argument have no fractional part. (To round a real-based
numeric value down to the nearest integer, use the .Floor
member,
see Section 5.2.2.)
To illustrate using the example from above, if lo
and hi
have type
int32
, then the code fragment can legally be written as follows:
var mid := (lo as int + hi as int) / 2;
where the type of mid
is inferred to be int
. Since the result
value of the division is a member of type int32
, one can introduce
yet another conversion operation to make the type of mid
be int32
:
var mid := ((lo as int + hi as int) / 2) as int32;
If the compiler does specialize the run-time representation for
int32
, then these statements come at the expense of two,
respectively three, run-time conversions.
The as N
conversion operation is grammatically a suffix operation like
.
field and array indexing, but binds less tightly than unary operations:
- x as int
is (- x) as int
; a + b as int
is a + (b as int)
.
The as N
conversion can also be used with reference types. For example,
if C
is a class, c
is an expression of type C
, and o
is an expression
of type object
, then c as object
and c as object?
are upcasts
and o is C
is a downcast. A downcast requires the LHS expression to
have the RHS type, as is enforced by the verifier.
For some types (in particular, reference types), there is also a
corresponding is
operation (Section 9.10) that
tests whether a value is valid for a given type.
5.8. Class types (grammar)
Examples:
trait T {}
class A {}
class B extends T {
const b: B?
var v: int
constructor (vv: int) { v := vv; b := null; }
function toString(): string { "a B" }
method m(i: int) { var x := new B(0); }
static method q() {}
}
Declarations within a class all begin with keywords and do not end with semicolons.
A class C
is a reference type declared as follows:
class C<T> extends J1, ..., Jn
{
_members_
}
where the <>-enclosed list of one-or-more type parameters T
is optional. The text
“extends J1, ..., Jn
” is also optional and says that the class extends traits J1
… Jn
.
The members of a class are fields, constant fields, functions, and
methods. These are accessed or invoked by dereferencing a reference
to a C
instance.
A function or method is invoked on an instance
of C
, unless the function or method is declared static
.
A function or method that is not static
is called an
instance function or method.
An instance function or method takes an implicit receiver
parameter, namely, the instance used to access the member. In the
specification and body of an instance function or method, the receiver
parameter can be referred to explicitly by the keyword this
.
However, in such places, members of this
can also be mentioned
without any qualification. To illustrate, the qualified this.f
and
the unqualified f
refer to the same field of the same object in the
following example:
class C {
var f: int
var x: int
method Example() returns (b: bool)
{
var x: int;
b := f == this.f;
}
}
so the method body always assigns true
to the out-parameter b
.
However, in this example, x
and this.x
are different because
the field x
is shadowed by the declaration of the local variable x
.
There is no semantic difference between qualified and
unqualified accesses to the same receiver and member.
A C
instance is created using new
. There are three forms of new
,
depending on whether or not the class declares any constructors
(see Section 6.3.2):
c := new C;
c := new C.Init(args);
c := new C(args);
For a class with no constructors, the first two forms can be used.
The first form simply allocates a new instance of a C
object, initializing
its fields to values of their respective types (and initializing each const
field
with a RHS to its specified value). The second form additionally invokes
an initialization method (here, named Init
) on the newly allocated object
and the given arguments. It is therefore a shorthand for
c := new C;
c.Init(args);
An initialization method is an ordinary method that has no out-parameters and
that modifies no more than this
.
For a class that declares one or more constructors, the second and third forms
of new
can be used. For such a class, the second form invokes the indicated
constructor (here, named Init
), which allocates and initializes the object.
The third form is the same as the second, but invokes the anonymous constructor
of the class (that is, a constructor declared with the empty-string name).
The details of constructors and other class members are described in Section 6.3.2.
5.9. Trait types (grammar)
A trait is an abstract superclass, similar to an “interface” or “mixin”. A trait can be extended only by another trait or by a class (and in the latter case we say that the class implements the trait). More specifically, algebraic datatypes cannot extend traits.4
The declaration of a trait is much like that of a class:
trait J
{
_members_
}
where members can include fields, constant fields, functions, methods and declarations of nested traits, but
no constructor methods. The functions and methods are allowed to be
declared static
.
A reference type C
that extends a trait J
is assignable to a variable of
type J
;
a value of type J
is assignable to a variable of a reference type C
that
extends J
only if the verifier can prove that the reference does
indeed refer to an object of allocated type C
.
The members of J
are available as members
of C
. A member in J
is not allowed to be redeclared in C
,
except if the member is a non-static
function or method without a
body in J
. By doing so, type C
can supply a stronger
specification and a body for the member. There is further discussion on
this point in Section 5.9.2.
new
is not allowed to be used with traits. Therefore, there is no
object whose allocated type is a trait. But there can of course be
objects of a class C
that implement a trait J
, and a reference to
such a C
object can be used as a value of type J
.
5.9.1. Type object
(grammar)
There is a built-in trait object
that is implicitly extended by all classes and traits.
It produces two types: the type object?
that is a supertype of all
reference types and a subset type object
that is a supertype of all non-null reference types.
This includes reference types like arrays and iterators that do not permit
explicit extending of traits. The purpose of type object
is to enable a uniform treatment of dynamic frames. In particular, it
is useful to keep a ghost field (typically named Repr
for
“representation”) of type set<object>
.
It serves no purpose (but does no harm) to explicitly list the trait object
as
an extendee in a class or trait declaration.
Traits object?
and object
contain no members.
The dynamic allocation of objects is done using new C
…,
where C
is the name of a class.
The name C
is not allowed to be a trait,
except that it is allowed to be object
.
The construction new object
allocates a new object (of an unspecified class type).
The construction can be used to create unique references, where no other properties of those references are needed.
(new object?
makes no sense; always use new object
instead because the result of
new
is always non-null.)
5.9.2. Inheritance
The purpose of traits is to be able to express abstraction: a trait encapsulates a set of behaviors; classes and traits that extend it inherit those behaviors, perhaps specializing them.
A trait or class may extend multiple other traits.
The traits syntactically listed in a trait or class’s extends
clause
are called its direct parents; the transitive parents of a trait or class
are its direct parents, the transitive parents of its direct parents, and
the object
trait (if it is not itself object
).
These are sets of traits, in that it does not matter if
there are repetitions of a given trait in a class or trait’s direct or
transitive parents. However, if a trait with type parameters is repeated,
it must have the same actual type parameters in each instance.
Furthermore, a trait may not be in its own set of transitive parents; that is,
the graph of traits connected by the directed extends relationship may not
have any cycles.
A class or trait inherits (as if they are copied) all the instance members of its transitive parents. However, since names may not be overloaded in Dafny, different members (that is, members with different type signatures) within the set of transitive parents and the class or trait itself must have different names.5 This restriction does mean that traits from different sources that coincidentally use the same name for different purposes cannot be combined by being part of the set of transitive parents for some new trait or class.
A declaration of member C.M
in a class or trait overrides any other declarations
of the same name (and signature) in a transitive parent. C.M
is then called an
override; a declaration that
does not override anything is called an original declaration.
Static members of a trait may not be redeclared; thus, if there is a body it must be declared in the trait; the compiler will require a body, though the verifier will not.
Where traits within an extension hierarchy do declare instance members with the same name (and thus the same signature), some rules apply. Recall that, for methods, every declaration includes a specification; if no specification is given explicitly, a default specification applies. Instance method declarations in traits, however, need not have a body, as a body can be declared in an override.
For a given non-static method M,
- A trait or class may not redeclare M if it has a transitive parent that declares M and provides a body.
- A trait may but need not provide a body if all its transitive parents that declare M do not declare a body.
- A trait or class may not have more than one transitive parent that declares M with a body.
- A class that has one or more transitive parents that declare M without a body and no transitive parent that declares M with a body must itself redeclare M with a body if it is compiled. (The verifier alone does not require a body.)
- Currently (and under debate), the following restriction applies:
if
M
overrides two (or more) declarations,P.M
andQ.M
, then eitherP.M
must overrideQ.M
orQ.M
must overrideP.M
.
The last restriction above is the current implementation. It effectively limits inheritance of a method M to a single “chain” of declarations and does not permit mixins.
Each of any method declarations explicitly or implicitly
includes a specification. In simple cases, those syntactically separate
specifications will be copies of each other (up to renaming to take account
of differing formal parameter names). However they need not be. The rule is
that the specifications of M in a given class or trait must be as strong as
M’s specifications in a transitive parent.
Here as strong as means that it
must be permitted to call the subtype’s M in the context of the supertype’s M.
Stated differently, where P and C are a parent trait and a child class or trait,
respectively, then, under the precondition of P.M
,
- C.M’s
requires
clause must be implied by P.M’srequires
clause - C.M’s
ensures
clause must imply P.M’sensures
clause - C.M’s
reads
set must be a subset of P.M’sreads
set - C.M’s
modifies
set must be a subset of P.M’smodifies
set - C.M’s
decreases
expression must be smaller than or equal to P.M’sdecreases
expression
Non-static const and field declarations are also inherited from parent traits. These may not be redeclared in extending traits and classes. However, a trait need not initialize a const field with a value. The class that extends a trait that declares such a const field without an initializer can initialize the field in a constructor. If the declaring trait does give an initial value in the declaration, the extending class or trait may not either redeclare the field or give it a value in a constructor.
When names are inherited from multiple traits, they must be different. If two traits declare a common name (even with the same signature), they cannot both be extendees of the same class or trait.
5.9.3. Example of traits
As an example, the following trait represents movable geometric shapes:
trait Shape
{
function Width(): real
reads this
decreases 1
method Move(dx: real, dy: real)
modifies this
method MoveH(dx: real)
modifies this
{
Move(dx, 0.0);
}
}
Members Width
and Move
are abstract (that is, body-less) and can
be implemented differently by different classes that extend the trait.
The implementation of method MoveH
is given in the trait and thus
is used by all classes that extend Shape
. Here are two classes
that each extend Shape
:
class UnitSquare extends Shape
{
var x: real, y: real
function Width(): real
decreases 0
{ // note the empty reads clause
1.0
}
method Move(dx: real, dy: real)
modifies this
{
x, y := x + dx, y + dy;
}
}
class LowerRightTriangle extends Shape
{
var xNW: real, yNW: real, xSE: real, ySE: real
function Width(): real
reads this
decreases 0
{
xSE - xNW
}
method Move(dx: real, dy: real)
modifies this
{
xNW, yNW, xSE, ySE := xNW + dx, yNW + dy, xSE + dx, ySE + dy;
}
}
Note that the classes can declare additional members, that they supply implementations for the abstract members of the trait, that they repeat the member signatures, and that they are responsible for providing their own member specifications that both strengthen the corresponding specification in the trait and are satisfied by the provided body. Finally, here is some code that creates two class instances and uses them together as shapes:
method m() {
var myShapes: seq<Shape>;
var A := new UnitSquare;
myShapes := [A];
var tri := new LowerRightTriangle;
// myShapes contains two Shape values, of different classes
myShapes := myShapes + [tri];
// move shape 1 to the right by the width of shape 0
myShapes[1].MoveH(myShapes[0].Width());
}
5.10. Array types (grammar)
Dafny supports mutable fixed-length array types of any positive dimension. Array types are (heap-based) reference types.
arrayToken
is a kind of reserved token,
such as array
, array?
, array2
, array2?
, array3
, and so on (but not array1
).
The type parameter suffix giving the element type can be omitted if the element type can be inferred, though in that case it is likely that the arrayToken
itself is also
inferrable and can be omitted.
5.10.1. One-dimensional arrays
A one-dimensional array of n
T
elements may be initialized by
any expression that returns a value of the desired type.
Commonly, array allocation expressions are used.
Some examples are shown here:
type T(0)
method m(n: nat) {
var a := new T[n];
var b: array<int> := new int[8];
var c: array := new T[9];
}
The initial values of the array elements are arbitrary values of type
T
.
A one-dimensional array value can also be assigned using an ordered list of expressions enclosed in square brackets, as follows:
a := new T[] [t1, t2, t3, t4];
The initialization can also use an expression that returns a function of type nat -> T
:
a := new int[5](i => i*i);
In fact, the initializer can simply be a function name for the right type of function:
a := new int[5](Square);
The length of an array is retrieved using the immutable Length
member. For example, the array allocated with a := new T[n];
satisfies:
a.Length == n
Once an array is allocated, its length cannot be changed.
For any integer-based numeric i
in the range 0 <= i < a.Length
,
the array selection expression a[i]
retrieves element i
(that
is, the element preceded by i
elements in the array). The
element stored at i
can be changed to a value t
using the array
update statement:
a[i] := t;
Caveat: The type of the array created by new T[n]
is
array<T>
. A mistake that is simple to make and that can lead to
befuddlement is to write array<T>
instead of T
after new
.
For example, consider the following:
type T(0)
method m(n: nat) {
var a := new array<T>;
var b := new array<T>[n];
var c := new array<T>(n); // resolution error
var d := new array(n); // resolution error
}
The first statement allocates an array of type array<T>
, but of
unknown length. The second allocates an array of type
array<array<T>>
of length n
, that is, an array that holds n
values of type array<T>
. The third statement allocates an
array of type array<T>
and then attempts to invoke an anonymous
constructor on this array, passing argument n
. Since array
has no
constructors, let alone an anonymous constructor, this statement
gives rise to an error. If the type-parameter list is omitted for a
type that expects type parameters, Dafny will attempt to fill these
in, so as long as the array
type parameter can be inferred, it is
okay to leave off the “<T>
” in the fourth statement above. However,
as with the third statement, array
has no anonymous constructor, so
an error message is generated.
5.10.2. Converting arrays to sequences
One-dimensional arrays support operations that convert a stretch of
consecutive elements into a sequence. For any array a
of type
array<T>
, integer-based numerics lo
and hi
satisfying
0 <= lo <= hi <= a.Length
, the following operations each yields a
seq<T>
:
expression | description |
---|---|
a[lo..hi] |
subarray conversion to sequence |
a[lo..] |
drop |
a[..hi] |
take |
a[..] |
array conversion to sequence |
The expression a[lo..hi]
takes the first hi
elements of the array,
then drops the first lo
elements thereof and returns what remains as
a sequence, with length hi - lo
.
The other operations are special instances of the first. If lo
is
omitted, it defaults to 0
and if hi
is omitted, it defaults to
a.Length
.
In the last operation, both lo
and hi
have been omitted, thus
a[..]
returns the sequence consisting of all the array elements of
a
.
The subarray operations are especially useful in specifications. For
example, the loop invariant of a binary search algorithm that uses
variables lo
and hi
to delimit the subarray where the search key
may still be found can be expressed as follows:
key !in a[..lo] && key !in a[hi..]
Another use is to say that a certain range of array elements have not been changed since the beginning of a method:
a[lo..hi] == old(a[lo..hi])
or since the beginning of a loop:
ghost var prevElements := a[..];
while // ...
invariant a[lo..hi] == prevElements[lo..hi]
{
// ...
}
Note that the type of prevElements
in this example is seq<T>
, if
a
has type array<T>
.
A final example of the subarray operation lies in expressing that an array’s elements are a permutation of the array’s elements at the beginning of a method, as would be done in most sorting algorithms. Here, the subarray operation is combined with the sequence-to-multiset conversion:
multiset(a[..]) == multiset(old(a[..]))
5.10.3. Multi-dimensional arrays
An array of 2 or more dimensions is mostly like a one-dimensional
array, except that new
takes more length arguments (one for each
dimension), and the array selection expression and the array update
statement take more indices. For example:
matrix := new T[m, n];
matrix[i, j], matrix[x, y] := matrix[x, y], matrix[i, j];
create a 2-dimensional array whose dimensions have lengths m
and
n
, respectively, and then swaps the elements at i,j
and x,y
.
The type of matrix
is array2<T>
, and similarly for
higher-dimensional arrays (array3<T>
, array4<T>
, etc.). Note,
however, that there is no type array0<T>
, and what could have been
array1<T>
is actually named just array<T>
. (Accordingly, array0
and array1
are just
normal identifiers, not type names.)
The new
operation above requires m
and n
to be non-negative
integer-based numerics. These lengths can be retrieved using the
immutable fields Length0
and Length1
. For example, the following
holds for the array created above:
matrix.Length0 == m && matrix.Length1 == n
Higher-dimensional arrays are similar (Length0
, Length1
,
Length2
, …). The array selection expression and array update
statement require that the indices are in bounds. For example, the
swap statement above is well-formed only if:
0 <= i < matrix.Length0 && 0 <= j < matrix.Length1 &&
0 <= x < matrix.Length0 && 0 <= y < matrix.Length1
In contrast to one-dimensional arrays, there is no operation to convert stretches of elements from a multi-dimensional array to a sequence.
There is however syntax to create a multi-dimensional array value using a function: see Section 9.16.
5.11. Iterator types (grammar)
See Section 7.5 for a description of iterator specifications.
An iterator provides a programming abstraction for writing code that iteratively returns elements. These CLU-style iterators are co-routines in the sense that they keep track of their own program counter and control can be transferred into and out of the iterator body.
An iterator is declared as follows:
iterator Iter<T>(_in-params_) yields (_yield-params_)
_specification_
{
_body_
}
where T
is a list of type parameters (as usual, if there are no type
parameters, “<T>
” is omitted). This declaration gives rise to a
reference type with the same name, Iter<T>
. In the signature,
in-parameters and yield-parameters are the iterator’s analog of a
method’s in-parameters and out-parameters. The difference is that the
out-parameters of a method are returned to a caller just once, whereas
the yield-parameters of an iterator are returned each time the iterator
body performs a yield
. The body consists of statements, like in a
method body, but with the availability also of yield
statements.
From the perspective of an iterator client, the iterator
declaration
can be understood as generating a class Iter<T>
with various
members, a simplified version of which is described next.
The Iter<T>
class contains an anonymous constructor whose parameters
are the iterator’s in-parameters:
predicate Valid()
constructor (_in-params_)
modifies this
ensures Valid()
An iterator is created using new
and this anonymous constructor.
For example, an iterator willing to return ten consecutive integers
from start
can be declared as follows:
iterator Gen(start: int) yields (x: int)
yield ensures |xs| <= 10 && x == start + |xs| - 1
{
var i := 0;
while i < 10 invariant |xs| == i {
x := start + i;
yield;
i := i + 1;
}
}
An instance of this iterator is created using
iter := new Gen(30);
It is used like this:
method Main() {
var i := new Gen(30);
while true
invariant i.Valid() && fresh(i._new)
decreases 10 - |i.xs|
{
var m := i.MoveNext();
if (!m) {break; }
print i.x;
}
}
The predicate Valid()
says when the iterator is in a state where one
can attempt to compute more elements. It is a postcondition of the
constructor and occurs in the specification of the MoveNext
member:
method MoveNext() returns (more: bool)
requires Valid()
modifies this
ensures more ==> Valid()
Note that the iterator remains valid as long as MoveNext
returns
true
. Once MoveNext
returns false
, the MoveNext
method can no
longer be called. Note, the client is under no obligation to keep
calling MoveNext
until it returns false
, and the body of the
iterator is allowed to keep returning elements forever.
The in-parameters of the iterator are stored in immutable fields of
the iterator class. To illustrate in terms of the example above, the
iterator class Gen
contains the following field:
const start: int
The yield-parameters also result in members of the iterator class:
var x: int
These fields are set by the MoveNext
method. If MoveNext
returns
true
, the latest yield values are available in these fields and the
client can read them from there.
To aid in writing specifications, the iterator class also contains
ghost members that keep the history of values returned by
MoveNext
. The names of these ghost fields follow the names of the
yield-parameters with an “s
” appended to the name (to suggest
plural). Name checking rules make sure these names do not give rise
to ambiguities. The iterator class for Gen
above thus contains:
ghost var xs: seq<int>
These history fields are changed automatically by MoveNext
, but are
not assignable by user code.
Finally, the iterator class contains some special fields for use in specifications. In particular, the iterator specification is recorded in the following immutable fields:
ghost var _reads: set<object>
ghost var _modifies: set<object>
ghost var _decreases0: T0
ghost var _decreases1: T1
// ...
where there is a _decreases(
i): T(
i)
field for each
component of the iterator’s decreases
clause.6
In addition, there is a field:
ghost var _new: set<object>;
to which any objects allocated on behalf of the iterator body are
added. The iterator body is allowed to remove elements from the
_new
set, but cannot by assignment to _new
add any elements.
Note, in the precondition of the iterator, which is to hold upon
construction of the iterator, the in-parameters are indeed
in-parameters, not fields of this
.
It is regrettably tricky to use iterators. The language really
ought to have a foreach
statement to make this easier.
Here is an example showing a definition and use of an iterator.
iterator Iter<T(0)>(s: set<T>) yields (x: T)
yield ensures x in s && x !in xs[..|xs|-1]
ensures s == set z | z in xs
{
var r := s;
while (r != {})
invariant r !! set z | z in xs
invariant s == r + set z | z in xs
{
var y :| y in r;
assert y !in xs;
r, x := r - {y}, y;
assert y !in xs;
yield;
assert y == xs[|xs|-1]; // a lemma to help prove loop invariant
}
}
method UseIterToCopy<T(0)>(s: set<T>) returns (t: set<T>)
ensures s == t
{
t := {};
var m := new Iter(s);
while (true)
invariant m.Valid() && fresh(m._new)
invariant t == set z | z in m.xs
decreases s - t
{
var more := m.MoveNext();
if (!more) { break; }
t := t + {m.x};
}
}
The design of iterators is under discussion and may change.
async method AM<T>(\(_in-params_\)) returns (\(_out-params_\))
also gives rise to an async-task type AM<T>
(outside the enclosing
class, the name of the type needs the qualification C.AM<T>
). The
async-task type is a reference type and can be understood as a class
with various members, a simplified version of which is described next.
Each in-parameter x
of type X
of the asynchronous method gives
rise to a immutable ghost field of the async-task type:
ghost var x: X;
Each out-parameter y
of type Y
gives rise to a field
var y: Y;
These fields are changed automatically by the time the asynchronous method is successfully awaited, but are not assignable by user code.
The async-task type also gets a number of special fields that are used to keep track of dependencies, outstanding tasks, newly allocated objects, etc. These fields will be described in more detail as the design of asynchronous methods evolves.
–>
5.12. Arrow types (grammar)
Examples:
(int) -> int
(bool,int) ~> bool
() --> object?
Functions are first-class values in Dafny. The types of function values
are called arrow types (aka, function types).
Arrow types have the form (TT) ~> U
where TT
is a (possibly empty)
comma-delimited list of types and U
is a type.
TT
is called the function’s domain type(s) and U
is its
range type. For example, the type of a function
function F(x: int, arr: array<bool>): real
requires x < 1000
reads arr
is (int, array<bool>) ~> real
.
As seen in the example above, the functions that are values of a type
(TT) ~> U
can have a precondition (as indicated by the requires
clause)
and can read values in the heap (as indicated by the reads
clause).
As described in Section 5.6.3.3,
- the subset type
(TT) --> U
denotes partial (but heap-independent) functions - and the subset type
(TT) -> U
denotes total functions.
A function declared without a reads
clause is known by the type
checker to be a partial function. For example, the type of
function F(x: int, b: bool): real
requires x < 1000
is (int, bool) --> real
.
Similarly, a function declared with neither a reads
clause nor a
requires
clause is known by the type checker to be a total function.
For example, the type of
function F(x: int, b: bool): real
is (int, bool) -> real
.
In addition to functions declared by name, Dafny also supports anonymous
functions by means of lambda expressions (see Section 9.13).
To simplify the appearance of the basic case where a function’s
domain consists of a list of exactly one non-function, non-tuple type, the parentheses around
the domain type can be dropped in this case. For example, you may
write just T -> U
for a total arrow type.
This innocent simplification requires additional explanation in the
case where that one type is a tuple type, since tuple types are also
written with enclosing parentheses.
If the function takes a single argument that is a tuple, an additional
set of parentheses is needed. For example, the function
function G(pair: (int, bool)): real
has type ((int, bool)) -> real
. Note the necessary double
parentheses. Similarly, a function that takes no arguments is
different from one that takes a 0-tuple as an argument. For instance,
the functions
function NoArgs(): real
function Z(unit: ()): real
have types () -> real
and (()) -> real
, respectively.
The function arrows are right associative.
For example, A -> B -> C
means A -> (B -> C)
, whereas
the other association requires explicit parentheses: (A -> B) -> C
.
As another example, A -> B --> C ~> D
means
A -> (B --> (C ~> D))
.
Note that the receiver parameter of a named function is not part of
the type. Rather, it is used when looking up the function and can
then be thought of as being captured into the function definition.
For example, suppose function F
above is declared in a class C
and
that c
references an object of type C
; then, the following is type
correct:
var f: (int, bool) -> real := c.F;
whereas it would have been incorrect to have written something like:
var f': (C, int, bool) -> real := F; // not correct
The arrow types themselves do not divide a function’s parameters into ghost versus non-ghost. Instead, a function used as a first-class value is considered to be ghost if either the function or any of its arguments is ghost. The following example program illustrates:
function F(x: int, ghost y: int): int
{
x
}
method Example() {
ghost var f: (int, int) -> int;
var g: (int, int) -> int;
var h: (int) -> int;
var x: int;
f := F;
x := F(20, 30);
g := F; // error: tries to assign ghost to non-ghost
h := F; // error: wrong arity (and also tries to assign ghost to non-ghost)
}
In addition to its type signature, each function value has three properties, described next.
Every function implicitly takes the heap as an argument. No function
ever depends on the entire heap, however. A property of the
function is its declared upper bound on the set of heap locations it
depends on for a given input. This lets the verifier figure out that
certain heap modifications have no effect on the value returned by a
certain function. For a function f: T ~> U
and a value t
of type
T
, the dependency set is denoted f.reads(t)
and has type
set<object>
.
The second property of functions stems from the fact that every function
is potentially partial. In other words, a property of a function is its
precondition. For a function f: T ~> U
, the precondition of f
for a
parameter value t
of type T
is denoted f.requires(t)
and has type
bool
.
The third property of a function is more obvious—the function’s
body. For a function f: T ~> U
, the value that the function yields
for an input t
of type T
is denoted f(t)
and has type U
.
Note that f.reads
and f.requires
are themselves functions.
Suppose f
has type T ~> U
and t
has type T
. Then, f.reads
is a function of type T ~> set<object?>
whose reads
and requires
properties are:
f.reads.reads(t) == f.reads(t)
f.reads.requires(t) == true
f.requires
is a function of type T ~> bool
whose reads
and
requires
properties are:
f.requires.reads(t) == f.reads(t)
f.requires.requires(t) == true
In these examples, if f
instead had type T --> U
or T -> U
,
then the type of f.reads
is T -> set<object?>
and the type
of f.requires
is T -> bool
.
Dafny also supports anonymous functions by means of lambda expressions. See Section 9.13.
5.13. Tuple types
TupleType = "(" [ [ "ghost" ] Type { "," [ "ghost" ] Type } ] ")"
Dafny builds in record types that correspond to tuples and gives these a convenient special syntax, namely parentheses. For example, for what might have been declared as
datatype Pair<T,U> = Pair(0: T, 1: U)
Dafny provides the type (T, U)
and the constructor (t, u)
, as
if the datatype’s name were “” (i.e., an empty string)
and its type arguments are given in
round parentheses, and as if the constructor name were the empty string.
Note that
the destructor names are 0
and 1
, which are legal identifier names
for members. For an example showing the use of a tuple destructor, here
is a property that holds of 2-tuples (that is, pairs):
method m(){
assert (5, true).1 == true;
}
Dafny declares n-tuples where n is 0 or 2 or more. There are no
1-tuples, since parentheses around a single type or a single value have
no semantic meaning. The 0-tuple type, ()
, is often known as the
unit type and its single value, also written ()
, is known as unit.
The ghost
modifier can be used to mark tuple components as being used for specification only:
const pair: (int, ghost int) := (1, ghost 2)
5.14. Algebraic Datatypes (grammar)
Dafny offers two kinds of algebraic datatypes, those defined
inductively (with datatype
) and those defined coinductively (with codatatype
).
The salient property of
every datatype is that each value of the type uniquely identifies one
of the datatype’s constructors and each constructor is injective in
its parameters.
5.14.1. Inductive datatypes
The values of inductive datatypes can be seen as finite trees where
the leaves are values of basic types, numeric types, reference types,
coinductive datatypes, or arrow types. Indeed, values of
inductive datatypes can be compared using Dafny’s well-founded
<
ordering.
An inductive datatype is declared as follows:
datatype D<T> = _Ctors_
where Ctors is a nonempty |
-separated list of
(datatype) constructors for the datatype. Each constructor has the
form:
C(_params_)
where params is a comma-delimited list of types, optionally
preceded by a name for the parameter and a colon, and optionally
preceded by the keyword ghost
. If a constructor has no parameters,
the parentheses after the constructor name may be omitted. If no
constructor takes a parameter, the type is usually called an
enumeration; for example:
datatype Friends = Agnes | Agatha | Jermaine | Jack
For every constructor C
, Dafny defines a discriminator C?
, which
is a member that returns true
if and only if the datatype value has
been constructed using C
. For every named parameter p
of a
constructor C
, Dafny defines a destructor p
, which is a member
that returns the p
parameter from the C
call used to construct the
datatype value; its use requires that C?
holds. For example, for
the standard List
type
datatype List<T> = Nil | Cons(head: T, tail: List<T>)
the following holds:
method m() {
assert Cons(5, Nil).Cons? && Cons(5, Nil).head == 5;
}
Note that the expression
Cons(5, Nil).tail.head
is not well-formed, since Cons(5, Nil).tail
does not necessarily satisfy
Cons?
.
A constructor can have the same name as the enclosing datatype; this is especially useful for single-constructor datatypes, which are often called record types. For example, a record type for black-and-white pixels might be represented as follows:
datatype Pixel = Pixel(x: int, y: int, on: bool)
To call a constructor, it is usually necessary only to mention the
name of the constructor, but if this is ambiguous, it is always
possible to qualify the name of constructor by the name of the
datatype. For example, Cons(5, Nil)
above can be written
List.Cons(5, List.Nil)
As an alternative to calling a datatype constructor explicitly, a
datatype value can be constructed as a change in one parameter from a
given datatype value using the datatype update expression. For any
d
whose type is a datatype that includes a constructor C
that has
a parameter (destructor) named f
of type T
, and any expression t
of type T
,
d.(f := t)
constructs a value like d
but whose f
parameter is t
. The
operation requires that d
satisfies C?
. For example, the
following equality holds:
method m(){
assert Cons(4, Nil).(tail := Cons(3, Nil)) == Cons(4, Cons(3, Nil));
}
The datatype update expression also accepts multiple field names, provided these are distinct. For example, a node of some inductive datatype for trees may be updated as follows:
node.(left := L, right := R)
The operator <
is defined for two operands of the same datataype.
It means is properly contained in. For example, in the code
datatype X = T(t: X) | I(i: int)
method comp() {
var x := T(I(0));
var y := I(0);
var z := I(1);
assert x.t < x;
assert y < x;
assert !(x < x);
assert z < x; // FAILS
}
x
is a datatype value that holds a T
variant, which holds a I
variant, which holds an integer 0
.
The value x.t
is a portion of the datatype structure denoted by x
, so x.t < x
is true.
Datatype values are immutable mathematical values, so the value of y
is identical to the value of
x.t
, so y < x
is true also, even though y
is constructed from the ground up, rather than as
a portion of x
. However, z
is different than either y
or x.t
and consequently z < x
is not provable.
Furthermore, <
does not include ==
, so x < x
is false.
Note that only <
is defined; not <=
or >
or >=
.
Also, <
is underspecified. With the above code, one can prove neither z < x
nor !(z < x)
and neither
z < y
nor !(z < y)
. In each pair, though, one or the other is true, so (z < x) || !(z < x)
is provable.
5.14.2. Coinductive datatypes
Whereas Dafny insists that there is a way to construct every inductive
datatype value from the ground up, Dafny also supports
coinductive datatypes, whose constructors are evaluated lazily, and
hence the language allows infinite structures.
A coinductive datatype is declared
using the keyword codatatype
; other than that, it is declared and
used like an inductive datatype.
For example,
codatatype IList<T> = Nil | Cons(head: T, tail: IList<T>)
codatatype Stream<T> = More(head: T, tail: Stream<T>)
codatatype Tree<T> = Node(left: Tree<T>, value: T, right: Tree<T>)
declare possibly infinite lists (that is, lists that can be either finite or infinite), infinite streams (that is, lists that are always infinite), and infinite binary trees (that is, trees where every branch goes on forever), respectively.
The paper Co-induction Simply, by Leino and Moskal[@LEINO:Dafny:Coinduction], explains Dafny’s implementation and verification of coinductive types. We capture the key features from that paper in the following section but the reader is referred to that paper for more complete details and to supply bibliographic references that are omitted here.
5.14.3. Coinduction
Mathematical induction is a cornerstone of programming and program verification. It arises in data definitions (e.g., some algebraic data structures can be described using induction), it underlies program semantics (e.g., it explains how to reason about finite iteration and recursion), and it is used in proofs (e.g., supporting lemmas about data structures use inductive proofs). Whereas induction deals with finite things (data, behavior, etc.), its dual, coinduction, deals with possibly infinite things. Coinduction, too, is important in programming and program verification: it arises in data definitions (e.g., lazy data structures), semantics (e.g., concurrency), and proofs (e.g., showing refinement in a coinductive big-step semantics). It is thus desirable to have good support for both induction and coinduction in a system for constructing and reasoning about programs.
Co-datatypes and co-recursive functions make it possible to use lazily evaluated data structures (like in Haskell or Agda). Greatest predicates, defined by greatest fix-points, let programs state properties of such data structures (as can also be done in, for example, Coq). For the purpose of writing coinductive proofs in the language, we introduce greatest and least lemmas. A greatest lemma invokes the coinduction hypothesis much like an inductive proof invokes the induction hypothesis. Underneath the hood, our coinductive proofs are actually approached via induction: greatest and least lemmas provide a syntactic veneer around this approach.
The following example gives a taste of how the coinductive features in Dafny come together to give straightforward definitions of infinite matters.
// infinite streams
codatatype IStream<T> = ICons(head: T, tail: IStream<T>)
// pointwise product of streams
function Mult(a: IStream<int>, b: IStream<int>): IStream<int>
{ ICons(a.head * b.head, Mult(a.tail, b.tail)) }
// lexicographic order on streams
greatest predicate Below(a: IStream<int>, b: IStream<int>)
{ a.head <= b.head &&
((a.head == b.head) ==> Below(a.tail, b.tail))
}
// a stream is Below its Square
greatest lemma Theorem_BelowSquare(a: IStream<int>)
ensures Below(a, Mult(a, a))
{ assert a.head <= Mult(a, a).head;
if a.head == Mult(a, a).head {
Theorem_BelowSquare(a.tail);
}
}
// an incorrect property and a bogus proof attempt
greatest lemma NotATheorem_SquareBelow(a: IStream<int>)
ensures Below(Mult(a, a), a) // ERROR
{
NotATheorem_SquareBelow(a);
}
The example defines a type IStream
of infinite streams, with constructor ICons
and
destructors head
and tail
. Function Mult
performs pointwise
multiplication on infinite streams of integers, defined using a
co-recursive call (which is evaluated lazily). Greatest predicate Below
is
defined as a greatest fix-point, which intuitively means that the
co-predicate will take on the value true if the recursion goes on forever
without determining a different value. The greatest lemma states the theorem
Below(a, Mult(a, a))
. Its body gives the proof, where the recursive
invocation of the co-lemma corresponds to an invocation of the
coinduction hypothesis.
The proof of the theorem stated by the first co-lemma lends
itself to the following intuitive reading: To prove that a
is below
Mult(a, a)
, check that their heads are ordered and, if the heads are
equal, also prove that the tails are ordered. The second co-lemma states
a property that does not always hold; the verifier is not fooled by the
bogus proof attempt and instead reports the property as unproved.
We argue that these definitions in Dafny are simple enough to level the
playing field between induction (which is familiar) and coinduction
(which, despite being the dual of induction, is often perceived as eerily
mysterious). Moreover, the automation provided by our SMT-based verifier
reduces the tedium in writing coinductive proofs. For example, it
verifies Theorem_BelowSquare
from the program text given above—no
additional lemmas or tactics are needed. In fact, as a consequence of the
automatic-induction heuristic in Dafny, the verifier will
automatically verify Theorem_BelowSquare
even given an empty body.
Just like there are restrictions on when an inductive hypothesis can be invoked, there are restrictions on how a coinductive hypothesis can be used. These are, of course, taken into consideration by Dafny’s verifier. For example, as illustrated by the second greatest lemma above, invoking the coinductive hypothesis in an attempt to obtain the entire proof goal is futile. (We explain how this works in the section about greatest lemmas) Our initial experience with coinduction in Dafny shows it to provide an intuitive, low-overhead user experience that compares favorably to even the best of today’s interactive proof assistants for coinduction. In addition, the coinductive features and verification support in Dafny have other potential benefits. The features are a stepping stone for verifying functional lazy programs with Dafny. Coinductive features have also shown to be useful in defining language semantics, as needed to verify the correctness of a compiler, so this opens the possibility that such verifications can benefit from SMT automation.
5.14.3.1. Well-Founded Function/Method Definitions
The Dafny programming language supports functions and methods. A function
in Dafny is a mathematical function (i.e., it is well-defined,
deterministic, and pure), whereas a method is a body of statements that
can mutate the state of the program. A function is defined by its given
body, which is an expression. To ensure that function definitions
are mathematically consistent, Dafny insists that recursive calls be well-founded,
enforced as follows: Dafny computes the call graph of functions. The strongly connected
components within it are clusters of mutually recursive definitions; the clusters are arranged in
a DAG. This stratifies the functions so that a call from one cluster in the DAG to a
lower cluster is allowed arbitrarily. For an intra-cluster call, Dafny prescribes a proof
obligation that is taken through the program verifier’s reasoning engine. Semantically,
each function activation is labeled by a rank—a lexicographic tuple determined
by evaluating the function’s decreases
clause upon invocation of the function. The
proof obligation for an intra-cluster call is thus that the rank of the callee is strictly less
(in a language-defined well-founded relation) than the rank of the caller. Because
these well-founded checks correspond to proving termination of executable code, we
will often refer to them as “termination checks”. The same process applies to methods.
Lemmas in Dafny are commonly introduced by declaring a method, stating
the property of the lemma in the postcondition (keyword ensures
) of
the method, perhaps restricting the domain of the lemma by also giving a
precondition (keyword requires
), and using the lemma by invoking
the method. Lemmas are stated, used, and proved as methods, but
since they have no use at run time, such lemma methods are typically
declared as ghost, meaning that they are not compiled into code. The
keyword lemma
introduces such a method. Control flow statements
correspond to proof techniques—case splits are introduced with if
statements, recursion and loops are used for induction, and method calls
for structuring the proof. Additionally, the statement:
forall x | P(x) { Lemma(x); }
is used to invoke Lemma(x)
on all x
for which P(x)
holds. If
Lemma(x)
ensures Q(x)
, then the forall statement establishes
forall x :: P(x) ==> Q(x).
5.14.3.2. Defining Coinductive Datatypes
Each value of an inductive datatype is finite, in the sense that it can be constructed by a finite number of calls to datatype constructors. In contrast, values of a coinductive datatype, or co-datatype for short, can be infinite. For example, a co-datatype can be used to represent infinite trees.
Syntactically, the declaration of a co-datatype in Dafny looks like that of a datatype, giving prominence to the constructors (following Coq). The following example defines a co-datatype Stream of possibly infinite lists.
codatatype Stream<T> = SNil | SCons(head: T, tail: Stream)
function Up(n: int): Stream<int> { SCons(n, Up(n+1)) }
function FivesUp(n: int): Stream<int>
decreases 4 - (n - 1) % 5
{
if (n % 5 == 0) then
SCons(n, FivesUp(n+1))
else
FivesUp(n+1)
}
Stream
is a coinductive datatype whose values are possibly infinite
lists. Function Up
returns a stream consisting of all integers upwards
of n
and FivesUp
returns a stream consisting of all multiples of 5
upwards of n
. The self-call in Up
and the first self-call in FivesUp
sit in productive positions and are therefore classified as co-recursive
calls, exempt from termination checks. The second self-call in FivesUp
is
not in a productive position and is therefore subject to termination
checking; in particular, each recursive call must decrease the rank
defined by the decreases
clause.
Analogous to the common finite list datatype, Stream
declares two
constructors, SNil
and SCons
. Values can be destructed using match
expressions and statements. In addition, like for inductive datatypes,
each constructor C
automatically gives rise to a discriminator C?
and
each parameter of a constructor can be named in order to introduce a
corresponding destructor. For example, if xs
is the stream
SCons(x, ys)
, then xs.SCons?
and xs.head == x
hold. In contrast
to datatype declarations, there is no grounding check for
co-datatypes—since a codatatype admits infinite values, the type is
nevertheless inhabited.
5.14.3.3. Creating Values of Co-datatypes
To define values of co-datatypes, one could imagine a “co-function”
language feature: the body of a “co-function” could include possibly
never-ending self-calls that are interpreted by a greatest fix-point
semantics (akin to a CoFixpoint in Coq). Dafny uses a different design:
it offers only functions (not “co-functions”), but it classifies each
intra-cluster call as either recursive or co-recursive. Recursive calls
are subject to termination checks. Co-recursive calls may be
never-ending, which is what is needed to define infinite values of a
co-datatype. For example, function Up(n)
in the preceding example is defined as the
stream of numbers from n
upward: it returns a stream that starts with n
and continues as the co-recursive call Up(n + 1)
.
To ensure that co-recursive calls give rise to mathematically consistent definitions, they must occur only in productive positions. This says that it must be possible to determine each successive piece of a co-datatype value after a finite amount of work. This condition is satisfied if every co-recursive call is syntactically guarded by a constructor of a co-datatype, which is the criterion Dafny uses to classify intra-cluster calls as being either co-recursive or recursive. Calls that are classified as co-recursive are exempt from termination checks.
A consequence of the productivity checks and termination checks is that, even in the
absence of talking about least or greatest fix-points of self-calling functions, all functions
in Dafny are deterministic. Since there cannot be multiple fix-points,
the language allows one function to be involved in both recursive and co-recursive calls,
as we illustrate by the function FivesUp
.
5.14.3.4. Co-Equality
Equality between two values of a co-datatype is a built-in co-predicate.
It has the usual equality syntax s == t
, and the corresponding prefix
equality is written s ==#[k] t
. And similarly for s != t
and s !=#[k] t
.
5.14.3.5. Greatest predicates
Determining properties of co-datatype values may require an infinite
number of observations. To that end, Dafny provides greatest predicates
which are function declarations that use the greatest predicate
keyword phrase.
Self-calls to a greatest predicate need not terminate. Instead, the value
defined is the greatest fix-point of the given recurrence equations.
Continuing the preceding example, the following code defines a
greatest predicate that holds for exactly those streams whose payload consists
solely of positive integers. The greatest predicate definition implicitly also
gives rise to a corresponding prefix predicate, Pos#
. The syntax for
calling a prefix predicate sets apart the argument that specifies the
prefix length, as shown in the last line; for this figure, we took the
liberty of making up a coordinating syntax for the signature of the
automatically generated prefix predicate (which is not part of
Dafny syntax).
greatest predicate Pos[nat](s: Stream<int>)
{
match s
case SNil => true
case SCons(x, rest) => x > 0 && Pos(rest)
}
The following code is automatically generated by the Dafny compiler:
predicate Pos#[_k: nat](s: Stream<int>)
decreases _k
{ if _k == 0 then true else
match s
case SNil => true
case SCons(x, rest) => x > 0 && Pos#[_k-1](rest)
}
Some restrictions apply. To guarantee that the greatest fix-point always exists, the (implicit functor defining the) greatest predicate must be monotonic. This is enforced by a syntactic restriction on the form of the body of greatest predicates: after conversion to negation normal form (i.e., pushing negations down to the atoms), intra-cluster calls of greatest predicates must appear only in positive positions—that is, they must appear as atoms and must not be negated. Additionally, to guarantee soundness later on, we require that they appear in continous positions—that is, in negation normal form, when they appear under existential quantification, the quantification needs to be limited to a finite range7. Since the evaluation of a greatest predicate might not terminate, greatest predicates are always ghost. There is also a restriction on the call graph that a cluster containing a greatest predicate must contain only greatest predicates, no other kinds of functions.
extreme predicates and lemmas, one in which _k
has type nat
and one in
which it has type ORDINAL
(the default). The continuous restriction
applies only when _k
is nat
. Also, higher-order function support in Dafny is
rather modest and typical reasoning patterns do not involve them, so this
restriction is not as limiting as it would have been in, e.g., Coq.
A greatest predicate declaration of P
defines not just a greatest predicate, but
also a corresponding prefix predicate P#
. A prefix predicate is a
finite unrolling of a co-predicate. The prefix predicate is constructed
from the co-predicate by
-
adding a parameter
_k
of typenat
to denote the prefix length, -
adding the clause
decreases _k;
to the prefix predicate (the greatest predicate itself is not allowed to have a decreases clause), -
replacing in the body of the greatest predicate every intra-cluster call
Q(args)
to a greatest predicate by a callQ#[_k - 1](args)
to the corresponding prefix predicate, and then -
prepending the body with
if _k == 0 then true else
.
For example, for greatest predicate Pos
, the definition of the prefix
predicate Pos#
is as suggested above. Syntactically, the prefix-length
argument passed to a prefix predicate to indicate how many times to
unroll the definition is written in square brackets, as in Pos#[k](s)
.
In the Dafny grammar this is called a HashCall
. The definition of
Pos#
is available only at clusters strictly higher than that of Pos
;
that is, Pos
and Pos#
must not be in the same cluster. In other
words, the definition of Pos
cannot depend on Pos#
.
5.14.3.6. Coinductive Proofs
From what we have said so far, a program can make use of properties of
co-datatypes. For example, a method that declares Pos(s)
as a
precondition can rely on the stream s
containing only positive integers.
In this section, we consider how such properties are established in the
first place.
5.14.3.6.1. Properties of Prefix Predicates
Among other possible strategies for establishing coinductive properties
we take the time-honored approach of reducing coinduction to
induction. More precisely, Dafny passes to the SMT solver an
assumption D(P)
for every greatest predicate P
, where:
D(P) = forall x • P(x) <==> forall k • P#[k](x)
In other words, a greatest predicate is true iff its corresponding prefix predicate is true for all finite unrollings.
In Sec. 4 of the paper [Co-induction Simply] a soundness theorem of such
assumptions is given, provided the greatest predicates meet the continous
restrictions. An example proof of Pos(Up(n))
for every n > 0
is
shown here:
lemma UpPosLemma(n: int)
requires n > 0
ensures Pos(Up(n))
{
forall k | 0 <= k { UpPosLemmaK(k, n); }
}
lemma UpPosLemmaK(k: nat, n: int)
requires n > 0
ensures Pos#[k](Up(n))
decreases k
{
if k != 0 {
// this establishes Pos#[k-1](Up(n).tail)
UpPosLemmaK(k-1, n+1);
}
}
The lemma UpPosLemma
proves Pos(Up(n))
for every n > 0
. We first
show Pos#[k](Up(n ))
, for n > 0
and an arbitrary k
, and then use
the forall statement to show forall k • Pos#[k](Up(n))
. Finally, the axiom
D(Pos)
is used (automatically) to establish the greatest predicate.
5.14.3.6.2. Greatest lemmas
As we just showed, with help of the D
axiom we can now prove a
greatest predicate by inductively proving that the corresponding prefix
predicate holds for all prefix lengths k
. In this section, we introduce
greatest lemma declarations, which bring about two benefits. The first benefit
is that greatest lemmas are syntactic sugar and reduce the tedium of having to
write explicit quantifications over k
. The second benefit is that, in
simple cases, the bodies of greatest lemmas can be understood as coinductive
proofs directly. As an example consider the following greatest lemma.
greatest lemma UpPosLemma(n: int)
requires n > 0
ensures Pos(Up(n))
{
UpPosLemma(n+1);
}
This greatest lemma can be understood as follows: UpPosLemma
invokes itself
co-recursively to obtain the proof for Pos(Up(n).tail)
(since Up(n).tail
equals Up(n+1)
). The proof glue needed to then conclude Pos(Up(n))
is
provided automatically, thanks to the power of the SMT-based verifier.
5.14.3.6.3. Prefix Lemmas
To understand why the above UpPosLemma
greatest lemma code is a sound proof,
let us now describe the details of the desugaring of greatest lemmas. In
analogy to how a greatest predicate declaration defines both a greatest predicate and
a prefix predicate, a greatest lemma declaration defines both a greatest lemma and
prefix lemma. In the call graph, the cluster containing a greatest lemma must
contain only greatest lemmas and prefix lemmas, no other methods or function.
By decree, a greatest lemma and its corresponding prefix lemma are always
placed in the same cluster. Both greatest lemmas and prefix lemmas are always
ghost code.
The prefix lemma is constructed from the greatest lemma by
-
adding a parameter
_k
of typenat
to denote the prefix length, -
replacing in the greatest lemma’s postcondition the positive continuous occurrences of greatest predicates by corresponding prefix predicates, passing in
_k
as the prefix-length argument, -
prepending
_k
to the (typically implicit) decreases clause of the greatest lemma, -
replacing in the body of the greatest lemma every intra-cluster call
M(args)
to a greatest lemma by a callM#[_k - 1](args)
to the corresponding prefix lemma, and then -
making the body’s execution conditional on
_k != 0
.
Note that this rewriting removes all co-recursive calls of greatest lemmas,
replacing them with recursive calls to prefix lemmas. These recursive
calls are, as usual, checked to be terminating. We allow the pre-declared
identifier _k
to appear in the original body of the
greatest lemma.8
We can now think of the body of the greatest lemma as being replaced by a
forall call, for every k , to the prefix lemma. By construction,
this new body will establish the greatest lemma’s declared postcondition (on
account of the D
axiom, and remembering that only the positive
continuous occurrences of greatest predicates in the greatest lemma’s postcondition
are rewritten), so there is no reason for the program verifier to check
it.
The actual desugaring of our greatest lemma UpPosLemma
is in fact the
previous code for the UpPosLemma
lemma except that UpPosLemmaK
is
named UpPosLemma#
and modulo a minor syntactic difference in how the
k
argument is passed.
In the recursive call of the prefix lemma, there is a proof obligation
that the prefixlength argument _k - 1
is a natural number.
Conveniently, this follows from the fact that the body has been wrapped
in an if _k != 0
statement. This also means that the postcondition must
hold trivially when _k == 0
, or else a postcondition violation will be
reported. This is an appropriate design for our desugaring, because
greatest lemmas are expected to be used to establish greatest predicates, whose
corresponding prefix predicates hold trivially when _k = 0
. (To prove
other predicates, use an ordinary lemma, not a greatest lemma.)
It is interesting to compare the intuitive understanding of the coinductive proof in using a greatest lemma with the inductive proof in using a lemma. Whereas the inductive proof is performing proofs for deeper and deeper equalities, the greatest lemma can be understood as producing the infinite proof on demand.
5.14.3.7. Abstemious and voracious functions
Some functions on codatatypes are abstemious, meaning that they do not
need to unfold a datatype instance very far (perhaps just one destructor call)
to prove a relevant property. Knowing this is the case can aid the proofs of
properties about the function. The attribute {:abstemious}
can be applied to
a function definition to indicate this.
TODO: Say more about the effect of this attribute and when it should be applied (and likely, correct the paragraph above).
6. Member declarations
Members are the various kinds of methods, the various kinds of functions, mutable fields, and constant fields. These are usually associated with classes, but they also may be declared (with limitations) in traits, newtypes and datatypes (but not in subset types or type synonyms).
6.1. Field Declarations (grammar)
Examples:
class C {
var c: int // no initialization
ghost var 123: bv10 // name may be a sequence of digits
var d: nat, e: real // type is required
}
A field declaration is not permitted in a value type nor as a member of a module (despite there being an implicit unnamed class).
The field name is either an identifier (that is not allowed to start with a leading underscore) or some digits. Digits are used if you want to number your fields, e.g. “0”, “1”, etc. The digits do not denote numbers but sequences of digits, so 0, 00, 0_0 are all different.
A field x of some type T is declared as:
var x: T
A field declaration declares one or more fields of the enclosing class. Each field is a named part of the state of an object of that class. A field declaration is similar to but distinct from a variable declaration statement. Unlike for local variables and bound variables, the type is required and will not be inferred.
Unlike method and function declarations, a field declaration is not permitted as a member of a module, even though there is an implicit class. Fields can be declared in either an explicit class or a trait. A class that inherits from multiple traits will have all the fields declared in any of its parent traits.
Fields that are declared as ghost
can only be used in specifications,
not in code that will be compiled into executable code.
Fields may not be declared static.
6.2. Constant Field Declarations (grammar)
Examples:
const c: int
ghost const d := 5
class A {
const e: bool
static const f: int
}
A const
declaration declares a name bound to a value,
which value is fixed after initialization.
The declaration must either have a type or an initializing expression (or both). If the type is omitted, it is inferred from the initializing expression.
- A const declaration may include the
ghost
,static
, andopaque
modifiers, but no others. - A const declaration may appear within a module or within any declaration that may contain members (class, trait, datatype, newtype).
- If it is in a module, it is implicitly
static
, and may not also be declaredstatic
. - If the declaration has an initializing expression that is a ghost
expression, then the ghost-ness of the declaration is inferred; the
ghost
modifier may be omitted. - If the declaration includes the
opaque
modifier, then uses of the declared variable know its name and type but not its value. The value can be made known for reasoning purposes by using the reveal statement. - The initialization expression may refer to other constant fields that are in scope and declared either before or after this declaration, but circular references are not allowed.
6.3. Method Declarations (grammar)
Examples:
method m(i: int) requires i > 0 {}
method p() returns (r: int) { r := 0; }
method q() returns (r: int, s: int, t: nat) ensures r < s < t { r := 0; s := 1; t := 2; }
ghost method g() {}
class A {
method f() {}
constructor Init() {}
static method g<T>(t: T) {}
}
lemma L(p: bool) ensures p || !p {}
twostate lemma TL(p: bool) ensures p || !p {}
least lemma LL[nat](p: bool) ensures p || !p {}
greatest lemma GL(p: bool) ensures p || !p {}
abstract module M { method m(i: int) }
module N refines M { method m ... {} }
Method declarations include a variety of related types of methods:
- method
- constructor
- lemma
- twostate lemma
- least lemma
- greatest lemma
A method signature specifies the method generic parameters,
input parameters and return parameters.
The formal parameters are not allowed to have ghost
specified
if ghost
was already specified for the method.
Within the body of a method, formal (input) parameters are immutable, that is,
they may not be assigned to, though their array elements or fields may be
assigned, if otherwise permitted.
The out-parameters are mutable and must be assigned in the body of the method.
An ellipsis
is used when a method or function is being redeclared
in a module that refines another module. (cf. Section 10)
In that case the signature is
copied from the module that is being refined. This works because
Dafny does not support method or function overloading, so the
name of the class method uniquely identifies it without the
signature.
See Section 7.2 for a description of the method specification.
Here is an example of a method declaration.
method {:att1}{:att2} M<T1, T2>(a: A, b: B, c: C)
returns (x: X, y: Y, z: Z)
requires Pre
modifies Frame
ensures Post
decreases Rank
{
Body
}
where :att1
and :att2
are attributes of the method,
T1
and T2
are type parameters of the method (if generic),
a, b, c
are the method’s in-parameters, x, y, z
are the
method’s out-parameters, Pre
is a boolean expression denoting the
method’s precondition, Frame
denotes a set of objects whose fields may
be updated by the method, Post
is a boolean expression denoting the
method’s postcondition, Rank
is the method’s variant function, and
Body
is a list of statements that implements the method. Frame
can be a list
of expressions, each of which is a set of objects or a single object, the
latter standing for the singleton set consisting of that one object. The
method’s frame is the union of these sets, plus the set of objects
allocated by the method body. For example, if c
and d
are parameters
of a class type C
, then
modifies {c, d}
modifies {c} + {d}
modifies c, {d}
modifies c, d
all mean the same thing.
If the method is an extreme lemma ( a least
or greatest
lemma), then the
method signature may also state the type of the k parameter as either nat
or ORDINAL
.
These are described
in Section 12.5.3 and subsequent sections.
6.3.1. Ordinary methods
A method can be declared as ghost by preceding the declaration with the
keyword ghost
and as static by preceding the declaration with the keyword static
.
The default is non-static (i.e., instance) for methods declared in a type and non-ghost.
An instance method has an implicit receiver parameter, this
.
A static method M in a class C can be invoked by C.M(…)
.
An ordinary method is declared with the method
keyword;
the section about constructors explains methods that instead use the
constructor
keyword; the section about lemmas discusses methods that are
declared with the lemma
keyword. Methods declared with the
least lemma
or greatest lemma
keyword phrases
are discussed later in the context of extreme
predicates (see the section about greatest lemmas).
A method without a body is abstract. A method is allowed to be abstract under the following circumstances:
- It contains an
{:axiom}
attribute - It contains an
{:extern}
attribute (in this case, to be runnable, the method must have a body in non-Dafny compiled code in the target language.) - It is a declaration in an abstract module. Note that when there is no body, Dafny assumes that the ensures clauses are true without proof.
6.3.2. Constructors
To write structured object-oriented programs, one often relies on
objects being constructed only in certain ways. For this purpose, Dafny
provides constructor (method)s.
A constructor is declared with the keyword
constructor
instead of method
; constructors are permitted only in classes.
A constructor is allowed to be declared as ghost
, in which case it
can only be used in ghost contexts.
A constructor can only be called at the time an object is allocated (see
object-creation examples below). Moreover, when a class contains a
constructor, every call to new
for a class must be accompanied
by a call to one of its constructors. A class may
declare no constructors or one or more constructors.
In general, a constructor is responsible for initializating the instance fields of its class. However, any field that is given an initializer in its declaration may not be reassigned in the body of the constructor.
6.3.2.1. Classes with no explicit constructors
For a class that declares no constructors, an instance of the class is created with
c := new C;
This allocates an object and initializes its fields to values of their
respective types (and initializes each const
field with a RHS to its specified
value). The RHS of a const
field may depend on other const
or var
fields,
but circular dependencies are not allowed.
This simple form of new
is allowed only if the class declares no constructors,
which is not possible to determine in every scope.
It is easy to determine whether or not a class declares any constructors if the
class is declared in the same module that performs the new
. If the class is
declared in a different module and that module exports a constructor, then it is
also clear that the class has a constructor (and thus this simple form of new
cannot be used). (Note that an export set that reveals
a class C
also exports
the anonymous constructor of C
, if any.)
But if the module that declares C
does not export any constructors
for C
, then callers outside the module do not know whether or not C
has a
constructor. Therefore, this simple form of new
is allowed only for classes that
are declared in the same module as the use of new
.
The simple new C
is allowed in ghost contexts. Also, unlike the forms of new
that call a constructor or initialization method, it can be used in a simultaneous
assignment; for example
c, d, e := new C, new C, 15;
is legal.
As a shorthand for writing
c := new C;
c.Init(args);
where Init
is an initialization method (see the top of the section about class types),
one can write
c := new C.Init(args);
but it is more typical in such a case to declare a constructor for the class.
(The syntactic support for initialization methods is provided for historical reasons. It may be deprecated in some future version of Dafny. In most cases, a constructor is to be preferred.)
6.3.2.2. Classes with one or more constructors
Like other class members, constructors have names. And like other members,
their names must be distinct, even if their signatures are different.
Being able to name constructors promotes names like InitFromList
or
InitFromSet
(or just FromList
and FromSet
).
Unlike other members, one constructor is allowed to be anonymous;
in other words, an anonymous constructor is a constructor whose name is
essentially the empty string. For example:
class Item {
constructor I(xy: int) // ...
constructor (x: int, y: int)
// ...
}
The named constructor is invoked as
i := new Item.I(42);
The anonymous constructor is invoked as
m := new Item(45, 29);
dropping the “.
”.
6.3.2.3. Two-phase constructors
The body of a constructor contains two sections,
an initialization phase and a post-initialization phase, separated by a new;
statement.
If there is no new;
statement, the entire body is the initialization phase.
The initialization phase is intended to initialize field variables
that were not given values in their declaration; it may not reassign
to fields that do have initializers in their declarations.
In this phase, uses of the object reference this
are restricted;
a program may use this
- as the receiver on the LHS,
- as the entire RHS of an assignment to a field of
this
, - and as a member of a set on the RHS that is being assigned to a field of
this
.
A const
field with a RHS is not allowed to be assigned anywhere else.
A const
field without a RHS may be assigned only in constructors, and more precisely
only in the initialization phase of constructors. During this phase, a const
field
may be assigned more than once; whatever value the const
field has at the end of the
initialization phase is the value it will have forever thereafter.
For a constructor declared as ghost
, the initialization phase is allowed to assign
both ghost and non-ghost fields. For such an object, values of non-ghost fields at
the end of the initialization phase are in effect no longer changeable.
There are no restrictions on expressions or statements in the post-initialization phase.
6.3.3. Lemmas
Sometimes there are steps of logic required to prove a program correct,
but they are too complex for Dafny to discover and use on its own. When
this happens, we can often give Dafny assistance by providing a lemma.
This is done by declaring a method with the lemma
keyword.
Lemmas are implicitly ghost methods and the ghost
keyword cannot
be applied to them.
Syntactically, lemmas can be placed where ghost methods can be placed, but they serve
a significantly different function. First of all, a lemma is forbidden to have
modifies
clause: it may not change anything about even the ghost state; ghost methods
may have modifies
clauses and may change ghost (but not non-ghost) state.
Furthermore, a lemma is not allowed to allocate any new objects.
And a lemma may be used in the program text in places where ghost methods may not,
such as within expressions (cf. Section 21.1).
Lemmas may, but typically do not, have out-parameters.
In summary, a lemma states a logical fact, summarizing an inference that the verifier cannot do on its own. Explicitly “calling” a lemma in the program text tells the verifier to use that fact at that location with the actual arguments substituted for the formal parameters. The lemma is proved separately for all cases of its formal parameters that satisfy the preconditions of the lemma.
For an example, see the FibProperty
lemma in
Section 12.5.2.
See the Dafny Lemmas tutorial for more examples and hints for using lemmas.
6.3.4. Two-state lemmas and functions
The heap is an implicit parameter to every function, though a function is only allowed
to read those parts of the mutable heap that it admits to in its reads
clause.
Sometimes, it is useful for a function to take two heap parameters, for example, so
the function can return the difference between the value of a field in the two heaps.
Such a two-state function is declared by twostate function
(or twostate predicate
,
which is the same as a twostate function
that returns a bool
). A two-state function
is always ghost. It is appropriate to think of these two implicit heap parameters as
representing a “current” heap and an “old” heap.
For example, the predicate
class Cell { var data: int constructor(i: int) { data := i; } }
twostate predicate Increasing(c: Cell)
reads c
{
old(c.data) <= c.data
}
returns true
if the value of c.data
has not been reduced from the old state to the
current. Dereferences in the current heap are written as usual (e.g., c.data
) and
must, as usual, be accounted for in the function’s reads
clause. Dereferences in the
old heap are enclosed by old
(e.g., old(c.data)
), just like when one dereferences
a method’s initial heap. The function is allowed to read anything in the old heap;
the reads
clause only declares dependencies on locations in the current heap.
Consequently, the frame axiom for a two-state function is sensitive to any change
in the old-heap parameter; in other words, the frame axiom says nothing about two
invocations of the two-state function with different old-heap parameters.
At a call site, the two-state function’s current-heap parameter is always passed in
as the caller’s current heap. The two-state function’s old-heap parameter is by
default passed in as the caller’s old heap (that is, the initial heap if the caller
is a method and the old heap if the caller is a two-state function). While there is
never a choice in which heap gets passed as the current heap, the caller can use
any preceding heap as the argument to the two-state function’s old-heap parameter.
This is done by labeling a state in the caller and passing in the label, just like
this is done with the built-in old
function.
For example, the following assertions all hold:
method Caller(c: Cell)
modifies c
{
c.data := c.data + 10;
label L:
assert Increasing(c);
c.data := c.data - 2;
assert Increasing(c);
assert !Increasing@L(c);
}
The first call to Increasing
uses Caller
’s initial state as the old-heap parameter,
and so does the second call. The third call instead uses as the old-heap parameter
the heap at label L
, which is why the third call returns false
.
As shown in the example, an explicitly given old-heap parameter is given after
an @
-sign (which follows the name of the function and any explicitly given type
parameters) and before the open parenthesis (after which the ordinary parameters are
given).
A two-state function is allowed to be called only from a two-state context, which
means a method, a two-state lemma (see below), or another two-state function.
Just like a label used with an old
expression, any label used in a call to a
two-state function must denote a program point that dominates the call. This means
that any control leading to the call must necessarily have passed through the labeled
program point.
Any parameter (including the receiver parameter, if any) passed to a two-state function
must have been allocated already in the old state. For example, the second call to
Diff
in method M
is illegal, since d
was not allocated on entry to M
:
twostate function Diff(c: Cell, d: Cell): int
reads d
{
d.data - old(c.data)
}
method M(c: Cell) {
var d := new Cell(10);
label L:
ghost var x := Diff@L(c, d);
ghost var y := Diff(c, d); // error: d is not allocated in old state
}
A two-state function may declare that it only assumes a parameter to be allocated
in the current heap. This is done by preceding the parameter with the new
modifier,
as illustrated in the following example, where the first call to DiffAgain
is legal:
twostate function DiffAgain(c: Cell, new d: Cell): int
reads d
{
d.data - old(c.data)
}
method P(c: Cell) {
var d := new Cell(10);
ghost var x := DiffAgain(c, d);
ghost var y := DiffAgain(d, c); // error: d is not allocated in old state
}
A two-state lemma works in an analogous way. It is a lemma with both a current-heap
parameter and an old-heap parameter, it can use old
expressions in its
specification (including in the precondition) and body, its parameters may
use the new
modifier, and the old-heap parameter is by default passed in as
the caller’s old heap, which can be changed by using an @
-parameter.
Here is an example of something useful that can be done with a two-state lemma:
function SeqSum(s: seq<Cell>): int
reads s
{
if s == [] then 0 else s[0].data + SeqSum(s[1..])
}
twostate lemma IncSumDiff(s: seq<Cell>)
requires forall c :: c in s ==> Increasing(c)
ensures old(SeqSum(s)) <= SeqSum(s)
{
if s == [] {
} else {
calc {
old(SeqSum(s));
== // def. SeqSum
old(s[0].data + SeqSum(s[1..]));
== // distribute old
old(s[0].data) + old(SeqSum(s[1..]));
<= { assert Increasing(s[0]); }
s[0].data + old(SeqSum(s[1..]));
<= { IncSumDiff(s[1..]); }
s[0].data + SeqSum(s[1..]);
== // def. SeqSum
SeqSum(s);
}
}
}
A two-state function can be used as a first-class function value, where the receiver
(if any), type parameters (if any), and old-heap parameter are determined at the
time the first-class value is mentioned. While the receiver and type parameters can
be explicitly instantiated in such a use (for example, p.F<int>
for a two-state
instance function F
that takes one type parameter), there is currently no syntactic
support for giving the old-heap parameter explicitly. A caller can work
around this restriction by using (fancy-word alert!) eta-expansion, meaning
wrapping a lambda expression around the call, as in x => p.F<int>@L(x)
.
The following example illustrates using such an eta-expansion:
class P {
twostate function F<X>(x: X): X
}
method EtaExample(p: P) returns (ghost f: int -> int) {
label L:
f := x => p.F<int>@L(x);
}
6.4. Function Declarations (grammar)
6.4.1. Functions
Examples:
function f(i: int): real { i as real }
function g(): (int, int) { (2,3) }
function h(i: int, k: int): int requires i >= 0 { if i == 0 then 0 else 1 }
Functions may be declared as ghost. If so, all the formal parameters and return values are ghost; if it is not a ghost function, then individual parameters may be declared ghost as desired.
See Section 7.3 for a description of the function specification.
A Dafny function is a pure mathematical function. It is allowed to
read memory that was specified in its reads
expression but is not
allowed to have any side effects.
Here is an example function declaration:
function {:att1}{:att2} F<T1, T2>(a: A, b: B, c: C): T
requires Pre
reads Frame
ensures Post
decreases Rank
{
Body
}
where :att1
and :att2
are attributes of the function, if any, T1
and T2
are type parameters of the function (if generic), a, b, c
are
the function’s parameters, T
is the type of the function’s result,
Pre
is a boolean expression denoting the function’s precondition,
Frame
denotes a set of objects whose fields the function body may
depend on, Post
is a boolean expression denoting the function’s
postcondition, Rank
is the function’s variant function, and Body
is
an expression that defines the function’s return value. The precondition
allows a function to be partial, that is, the precondition says when the
function is defined (and Dafny will verify that every use of the function
meets the precondition).
The postcondition is usually not needed, since the body of the function gives the full definition. However, the postcondition can be a convenient place to declare properties of the function that may require an inductive proof to establish, such as when the function is recursive. For example:
function Factorial(n: int): int
requires 0 <= n
ensures 1 <= Factorial(n)
{
if n == 0 then 1 else Factorial(n-1) * n
}
says that the result of Factorial is always positive, which Dafny verifies inductively from the function body.
Within a postcondition, the result of the function is designated by
a call of the function, such as Factorial(n)
in the example above.
Alternatively, a name for the function result can be given in the signature,
as in the following rewrite of the example above.
function Factorial(n: int): (f: int)
requires 0 <= n
ensures 1 <= f
{
if n == 0 then 1 else Factorial(n-1) * n
}
Pre v4.0, a function is ghost
by default, and cannot be called from non-ghost
code. To make it non-ghost, replace the keyword function
with the two
keywords “function method
”. From v4.0 on, a function is non-ghost by
default. To make it ghost, replace the keyword function
with the two keywords “ghost function
”.
(See the –function-syntax option for a description
of the migration path for this change in behavior.}
Functions (including predicates, function-by-methods, two-state functions, and extreme predicates) may be
declared opaque
. In that case, only the signature and specification of the method
is known at its points of use, not its body. The body can be revealed for reasoning
purposes using the reveal statment.
Like methods, functions can be either instance (which they are by default when declared within a type) or
static (when the function declaration contains the keyword static
or is declared in a module).
An instance function, but not a static function, has an implicit receiver parameter, this
.
A static function F
in a class C
can be invoked
by C.F(…)
. This provides a convenient way to declare a number of helper
functions in a separate class.
As for methods, a ...
is used when declaring
a function in a module refinement (cf. Section 10).
For example, if module M0
declares
function F
, a module M1
can be declared to refine M0
and
M1
can then refine F
. The refinement function, M1.F
can have
a ...
which means to copy the signature from
M0.F
. A refinement function can furnish a body for a function
(if M0.F
does not provide one). It can also add ensures
clauses.
If a function definition does not have a body, the program that contains it may still be verified.
The function itself has nothing to verify.
However, any calls of a body-less function are treated as unverified assumptions by the caller,
asserting the preconditions and assuming the postconditions.
Because body-less functions are unverified assumptions, Dafny will not compile them and will complain if called by dafny translate
, dafny build
or even dafny run
6.4.2. Predicates
A function that returns a bool
result is called a predicate. As an
alternative syntax, a predicate can be declared by replacing the function
keyword with the predicate
keyword and possibly omitting a declaration of the
return type (if it is not named).
6.4.3. Function-by-method
A function with a by method
clause declares a function-by-method.
A function-by-method gives a way to implement a
(deterministic, side-effect free) function by a method (whose body may be
nondeterministic and may allocate objects that it modifies). This can
be useful if the best implementation uses nondeterminism (for example,
because it uses :|
in a nondeterministic way) in a way that does not
affect the result, or if the implementation temporarily makes use of some
mutable data structures, or if the implementation is done with a loop.
For example, here is the standard definition of the Fibonacci function
but with an efficient implementation that uses a loop:
function Fib(n: nat): nat {
if n < 2 then n else Fib(n - 2) + Fib(n - 1)
} by method {
var x, y := 0, 1;
for i := 0 to n
invariant x == Fib(i) && y == Fib(i + 1)
{
x, y := y, x + y;
}
return x;
}
The by method
clause is allowed only for non-ghost function
or predicate
declarations (without twostate
, least
, and greatest
, but
possibly with static
); it inherits the in-parameters, attributes, and requires
and decreases
clauses of the function. The method also gets one out-parameter, corresponding
to the function’s result value (and the name of it, if present). Finally,
the method gets an empty modifies
clause and a postcondition
ensures r == F(args)
, where r
is the name of the out-parameter and
F(args)
is the function with its arguments. In other words, the method
body must compute and return exactly what the function says, and must
do so without modifying any previously existing heap state.
The function body of a function-by-method is allowed to be ghost, but the method body must be compilable. In non-ghost contexts, the compiler turns a call of the function-by-method into a call that leads to the method body.
Note, the method body of a function-by-method may contain print
statements.
This means that the run-time evaluation of an expression may have print effects.
If --track-print-effects
is enabled, this use of print in a function context
will be disallowed.
6.4.4. Function Transparency
A function is said to be transparent in a location if the body of the function is visible at that point. A function is said to be opaque at a location if it is not transparent. However the specification of a function is always available.
A function is usually transparent up to some unrolling level (up to 1, or maybe 2 or 3). If its arguments are all literals it is transparent all the way.
But the transparency of a function is affected by
whether the function was declared with an [opaque
modifier]((#sec-opaque),
the (reveal statement),
and whether it was reveal
ed in an export set.
- Inside the module where the function is declared:
- if there is no
opaque
modifier, the function is transparent - if there is an
opaque
modifier, then the function is opaque, except if the function is mentioned in areveal
statement, then it is transparent between thatreveal
statement and the end of the block containing thereveal
statement.
- if there is no
- Outside the module where the function is declared, the function is
visible only if it was listed in the export set by which the contents
of its module was imported. In that case, if the function was exported
with
reveals
, the rules are the same within the importing module as when the function is used inside its declaring module. If the function is exported only withprovides
it is always opaque and is not permitted to be used in a reveal statement.
6.4.5. Extreme (Least or Greatest) Predicates and Lemmas
See Section 12.5.3 for descriptions of extreme predicates and lemmas.
6.4.6. older
parameters in predicates
A parameter of any predicate (more precisely, of any
boolean-returning, non-extreme function) can be marked as
older
. This specifies that the truth of the predicate implies that
the allocatedness of the parameter follows from the allocatedness of
the non-older
parameters.
To understand what this means and why this attribute is useful,
consider the following example, which specifies reachability between
nodes in a directed graph. A Node
is declared to have any number of
children:
class Node {
var children: seq<Node>
}
There are several ways one could specify reachability between
nodes. One way (which is used in Test/dafny1/SchorrWaite.dfy
in the
Dafny test suite) is to define a type Path
, representing lists of
Node
s, and to define a predicate that checks if a given list of
Node
s is indeed a path between two given nodes:
datatype Path = Empty | Extend(Path, Node)
predicate ReachableVia(source: Node, p: Path, sink: Node, S: set<Node>)
reads S
decreases p
{
match p
case Empty =>
source == sink
case Extend(prefix, n) =>
n in S && sink in n.children && ReachableVia(source, prefix, n, S)
}
In a nutshell, the definition of ReachableVia
says
- An empty path lets
source
reachsink
just whensource
andsink
are the same node. - A path
Extend(prefix, n)
letssource
reachsink
just when the pathprefix
letssource
reachn
andsink
is one of the children nodes ofn
.
To be admissible by Dafny, the recursive predicate must be shown to
terminate. Termination is assured by the specification decreases p
,
since every such datatype value has a finite structure and every
recursive call passes in a path that is structurally included in the
previous. Predicate ReachableVia
must also declare (an upper bound
on) which heap objects it depends on. For this purpose, the
predicate takes an additional parameter S
, which is used to limit
the set of intermediate nodes in the path. More precisely, predicate
ReachableVia(source, p, sink, S)
returns true
if and only if p
is a list of nodes in S
and source
can reach sink
via p
.
Using predicate ReachableVia
, we can now define reachability in S
:
predicate Reachable(source: Node, sink: Node, S: set<Node>)
reads S
{
exists p :: ReachableVia(source, p, sink, S)
}
This looks like a good definition of reachability, but Dafny won’t admit it. The reason is twofold:
-
Quantifiers and comprehensions are allowed to range only over allocated state. Ater all, Dafny is a type-safe language where every object reference is valid (that is, a pointer to allocated storage of the right type)—it should not be possible, not even through a bound variable in a quantifier or comprehension, for a program to obtain an object reference that isn’t valid.
-
This property is ensured by disallowing open-ended quantifiers. More precisely, the object references that a quantifier may range over must be shown to be confined to object references that were allocated before some of the non-
older
parameters passed to the predicate. Quantifiers that are not open-ended are called close-ended. Note that close-ended refers only to the object references that the quantification or comprehension ranges over—it does not say anything about values of other types, like integers.
Often, it is easy to show that a quantifier is close-ended. In fact, if the type of a bound variable does not contain any object references, then the quantifier is trivially close-ended. For example,
forall x: int :: x <= Square(x)
is trivially close-ended.
Another innocent-looking quantifier occurs in the following example:
predicate IsCommutative<X>(r: (X, X) -> bool)
{
forall x, y :: r(x, y) == r(y, x) // error: open-ended quantifier
}
Since nothing is known about type X
, this quantifier might be
open-ended. For example, if X
were passed in as a class type, then
the quantifier would be open-ended. One way to fix this predicate is
to restrict it to non-heap based types, which is indicated with the
(!new)
type characteristic (see Section 5.3.1.4):
ghost predicate IsCommutative<X(!new)>(r: (X, X) -> bool) // X is restricted to non-heap types
{
forall x, y :: r(x, y) == r(y, x) // allowed
}
Another way to make IsCommutative
close-ended is to constrain the values
of the bound variables x
and y
. This can be done by adding a parameter
to the predicate and limiting the quantified values to ones in the given set:
predicate IsCommutativeInS<X>(r: (X, X) -> bool, S: set<X>)
{
forall x, y :: x in S && y in S ==> r(x, y) == r(y, x) // close-ended
}
Through a simple syntactic analysis, Dafny detects the antecedents
x in S
and y in S
, and since S
is a parameter and thus can only be
passed in as something that the caller has already allocated, the
quantifier in IsCommutativeInS
is determined to be close-ended.
Note, the x in S
trick does not work for the motivating example,
Reachable
. If you try to write
predicate Reachable(source: Node, sink: Node, S: set<Node>)
reads S
{
exists p :: p in S && ReachableVia(source, p, sink, S) // type error: p
}
you will get a type error, because p in S
does not make sense if p
has type Path
. We need some other way to justify that the
quantification in Reachable
is close-ended.
Dafny offers a way to extend the x in S
trick to more situations.
This is where the older
modifier comes in. Before we apply older
in the Reachable
example, let’s first look at what older
does in a
less cluttered example.
Suppose we rewrite IsCommutativeInS
using a programmer-defined predicate In
:
predicate In<X>(x: X, S: set<X>) {
x in S
}
predicate IsCommutativeInS<X>(r: (X, X) -> bool, S: set<X>)
{
forall x, y :: In(x, S) && In(y, S) ==> r(x, y) == r(y, x) // error: open-ended?
}
The simple syntactic analysis that looks for x in S
finds nothing
here, because the in
operator is relegated to the body of predicate
In
. To inform the analysis that In
is a predicate that, in effect,
is like in
, you can mark parameter x
with older
:
predicate In<X>(older x: X, S: set<X>) {
x in S
}
This causes the simple syntactic analysis to accept the quantifier in
IsCommutativeInS
. Adding older
also imposes a semantic check on
the body of predicate In
, enforced by the verifier. The semantic
check is that all the object references in the value x
are older (or
equally old as) the object references that are part of the other
parameters, in the event that the predicate returns true. That is,
older
is designed to help the caller only if the predicate returns
true
, and the semantic check amounts to nothing if the predicate
returns false
.
Finally, let’s get back to the motivating example. To allow the quantifier
in Reachable
, mark parameter p
of ReachableVia
with older
:
class Node {
var children: seq<Node>
}
datatype Path = Empty | Extend(Path, Node)
ghost predicate Reachable(source: Node, sink: Node, S: set<Node>)
reads S
{
exists p :: ReachableVia(source, p, sink, S) // allowed because of 'older p' on ReachableVia
}
ghost predicate ReachableVia(source: Node, older p: Path, sink: Node, S: set<Node>)
reads S
decreases p
{
match p
case Empty =>
source == sink
case Extend(prefix, n) =>
n in S && sink in n.children && ReachableVia(source, prefix, n, S)
}
This example is more involved than the simpler In
example
above. Because of the older
modifier on the parameter, the quantifier in
Reachable
is allowed. For intuition, you can think of the effect of
older p
as adding an antecedent p in {source} + {sink} + S
(but, as we have seen, this is not type correct). The semantic check
imposed on the body of ReachableVia
makes sure that, if the
predicate returns true
, then every object reference in p
is as old
as some object reference in another parameter to the predicate.
6.5. Nameonly Formal Parameters and Default-Value Expressions
A formal parameter of a method, constructor in a class, iterator, function, or datatype constructor can be declared with an expression denoting a default value. This makes the parameter optional, as opposed to required.
For example,
function f(x: int, y: int := 10): int
may be called as either
const i := f(1, 2)
const j := f(1)
where f(1)
is equivalent to f(1, 10)
in this case.
The above function may also be called as
var k := f(y := 10, x := 2);
using names; actual arguments with names may be given in any order, though they must be after actual arguments without names.
Formal parameters may also be declared nameonly
, in which case a call site
must always explicitly name the formal when providing its actual argument.
For example, a function ff
declared as
function ff(x: int, nameonly y: int): int
must be called either by listing the value for x and then y with a name,
as in ff(0, y := 4)
or by giving both actuals by name (in any order).
A nameonly
formal may also have a default value and thus be optional.
Any formals after a nameonly
formal must either be nameonly
themselves or have default values.
The formals of datatype constructors are not required to have names. A nameless formal may not have a default value, nor may it follow a formal that has a default value.
The default-value expression for a parameter is allowed to mention the
other parameters, including this
(for instance methods and instance
functions), but not the implicit _k
parameter in least and greatest
predicates and lemmas. The default value of a parameter may mention
both preceding and subsequent parameters, but there may not be any
dependent cycle between the parameters and their default-value
expressions.
The well-formedness of default-value expressions is checked independent
of the precondition of the enclosing declaration. For a function, the
parameter default-value expressions may only read what the function’s
reads
clause allows. For a datatype constructor, parameter default-value
expressions may not read anything. A default-value expression may not be
involved in any recursive or mutually recursive calls with the enclosing
declaration.
7. Specifications
Specifications describe logical properties of Dafny methods, functions, lambdas, iterators and loops. They specify preconditions, postconditions, invariants, what memory locations may be read or modified, and termination information by means of specification clauses. For each kind of specification, zero or more specification clauses (of the type accepted for that type of specification) may be given, in any order.
We document specifications at these levels:
- At the lowest level are the various kinds of specification clauses,
e.g., a
RequiresClause
. - Next are the specifications for entities that need them,
e.g., a
MethodSpec
, which typically consist of a sequence of specification clauses. - At the top level are the entity declarations that include
the specifications, e.g.,
MethodDecl
.
This section documents the first two of these in a bottom-up manner. We first document the clauses and then the specifications that use them.
Specification clauses typically appear in a sequence. They all begin with a keyword and do not end with semicolons.
7.1. Specification Clauses
Within expressions in specification clauses, you can use specification expressions along with any other expressions you need.
7.1.1. Requires Clause (grammar)
Examples:
method m(i: int)
requires true
requires i > 0
requires L: 0 < i < 10
The requires
clauses specify preconditions for methods,
functions, lambda expressions and iterators. Dafny checks
that the preconditions are met at all call sites. The
callee may then assume the preconditions hold on entry.
If no requires
clause is specified, then a default implicit
clause requires true
is used.
If more than one requires
clause is given, then the
precondition is the conjunction of all of the expressions
from all of the requires
clauses, with a collected list
of all the given Attributes. The order of conjunctions
(and hence the order of requires
clauses with respect to each other)
can be important: earlier conjuncts can set conditions that
establish that later conjuncts are well-defined.
The attributes recognized for requires clauses are discussed in Section 11.3.
A requires clause can have custom error and success messages.
7.1.2. Ensures Clause (grammar)
Examples:
method m(i: int) returns (r: int)
ensures r > 0
An ensures
clause specifies the post condition for a
method, function or iterator.
If no ensures
clause is specified, then a default implicit
clause ensures true
is used.
If more than one ensures
clause is given, then the
postcondition is the conjunction of all of the expressions
from all of the ensures
clauses, with a
collected list of all the given Attributes.
The order of conjunctions
(and hence the order of ensures
clauses with respect to each other)
can be important: earlier conjuncts can set conditions that
establish that later conjuncts are well-defined.
The attributes recognized for ensures clauses are discussed in Section 11.3.
An ensures clause can have custom error and success messages.
7.1.3. Decreases Clause (grammar)
Examples:
method m(i: int, j: int) returns (r: int)
decreases i, j
method n(i: int) returns (r: int)
decreases *
Decreases clauses are used to prove termination in the
presence of recursion. If more than one decreases
clause is given
it is as if a single decreases
clause had been given with the
collected list of arguments and a collected list of Attributes. That is,
decreases A, B
decreases C, D
is equivalent to
decreases A, B, C, D
Note that changing the order of multiple decreases
clauses will change
the order of the expressions within the equivalent single decreases
clause, and will therefore have different semantics.
Loops and compiled methods (but not functions and not ghost methods,
including lemmas) can be specified to be possibly non-terminating.
This is done by declaring the method or loop with decreases *
, which
causes the proof of termination to be skipped. If a *
is present
in a decreases
clause, no other expressions are allowed in the
decreases
clause. A method that contains a possibly non-terminating
loop or a call to a possibly non-terminating method must itself be
declared as possibly non-terminating.
Termination metrics in Dafny, which are declared by decreases
clauses,
are lexicographic tuples of expressions. At each recursive (or mutually
recursive) call to a function or method, Dafny checks that the effective
decreases
clause of the callee is strictly smaller than the effective
decreases
clause of the caller.
What does “strictly smaller” mean? Dafny provides a built-in
well-founded order for every type and, in some cases, between types. For
example, the Boolean false
is strictly smaller than true
, the
integer 78
is strictly smaller than 102
, the set {2,5}
is strictly
smaller than (because it is a proper subset of) the set {2,3,5}
, and for s
of type seq<Color>
where
Color
is some inductive datatype, the color s[0]
is strictly less than
s
(provided s
is nonempty).
What does “effective decreases clause” mean? Dafny always appends a
“top” element to the lexicographic tuple given by the user. This top
element cannot be syntactically denoted in a Dafny program and it never
occurs as a run-time value either. Rather, it is a fictitious value,
which here we will denote $\top$, such that each value that can ever occur
in a Dafny program is strictly less than $\top$. Dafny sometimes also
prepends expressions to the lexicographic tuple given by the user. The
effective decreases clause is any such prefix, followed by the
user-provided decreases clause, followed by $\top$. We said “user-provided
decreases clause”, but if the user completely omits a decreases
clause,
then Dafny will usually make a guess at one, in which case the effective
decreases clause is any prefix followed by the guess followed by $\top$.
Here is a simple but interesting example: the Fibonacci function.
function Fib(n: nat) : nat
{
if n < 2 then n else Fib(n-2) + Fib(n-1)
}
In this example, Dafny supplies a decreases n
clause.
Let’s take a look at the kind of example where a mysterious-looking decreases clause like “Rank, 0” is useful.
Consider two mutually recursive methods, A
and B
:
method A(x: nat)
{
B(x);
}
method B(x: nat)
{
if x != 0 { A(x-1); }
}
To prove termination of A
and B
, Dafny needs to have effective
decreases clauses for A and B such that:
-
the measure for the callee
B(x)
is strictly smaller than the measure for the callerA(x)
, and -
the measure for the callee
A(x-1)
is strictly smaller than the measure for the callerB(x)
.
Satisfying the second of these conditions is easy, but what about the
first? Note, for example, that declaring both A
and B
with “decreases x”
does not work, because that won’t prove a strict decrease for the call
from A(x)
to B(x)
.
Here’s one possibility:
method A(x: nat)
decreases x, 1
{
B(x);
}
method B(x: nat)
decreases x, 0
{
if x != 0 { A(x-1); }
}
For the call from A(x)
to B(x)
, the lexicographic tuple "x, 0"
is
strictly smaller than "x, 1"
, and for the call from B(x)
to A(x-1)
, the
lexicographic tuple "x-1, 1"
is strictly smaller than "x, 0"
.
Two things to note: First, the choice of “0” and “1” as the second
components of these lexicographic tuples is rather arbitrary. It could
just as well have been “false” and “true”, respectively, or the sets
{2,5}
and {2,3,5}
. Second, the keyword decreases
often gives rise to
an intuitive English reading of the declaration. For example, you might
say that the recursive calls in the definition of the familiar Fibonacci
function Fib(n)
“decreases n”. But when the lexicographic tuple contains
constants, the English reading of the declaration becomes mysterious and
may give rise to questions like “how can you decrease the constant 0?”.
The keyword is just that—a keyword. It says “here comes a list of
expressions that make up the lexicographic tuple we want to use for the
termination measure”. What is important is that one effective decreases
clause is compared against another one, and it certainly makes sense to
compare something to a constant (and to compare one constant to
another).
We can simplify things a little bit by remembering that Dafny appends
$\top$ to the user-supplied decreases clause. For the A-and-B example,
this lets us drop the constant from the decreases
clause of A:
method A(x: nat)
decreases x
{
B(x);
}
method B(x: nat)
decreases x, 0
{
if x != 0 { A(x-1); }
}
The effective decreases clause of A
is $(x, \top)$ and the effective
decreases clause of B
is $(x, 0, \top)$. These tuples still satisfy the two
conditions $(x, 0, \top) < (x, \top)$ and $(x-1, \top) < (x, 0, \top)$. And
as before, the constant “0” is arbitrary; anything less than $\top$ (which
is any Dafny expression) would work.
Let’s take a look at one more example that better illustrates the utility
of $\top$. Consider again two mutually recursive methods, call them Outer
and Inner
, representing the recursive counterparts of what iteratively
might be two nested loops:
method Outer(x: nat)
{
// set y to an arbitrary non-negative integer
var y :| 0 <= y;
Inner(x, y);
}
method Inner(x: nat, y: nat)
{
if y != 0 {
Inner(x, y-1);
} else if x != 0 {
Outer(x-1);
}
}
The body of Outer
uses an assign-such-that statement to represent some
computation that takes place before Inner
is called. It sets “y” to some
arbitrary non-negative value. In a more concrete example, Inner
would do
some work for each “y” and then continue as Outer
on the next smaller
“x”.
Using a decreases
clause $(x, y)$ for Inner
seems natural, but if
we don’t have any bound on the size of the $y$ computed by Outer
,
there is no expression we can write in the decreases
clause of Outer
that is sure to lead to a strictly smaller value for $y$ when Inner
is called. $\top$ to the rescue. If we arrange for the effective
decreases clause of Outer
to be $(x, \top)$ and the effective decreases
clause for Inner
to be $(x, y, \top)$, then we can show the strict
decreases as required. Since $\top$ is implicitly appended, the two
decreases clauses declared in the program text can be:
method Outer(x: nat)
decreases x
{
// set y to an arbitrary non-negative integer
var y :| 0 <= y;
Inner(x, y);
}
method Inner(x: nat, y: nat)
decreases x,y
{
if y != 0 {
Inner(x, y-1);
} else if x != 0 {
Outer(x-1);
}
}
Moreover, remember that if a function or method has no user-declared
decreases
clause, Dafny will make a guess. The guess is (usually)
the list of arguments of the function/method, in the order given. This is
exactly the decreases clauses needed here. Thus, Dafny successfully
verifies the program without any explicit decreases
clauses:
method Outer(x: nat)
{
var y :| 0 <= y;
Inner(x, y);
}
method Inner(x: nat, y: nat)
{
if y != 0 {
Inner(x, y-1);
} else if x != 0 {
Outer(x-1);
}
}
The ingredients are simple, but the end result may seem like magic. For many users, however, there may be no magic at all – the end result may be so natural that the user never even has to be bothered to think about that there was a need to prove termination in the first place.
Dafny also prepends two expressions to the user-specified (or guessed) tuple of expressions in the decreases clause. The first expression is the ordering of the module containing the decreases clause in the dependence-ordering of modules. That is, a module that neither imports or defines (as submodules) any other modules has the lowest value in the order and every other module has a value that is higher than that of any module it defines or imports. As a module cannot call a method in a module that it does not depend on, this is an effective first component to the overall decreases tuple.
The second prepended expression represents the position of the method in the call graph within a module. Dafny analyzes the call-graph of the module, grouping all methods into mutually-recursive groups. Any method that calls nothing else is at the lowest level (say level 0). Absent recursion, every method has a level value strictly greater than any method it calls. Methods that are mutually recursive are at the same level and they are above the level of anything else they call. With this level value prepended to the decreases clause, the decreases tuple automatically decreases on any calls in a non-recursive context.
Though Dafny fixes a well-founded order that it uses when checking termination, Dafny does not surface this ordering directly in expressions. That is, syntactically, there is no single operator that stands for the well-founded ordering.
7.1.4. Framing (grammar)
Examples:
*
o
o`a
`a
{ o, p, q }
{}
Frame expressions are used to denote the set of memory locations
that a Dafny program element may read or write.
They are used in reads
and modifies
clauses.
A frame expression is a set expression. The form {}
is the empty set.
The type of the frame expression is set<object>
.
Note that framing only applies to the heap, or memory accessed through references. Local variables are not stored on the heap, so they cannot be mentioned (well, they are not in scope in the declaration) in frame annotations. Note also that types like sets, sequences, and multisets are value types, and are treated like integers or local variables. Arrays and objects are reference types, and they are stored on the heap (though as always there is a subtle distinction between the reference itself and the value it points to.)
The FrameField
construct is used to specify a field of a
class object. The identifier following the back-quote is the
name of the field being referenced.
If the FrameField
is preceded by an expression the expression
must be a reference to an object having that field.
If the FrameField
is not preceded by an expression then
the frame expression is referring to that field of the current
object (this
). This form is only used within a method of a class or trait.
A FrameField
can be useful in the following case:
When a method modifies only one field, rather than writing
class A {
var i: int
var x0: int
var x1: int
var x2: int
var x3: int
var x4: int
method M()
modifies this
ensures unchanged(`x0) && unchanged(`x1) && unchanged(`x2) && unchanged(`x3) && unchanged(`x4)
{ i := i + 1; }
}
one can write the more concise:
class A {
var i: int
var x0: int
var x1: int
var x2: int
var x3: int
var x4: int
method M()
modifies `i
{ i := i + 1; }
}
There’s (unfortunately) no form of it for array
elements – but to account for unchanged elements, you can always write
forall i | 0 <= i < |a| :: unchanged(a[i])
.
A FrameField
is not taken into consideration for
lambda expressions.
7.1.5. Reads Clause (grammar)
Examples:
const o: object
const o, oo: object
function f()
reads *
function g()
reads o, oo
function h()
reads { o }
Functions are not allowed to have side effects; they may also be restricted in what they can read. The reading frame of a function (or predicate) consists of all the heap memory locations that the function is allowed to read. The reason we might limit what a function can read is so that when we write to memory, we can be sure that functions that did not read that part of memory have the same value they did before. For example, we might have two arrays, one of which we know is sorted. If we did not put a reads annotation on the sorted predicate, then when we modify the unsorted array, we cannot determine whether the other array stopped being sorted. While we might be able to give invariants to preserve it in this case, it gets even more complex when manipulating data structures. In this case, framing is essential to making the verification process feasible.
It is not just the body of a function that is subject to reads
checks, but also its precondition and the reads
clause itself.
A reads
clause can list a wildcard *
, which allows the enclosing
function to read anything. In many cases, and in particular in all cases
where the function is defined recursively, this makes it next to
impossible to make any use of the function. Nevertheless, as an
experimental feature, the language allows it (and it is sound).
If a reads
clause uses *
, then the reads
clause is not allowed to
mention anything else (since anything else would be irrelevant, anyhow).
A reads
clause specifies the set of memory locations that a function,
lambda, or iterator may read. The readable memory locations are all the fields
of all of the references given in the set specified in the frame expression
and the single fields given in FrameField
elements of the frame expression.
For example, in
class C {
var x: int
var y: int
predicate f(c: C)
reads this, c`x
{
this.x == c.x
}
}
the reads
clause allows reading this.x
, this,y
, and c.x
(which may be the same
memory location as this.x
).
}
If more than one reads
clause is given
in a specification the effective read set is the union of the sets
specified. If there are no reads
clauses the effective read set is
empty. If *
is given in a reads
clause it means any memory may be
read.
If a reads
clause refers to a sequence or multiset, that collection
(call it c
) is converted to a set by adding an implicit set
comprehension of the form set o: object | o in c
before computing the
union of object sets from other reads
clauses.
An expression in a reads
clause is also allowed to be a function call whose value is
a collection of references. Such an expression is converted to a set by taking the
union of the function’s image over all inputs. For example, if F
is
a function from int
to set<object>
, then reads F
has the meaning
set x: int, o: object | o in F(x) :: o
For each function value f
, Dafny defines the function f.reads
,
which takes the same arguments as f
and returns that set of objects
that f
reads (according to its reads clause) with those arguments.
f.reads
has type T ~> set<object>
, where T
is the input type(s) of f
.
This is particularly useful when wanting to specify the reads set of
another function. For example, function Sum
adds up the values of
f(i)
where i
ranges from lo
to hi
:
function Sum(f: int ~> real, lo: int, hi: int): real
requires lo <= hi
requires forall i :: lo <= i < hi ==> f.requires(i)
reads f.reads
decreases hi - lo
{
if lo == hi then 0.0 else
f(lo) + Sum(f, lo + 1, hi)
}
Its reads
specification says that Sum(f, lo, hi)
may read anything
that f
may read on any input. (The specification
reads f.reads
gives an overapproximation of what Sum
will actually
read. More precise would be to specify that Sum
reads only what f
reads on the values from lo
to hi
, but the larger set denoted by
reads f.reads
is easier to write down and is often good enough.)
Note, only reads
clauses, not modifies
clauses, are allowed to
include functions as just described.
7.1.6. Modifies Clause (grammar)
Examples:
class A { var f: int }
const o: object?
const p: A?
method M()
modifies { o, p }
method N()
modifies { }
method Q()
modifies o, p`f
Frames also affect methods. Methods are not
required to list the things they read. Methods are allowed to read
whatever memory they like, but they are required to list which parts of
memory they modify, with a modifies
annotation. These are almost identical
to their reads
cousins, except they say what can be changed, rather than
what the value of the function depends on. In combination with reads,
modification restrictions allow Dafny to prove properties of code that
would otherwise be very difficult or impossible. Reads and modifies are
one of the tools that allow Dafny to work on one method at a time,
because they restrict what would otherwise be arbitrary modifications of
memory to something that Dafny can reason about.
Just as for a reads
clause, the memory locations allowed to be modified
in a method are all the fields of any object reference in the frame expression
set and any specific field denoted by a FrameField
in the modifies
clause.
For example, in
class C {
var next: C?
var value: int
method M()
modifies next
{
...
}
}
method M
is permitted to modify this.next.next
and this.next.value
but not this.next
. To be allowed to modify this.next
, the modifies clause
must include this
, or some expression that evaluates to this
, or this`next
.
If an object is newly allocated within the body of a method
or within the scope of a modifies
statement or a loop’s modifies
clause,
then the fields of that object may always be modified.
A modifies
clause specifies the set of memory locations that a
method, iterator or loop body may modify. If more than one modifies
clause is given in a specification, the effective modifies set is the
union of the sets specified. If no modifies
clause is given the
effective modifies set is empty. There is no wildcard (*
) allowed in
a modifies clause. A loop can also have a
modifies
clause. If none is given, the loop may modify anything
the enclosing context is allowed to modify.
Note that modifies here is used in the sense of writes. That is, a field
that may not be modified may not be written to, even with the same value it
already has or even if the value is restored later. The terminology and
semantics varies among specification languages. Some define frame conditions
in this sense (a) of writes and others in the sense (b) that allows writing
a field with the same value or changing the value so long as the original
value is restored by the end of the scope. For example, JML defines
assignable
and modifies
as synonyms in the sense (a), though KeY
interprets JML’s assigns/modifies
in sense (b).
ACSL and ACSL++ use the assigns
keyword, but with modify (b) semantics.
Ada/SPARK’s dataflow contracts encode write (a) semantics.
7.1.7. Invariant Clause (grammar)
Examples:
method m()
{
var i := 10;
while 0 < i
invariant 0 <= i < 10
}
An invariant
clause is used to specify an invariant
for a loop. If more than one invariant
clause is given for
a loop, the effective invariant is the conjunction of
the conditions specified, in the order given in the source text.
The invariant must hold on entry to the loop. And assuming it is valid on entry to a particular iteration of the loop, Dafny must be able to prove that it then holds at the end of that iteration of the loop.
An invariant can have custom error and success messages.
7.2. Method Specification (grammar)
Examples:
class C {
var next: C?
var value: int
method M(i: int) returns (r: int)
requires i >= 0
modifies next
decreases i
ensures r >= 0
{
...
}
}
A method specification consists of zero or more modifies
, requires
,
ensures
or decreases
clauses, in any order.
A method does not have reads
clauses because methods are allowed to
read any memory.
7.3. Function Specification (grammar)
Examples:
class C {
var next: C?
var value: int
function M(i: int): (r: int)
requires i >= 0
reads this
decreases i
ensures r >= 0
{
0
}
}
A function specification is zero or more reads
, requires
,
ensures
or decreases
clauses, in any order. A function
specification does not have modifies
clauses because functions are not
allowed to modify any memory.
7.4. Lambda Specification (grammar)
A lambda specification provides a specification for a lambda function expression;
it consists of zero or more reads
or requires
clauses.
Any requires
clauses may not have labels or attributes.
Lambda specifications do not have ensures
clauses because the body
is never opaque.
Lambda specifications do not have decreases
clauses because lambda expressions do not have names and thus cannot be recursive. A
lambda specification does not have modifies
clauses because lambdas
are not allowed to modify any memory.
7.5. Iterator Specification (grammar)
An iterator specification may contains reads
, modifies
,
decreases
, requires
, yield requires,
ensures
and
yield ensures` clauses.
An iterator specification applies both to the iterator’s constructor
method and to its MoveNext
method.
- The
reads
andmodifies
clauses apply to both of them. - The
requires
andensures
clauses apply to the constructor. - The
yield requires
andyield ensures
clauses apply to theMoveNext
method.
Examples of iterators, including iterator specifications, are given in Section 5.11. Briefly
- a requires clause gives a precondition for creating an iterator
- an ensures clause gives a postcondition when the iterator exits (after all iterations are complete)
- a decreases clause is used to show that the iterator will eventually terminate
- a yield requires clause is a precondition for calling
MoveNext
- a yield ensures clause is a postcondition for calling
MoveNext
7.6. Loop Specification (grammar)
A loop specification provides the information Dafny needs to
prove properties of a loop. It contains invariant
,
decreases
, and modifies
clauses.
The invariant
clause
is effectively a precondition and it along with the
negation of the loop test condition provides the postcondition.
The decreases
clause is used to prove termination.
7.7. Auto-generated boilerplate specifications
AutoContracts is an experimental feature that inserts much of the dynamic-frames boilerplate into a class. The user simply
- marks the class with
{:autocontracts}
and - declares a function (or predicate) called Valid().
AutoContracts then
- Declares, unless there already exist members with these names:
ghost var Repr: set(object) predicate Valid()
- For function/predicate
Valid()
, insertsreads this, Repr ensures Valid() ==> this in Repr
- Into body of
Valid()
, inserts (at the beginning of the body)this in Repr && null !in Repr
and also inserts, for every array-valued field
A
declared in the class:(A != null ==> A in Repr) &&
and for every field
F
of a class typeT
whereT
has a field calledRepr
, also inserts(F != null ==> F in Repr && F.Repr SUBSET Repr && this !in Repr && F.Valid())
except, if
A
orF
is declared with{:autocontracts false}
, then the implication will not be added. - For every constructor, inserts
ensures Valid() && fresh(Repr)
- At the end of the body of the constructor, adds
Repr := {this}; if (A != null) { Repr := Repr + {A}; } if (F != null) { Repr := Repr + {F} + F.Repr; }
In all the following cases, no modifies
clause or reads
clause is added if the user
has given one.
- For every non-static non-ghost method that is not a “simple query method”,
inserts
requires Valid() modifies Repr ensures Valid() && fresh(Repr - old(Repr))
- At the end of the body of the method, inserts
if (A != null && !(A in Repr)) { Repr := Repr + {A}; } if (F != null && !(F in Repr && F.Repr SUBSET Repr)) { Repr := Repr + {F} + F.Repr; }
- For every non-static non-twostate method that is either ghost or is a “simple query method”,
add:
requires Valid()
- For every non-static twostate method, inserts
requires old(Valid())
- For every non-“Valid” non-static function, inserts
requires Valid() reads Repr
8. Statements (grammar)
Many of Dafny’s statements are similar to those in traditional programming languages, but a number of them are significantly different. Dafny’s various kinds of statements are described in subsequent sections.
Statements have zero or more labels and end with either a semicolon (;
) or a closing curly brace (‘}’).
8.1. Labeled Statement (grammar)
Examples:
class A { var f: int }
method m(a: A) {
label x:
while true {
if (*) { break x; }
}
a.f := 0;
label y:
a.f := 1;
assert old@y(a.f) == 1;
}
A labeled statement is just
- the keyword
label
- followed by an identifier, which is the label,
- followed by a colon
- and a statement.
The label may be
referenced in a break
or continue
statement within the labeled statement
(see Section 8.14). That is, the break or continue that
mentions the label must be enclosed in the labeled statement.
The label may also be used in an old
expression (Section 9.22). In this case, the label
must have been encountered during the control flow en route to the old
expression. We say in this case that the (program point of the) label dominates
the (program point of the) use of the label.
Similarly, labels are used to indicate previous states in calls of two-state predicates,
fresh expressions, unchanged expressions,
and allocated expressions.
A statement can be given several labels. It makes no difference which of these labels is used to reference the statement—they are synonyms of each other. The labels must be distinct from each other, and are not allowed to be the same as any previous enclosing or dominating label.
8.2. Block Statement (grammar)
Examples:
{
print 0;
var x := 0;
}
A block statement is a sequence of zero or more statements enclosed by curly braces. Local variables declared in the block end their scope at the end of the block.
8.3. Return Statement (grammar)
Examples:
method m(i: int) returns (r: int) {
return i+1;
}
method n(i: int) returns (r: int, q: int) {
return i+1, i + 2;
}
method p() returns (i: int) {
i := 1;
return;
}
method q() {
return;
}
A return statement can only be used in a method. It is used to terminate the execution of the method.
To return a value from a method, the value is assigned to one of the named out-parameters sometime before a return statement. In fact, the out-parameters act very much like local variables, and can be assigned to more than once. Return statements are used when one wants to return before reaching the end of the body block of the method.
Return statements can be just the return
keyword (where the current values
of the out-parameters are used), or they can take a list of expressions to
return. If a list is given, the number of expressions given must be the same
as the number of named out-parameters. These expressions are
evaluated, then they are assigned to the out-parameters, and then the
method terminates.
8.4. Yield Statement (grammar)
A yield statement may only be used in an iterator. See iterator types for more details about iterators.
The body of an iterator is a co-routine. It is used
to yield control to its caller, signaling that a new
set of values for the iterator’s yield (out-)parameters (if any)
are available. Values are assigned to the yield parameters
at or before a yield statement.
In fact, the yield parameters act very much like local variables,
and can be assigned to more than once. Yield statements are
used when one wants to return new yield parameter values
to the caller. Yield statements can be just the
yield
keyword (where the current values of the yield parameters
are used), or they can take a list of expressions to yield.
If a list is given, the number of expressions given must be the
same as the number of named iterator out-parameters.
These expressions are then evaluated, then they are
assigned to the yield parameters, and then the iterator
yields.
8.5. Update and Call Statements (grammar)
Examples:
class C { var f: int }
class D {
var i: int
constructor(i: int) {
this.i := i;
}
}
method q(i: int, j: int) {}
method r() returns (s: int, t: int) { return 2,3; }
method m() {
var ss: int, tt: int, c: C?, a: array<int>, d: D?;
q(0,1);
ss, c.f := r();
c := new C;
d := new D(2);
a := new int[10];
ss, tt := 212, 33;
ss :| ss > 7;
ss := *;
}
This statement corresponds to familiar assignment or method call statements, with variations. If more than one left-hand side is used, these must denote different l-values, unless the corresponding right-hand sides also denote the same value.
The update statement serves several logical purposes.
8.5.1. Method call with no out-parameters
1) Examples of method calls take this form
m();
m(1,2,3) {:attr} ;
e.f().g.m(45);
As there are no left-hand-side locations to receive values, this form is allowed only for methods that have no out-parameters.
8.5.2. Method call with out-parameters
This form uses :=
to denote the assignment of the out-parameters of the method to the
corresponding number of LHS values.
a, b.e().f := m() {:attr};
In this case, the right-hand-side must be a method call and the number of
left-hand sides must match the number of out-parameters of the
method that is called or there must be just one Lhs
to the left of
the :=
, which then is assigned a tuple of the out-parameters.
Note that the result of a method call is not allowed to be used as an argument of
another method call, as if it were an expression.
8.5.3. Parallel assignment
A parallel-assignment has one-or-more right-hand-side expressions, which may be function calls but may not be method calls.
x, y := y, x;
The above example swaps the values of x
and y
. If more than one
left-hand side is used, these must denote different l-values, unless the
corresponding right-hand sides also denote the same value. There must
be an equal number of left-hand sides and right-hand sides.
The most common case has only one RHS and one LHS.
8.5.4. Havoc assignment
The form with a right-hand-side that is *
is a havoc assignment.
It assigns an arbitrary but type-correct value to the corresponding left-hand-side.
It can be mixed with other assignments of computed values.
a := *;
a, b, c := 4, *, 5;
8.5.5. Such-that assignment
This form has one or more left-hand-sides, a :|
symbol and then a boolean expression on the right.
The effect is to assign values to the left-hand-sides that satisfy the
RHS condition.
x, y :| 0 < x+y < 10;
This is read as assign values to x
and y
such that 0 < x+y < 10
is true.
The given boolean expression need not constrain the LHS values uniquely:
the choice of satisfying values is non-deterministic.
This can be used to make a choice as in the
following example where we choose an element in a set.
method Sum(X: set<int>) returns (s: int)
{
s := 0; var Y := X;
while Y != {}
decreases Y
{
var y: int;
y :| y in Y;
s, Y := s + y, Y - {y};
}
}
Dafny will report an error if it cannot prove that values exist that satisfy the condition.
In this variation, with an assume
keyword
y :| assume y in Y;
Dafny assumes without proof that an appropriate value exists.
Note that the syntax
Lhs ":"
is interpreted as a label in which the user forgot the label
keyword.
8.6. Update with Failure Statement (:-
) (grammar)
See the subsections below for examples.
A :-
9 statement is an alternate form of the :=
statement that allows for abrupt return if a failure is detected.
This is a language feature somewhat analogous to exceptions in other languages.
An update-with-failure statement uses failure-compatible types. A failure-compatible type is a type that has the following (non-static) members (each with no in-parameters and one out-parameter):
- a non-ghost function
IsFailure()
that returns abool
- an optional non-ghost function
PropagateFailure()
that returns a value assignable to the first out-parameter of the caller - an optional function
Extract()
(PropagateFailure and Extract were permitted to be methods (but deprecated) prior to Dafny 4. They will be required to be functions in Dafny 4.)
A failure-compatible type with an Extract
member is called value-carrying.
To use this form of update,
- if the RHS of the update-with-failure statement is a method call, the first out-parameter of the callee must be failure-compatible
- if instead, the RHS of the update-with-failure statement is one or more expressions, the first of these expressions must be a value with a failure-compatible type
- the caller must have a first out-parameter whose type matches the output of
PropagateFailure
applied to the first output of the callee, unless anexpect
,assume
, orassert
keyword is used after:-
(cf. Section 8.6.7). - if the failure-compatible type of the RHS does not have an
Extract
member, then the LHS of the:-
statement has one less expression than the RHS (or than the number of out-parameters from the method call), the value of the first out-parameter or expression being dropped (see the discussion and examples in Section 8.6.2) - if the failure-compatible type of the RHS does have an
Extract
member, then the LHS of the:-
statement has the same number of expressions as the RHS (or as the number of out-parameters from the method call) and the type of the first LHS expression must be assignable from the return type of theExtract
member - the
IsFailure
andPropagateFailure
methods may not be ghost - the LHS expression assigned the output of the
Extract
member is ghost precisely ifExtract
is ghost
The following subsections show various uses and alternatives.
8.6.1. Failure compatible types
A simple failure-compatible type is the following:
datatype Status =
| Success
| Failure(error: string)
{
predicate IsFailure() { this.Failure? }
function PropagateFailure(): Status
requires IsFailure()
{
Failure(this.error)
}
}
A commonly used alternative that carries some value information is something like this generic type:
datatype Outcome<T> =
| Success(value: T)
| Failure(error: string)
{
predicate IsFailure() {
this.Failure?
}
function PropagateFailure<U>(): Outcome<U>
requires IsFailure()
{
Failure(this.error) // this is Outcome<U>.Failure(...)
}
function Extract(): T
requires !IsFailure()
{
this.value
}
}
8.6.2. Simple status return with no other outputs
The simplest use of this failure-return style of programming is to have a method call that just returns a non-value-carrying Status
value:
method Callee(i: int) returns (r: Status)
{
if i < 0 { return Failure("negative"); }
return Success;
}
method Caller(i: int) returns (rr: Status)
{
:- Callee(i);
...
}
Note that there is no LHS to the :-
statement.
If Callee
returns Failure
, then the caller immediately returns,
not executing any statements following the call of Callee
.
The value returned by Caller
(the value of rr
in the code above) is the result of PropagateFailure
applied to the value returned by Callee
, which is often just the same value.
If Callee
does not return Failure
(that is, returns a value for which IsFailure()
is false
)
then that return value is forgotten and execution proceeds normally with the statements following the call of Callee
in the body of Caller
.
The desugaring of the :- Callee(i);
statement is
var tmp;
tmp := Callee(i);
if tmp.IsFailure() {
rr := tmp.PropagateFailure();
return;
}
In this and subsequent examples of desugaring, the tmp
variable is a new, unique variable, unused elsewhere in the calling member.
8.6.3. Status return with additional outputs
The example in the previous subsection affects the program only through side effects or the status return itself.
It may well be convenient to have additional out-parameters, as is allowed for :=
updates;
these out-parameters behave just as for :=
.
Here is an example:
method Callee(i: int) returns (r: Status, v: int, w: int)
{
if i < 0 { return Failure("negative"), 0, 0; }
return Success, i+i, i*i;
}
method Caller(i: int) returns (rr: Status, k: int)
{
var j: int;
j, k :- Callee(i);
k := k + k;
...
}
Here Callee
has two outputs in addition to the Status
output.
The LHS of the :-
statement accordingly has two l-values to receive those outputs.
The recipients of those outputs may be any sort of l-values;
here they are a local variable and an out-parameter of the caller.
Those outputs are assigned in the :-
call regardless of the Status
value:
- If
Callee
returns a failure value as its first output, then the other outputs are assigned, the caller’s first out-parameter (hererr
) is assigned the value ofPropagateFailure
, and the caller returns. - If
Callee
returns a non-failure value as its first output, then the other outputs are assigned and the caller continues execution as normal.
The desugaring of the j, k :- Callee(i);
statement is
var tmp;
tmp, j, k := Callee(i);
if tmp.IsFailure() {
rr := tmp.PropagateFailure();
return;
}
8.6.4. Failure-returns with additional data
The failure-compatible return value can carry additional data as shown in the Outcome<T>
example above.
In this case there is a (first) LHS l-value to receive this additional data. The type of that first LHS
value is one that is assignable from the result of the Extract
function, not the actual first out-parameter.
method Callee(i: int) returns (r: Outcome<nat>, v: int)
{
if i < 0 { return Failure("negative"), i+i; }
return Success(i), i+i;
}
method Caller(i: int) returns (rr: Outcome<int>, k: int)
{
var j: int;
j, k :- Callee(i);
k := k + k;
...
}
Suppose Caller
is called with an argument of 10
.
Then Callee
is called with argument 10
and returns r
and v
of Outcome<nat>.Success(10)
and 20
.
Here r.IsFailure()
is false
, so control proceeds normally.
The j
is assigned the result of r.Extract()
, which will be 10
,
and k
is assigned 20
.
Control flow proceeds to the next line, where k
now gets the value 40
.
Suppose instead that Caller
is called with an argument of -1
.
Then Callee
is called with the value -1
and returns r
and v
with values Outcome<nat>.Failure("negative")
and -2
.
k
is assigned the value of v
(-2).
But r.IsFailure()
is true
, so control proceeds directly to return from Caller
.
The first out-parameter of Caller
(rr
) gets the value of r.PropagateFailure()
,
which is Outcome<int>.Failure("negative")
; k
already has the value -2
.
The rest of the body of Caller
is skipped.
In this example, the first out-parameter of Caller
has a failure-compatible type
so the exceptional return will propagate up the call stack.
It will keep propagating up the call stack
as long as there are callers with this first special output type
and calls that use :-
and the return value keeps having IsFailure()
true.
The desugaring of the j, k :- Callee(i);
statement in this example is
var tmp;
tmp, k := Callee(i);
if tmp.IsFailure() {
rr := tmp.PropagateFailure();
return;
}
j := tmp.Extract();
8.6.5. RHS with expression list
Instead of a failure-returning method call on the RHS of the statement,
the RHS can instead be a list of expressions.
As for a :=
statement, in this form, the expressions on the left and right sides of :-
must correspond,
just omitting a LHS l-value for the first RHS expression if its type is not value-carrying.
The semantics is very similar to that in the previous subsection.
- The first RHS expression must have a failure-compatible type.
- All the assignments of RHS expressions to LHS values except for the first RHS value are made.
- If the first RHS value (say
r
) respondstrue
tor.IsFailure()
, thenr.PropagateFailure()
is assigned to the first out-parameter of the caller and the execution of the caller’s body is ended. - If the first RHS value (say
r
) respondsfalse
tor.IsFailure()
, then- if the type of
r
is value-carrying, thenr.Extract()
is assigned to the first LHS value of the:-
statement; ifr
is not value-carrying, then the corresponding LHS l-value is omitted - execution of the caller’s body continues with the statement following the
:-
statement.
- if the type of
A RHS with a method call cannot be mixed with a RHS containing multiple expressions.
For example, the desugaring of
method m(r: Status) returns (rr: Status) {
var k;
k :- r, 7;
...
}
is
var k;
var tmp;
tmp, k := r, 7;
if tmp.IsFailure() {
rr := tmp.PropagateFailure();
return;
}
8.6.6. Failure with initialized declaration.
The :-
syntax can also be used in initialization, as in
var s, t :- M();
This is equivalent to
var s, t;
s, t :- M();
with the semantics as described above.
8.6.7. Keyword alternative
In any of the above described uses of :-
, the :-
token may be followed immediately by the keyword expect
, assert
or assume
.
assert
means that the RHS evaluation is expected to be successful, but that the verifier should prove that this is so; that is, the verifier should proveassert !r.IsFailure()
(wherer
is the status return from the callee) (cf. Section 8.17)assume
means that the RHS evaluation should be assumed to be successful, as if the statementassume !r.IsFailure()
followed the evaluation of the RHS (cf. Section 8.18)expect
means that the RHS evaluation should be assumed to be successful (like usingassume
above), but that the compiler should include a run-time check for success. This is equivalent to includingexpect !r.IsFailure()
after the RHS evaluation; that is, if the status return is a failure, the program halts. (cf. Section 8.19)
In each of these cases, there is no abrupt return from the caller. Thus
there is no evaluation of PropagateFailure
. Consequently the first
out-parameter of the caller need not match the return type of
PropagateFailure
; indeed, the failure-compatible type returned by the
callee need not have a PropagateFailure
member.
The equivalent desugaring replaces
if tmp.IsFailure() {
rr := tmp.PropagateFailure();
return;
}
with
expect !tmp.IsFailure(), tmp;
or
assert !tmp.IsFailure();
or
assume !tmp.IsFailure();
There is a grammatical nuance that the user should be aware of.
The keywords assert
, assume
, and expect
can start an expression.
For example, assert P; E
can be an expression. However, in
e :- assert P; E;
the assert
is parsed as the keyword associated with
:-
. To have the assert
considered part of the expression use parentheses:
e :- (assert P; E);
.
8.6.8. Key points
There are several points to note.
- The first out-parameter of the callee is special. It has a special type and that type indicates that the value is inspected to see if an abrupt return from the caller is warranted. This type is often a datatype, as shown in the examples above, but it may be any type with the appropriate members.
- The restriction on the type of caller’s first out-parameter is
just that it must be possible (perhaps through generic instantiation and type inference, as in these examples)
for
PropagateFailure
applied to the failure-compatible output from the callee to produce a value of the caller’s first out-parameter type. If the caller’s first out-parameter type is failure-compatible (which it need not be), then failures can be propagated up the call chain. If the keyword form (e.g.assume
) of the statement is used, then noPropagateFailure
member is needed, because no failure can occur, and there is no restriction on the caller’s first out-parameter. - In the statement
j, k :- Callee(i);
, when the callee’s return value has anExtract
member, the type ofj
is not the type of the first out-parameter ofCallee
. Rather it is a type assignable from the output type ofExtract
applied to the first out-value ofCallee
. - A method like
Callee
with a special first out-parameter type can still be used in the normal way:r, k := Callee(i)
. Nowr
gets the first output value fromCallee
, of typeStatus
orOutcome<nat>
in the examples above. No special semantics or exceptional control paths apply. Subsequent code can do its own testing of the value ofr
and whatever other computations or control flow are desired. - The caller and callee can have any (positive) number of output arguments,
as long as the callee’s first out-parameter has a failure-compatible type
and the caller’s first out-parameter type matches
PropagateFailure
. - If there is more than one LHS, the LHSs must denote different l-values, unless the RHS is a list of expressions and the corresponding RHS values are equal.
- The LHS l-values are evaluated before the RHS method call, in case the method call has side-effects or return values that modify the l-values prior to assignments being made.
It is important to note the connection between the failure-compatible types used in the caller and callee,
if they both use them.
They do not have to be the same type, but they must be closely related,
as it must be possible for the callee’s PropagateFailure
to return a value of the caller’s failure-compatible type.
In practice this means that one such failure-compatible type should be used for an entire program.
If a Dafny program uses a library shared by multiple programs, the library should supply such a type
and it should be used by all the client programs (and, effectively, all Dafny libraries).
It is also the case that it is inconvenient to mix types such as Outcome
and Status
above within the same program.
If there is a mix of failure-compatible types, then the program will need to use :=
statements and code for
explicit handling of failure values.
8.6.9. Failure returns and exceptions
The :-
mechanism is like the exceptions used in other programming languages, with some similarities and differences.
- There is essentially just one kind of ‘exception’ in Dafny, the variations of the failure-compatible data type.
- Exceptions are passed up the call stack whether or not intervening methods are aware of the possibility of an exception,
that is, whether or not the intervening methods have declared that they throw exceptions.
Not so in Dafny: a failure is passed up the call stack only if each caller has a failure-compatible first out-parameter, is itself called in a
:-
statement, and returns a value that responds true toIsFailure()
. - All methods that contain failure-return callees must explicitly handle those failures
using either
:-
statements or using:=
statements with a LHS to receive the failure value.
8.7. Variable Declaration Statement (grammar)
Examples:
method m() {
var x, y: int; // x's type is inferred, not necessarily 'int'
var b: bool, k: int;
x := 1; // settles x's type
}
A variable declaration statement is used to declare one or more local variables in a method or function. The type of each local variable must be given unless its type can be inferred, either from a given initial value, or from other uses of the variable. If initial values are given, the number of values must match the number of variables declared.
The scope of the declared variable extends to the end of the block in which it is
declared. However, be aware that if a simple variable declaration is followed
by an expression (rather than a subsequent statement) then the var
begins a
Let Expression and the scope of the introduced variables is
only to the end of the expression. In this case, though, the var
is in an expression
context, not a statement context.
Note that the type of each variable must be given individually. The following code
var x, y : int;
var x, y := 5, 6;
var x, y :- m();
var x, y :| 0 < x + y < 10;
var (x, y) := makePair();
var Cons(x, y) = ConsMaker();
does not declare both x
and y
to be of type int
. Rather it will give an
error explaining that the type of x
is underspecified if it cannot be
inferred from uses of x.
The variables can be initialized with syntax similar to update statements (cf. Section 8.5).
If the RHS is a call, then any variable receiving the value of a
formal ghost out-parameter will automatically be declared as ghost, even
if the ghost
keyword is not part of the variable declaration statement.
The left-hand side can also contain a tuple of patterns that will be matched against the right-hand-side. For example:
function returnsTuple() : (int, int)
{
(5, 10)
}
function usesTuple() : int
{
var (x, y) := returnsTuple();
x + y
}
The initialization with failure operator :-
returns from the enclosing method if the initializer evaluates to a failure value of a failure-compatible type (see Section 8.6).
8.8. Guards (grammar)
Examples (in if
statements):
method m(i: int) {
if (*) { print i; }
if i > 0 { print i; }
}
Guards are used in if
and while
statements as boolean expressions. Guards
take two forms.
The first and most common form is just a boolean expression.
The second form is either *
or (*)
. These have the same meaning. An
unspecified boolean value is returned. The value returned
may be different each time it is executed.
8.9. Binding Guards (grammar)
Examples (in if
statements):
method m(i: int) {
ghost var k: int;
if i, j :| 0 < i+j < 10 {
k := 0;
} else {
k := 1;
}
}
An if
statement can also take a binding guard.
Such a guard checks if there exist values for the given variables that satisfy the given expression.
If so, it binds some satisfying values to the variables and proceeds
into the “then” branch; otherwise it proceeds with the “else” branch,
where the bound variables are not in scope.
In other words, the statement
if x :| P { S } else { T }
has the same meaning as
if exists x :: P { var x :| P; S } else { T }
The identifiers bound by the binding guard are ghost variables and cannot be assigned to non-ghost variables. They are only used in specification contexts.
Here is another example:
predicate P(n: int)
{
n % 2 == 0
}
method M1() returns (ghost y: int)
requires exists x :: P(x)
ensures P(y)
{
if x : int :| P(x) {
y := x;
}
}
8.10. If Statement (grammar)
Examples:
method m(i: int) {
var x: int;
if i > 0 {
x := i;
} else {
x := -i;
}
if * {
x := i;
} else {
x := -i;
}
if i: nat, j: nat :| i+j<10 {
assert i < 10;
}
if i == 0 {
x := 0;
} else if i > 0 {
x := 1;
} else {
x := -1;
}
if
case i == 0 => x := 0;
case i > 0 => x := 1;
case i < 0 => x := -1;
}
The simplest form of an if
statement uses a guard that is a boolean
expression. For example,
if x < 0 {
x := -x;
}
Unlike match
statements, if
statements do not have to be exhaustive:
omitting the else
block is the same as including an empty else
block. To ensure that an if
statement is exhaustive, use the
if-case
statement documented below.
If the guard is an asterisk then a non-deterministic choice is made:
if * {
print "True";
} else {
print "False";
}
The then alternative of the if-statement must be a block statement; the else alternative may be either a block statement or another if statement. The condition of the if statement need not (but may) be enclosed in parentheses.
An if statement with a binding guard is non-deterministic;
it will not be compiled if --enforce-determinism
is enabled
(even if it can be proved that there is a unique value).
An if statement with *
for a guard is non-deterministic and ghost.
The if-case
statement using the AlternativeBlock
form is similar to the
if ... fi
construct used in the book “A Discipline of Programming” by
Edsger W. Dijkstra. It is used for a multi-branch if
.
For example:
method m(x: int, y: int) returns (max: int)
{
if {
case x <= y => max := y;
case y <= x => max := x;
}
}
In this form, the expressions following the case
keyword are called
guards. The statement is evaluated by evaluating the guards in an
undetermined order until one is found that is true
and the statements
to the right of =>
for that guard are executed. The statement requires
at least one of the guards to evaluate to true
(that is, if-case
statements must be exhaustive: the guards must cover all cases).
In the if-with-cases, a sequence of statements may follow the =>
; it
may but need not be a block statement (a brace-enclosed sequence of statements).
The form that used ...
(a refinement feature) as the guard is deprecated.
8.11. Match Statement (grammar)
Examples:
match list {
case Nil => {}
case Cons(head,tail) => print head;
}
match x
case 1 =>
print x;
case 2 =>
var y := x*x;
print y;
case _ =>
print "Other";
// Any statement after is captured in this case.
The match
statement is used to do case analysis on a value of an expression.
The expression may be a value of a basic type (e.g. int
), a newtype, or
an inductive or coinductive datatype (which includes the built-in tuple types).
The expression after the match
keyword is called the selector.
The selector is evaluated and then matched against
each clause in order until a matching clause is found.
The process of matching the selector expression against the case patterns is the same as for match expressions and is described in Section 9.31.2.
The selector need not be enclosed in parentheses; the sequence of cases may but need not be enclosed in braces.
The cases need not be disjoint.
The cases must be exhaustive, but you can use a wild variable (_
) or a simple identifier to indicate “match anything”.
Please refer to the section about case patterns to learn more about shadowing, constants, etc.
The code below shows an example of a match statement.
datatype Tree = Empty | Node(left: Tree, data: int, right: Tree)
// Return the sum of the data in a tree.
method Sum(x: Tree) returns (r: int)
{
match x {
case Empty => r := 0;
case Node(t1, d, t2) =>
var v1 := Sum(t1);
var v2 := Sum(t2);
r := v1 + d + v2;
}
}
Note that the Sum
method is recursive yet has no decreases
annotation.
In this case it is not needed because Dafny is able to deduce that
t1
and t2
are smaller (structurally) than x
. If Tree
had been
coinductive this would not have been possible since x
might have been
infinite.
8.12. While Statement (grammar)
Examples:
method m() {
var i := 10;
while 0 < i
invariant 0 <= i <= 10
decreases i
{
i := i-1;
}
while * {}
i := *;
while
decreases if i < 0 then -i else i
{
case i < 0 => i := i + 1;
case i > 0 => i := i - 1;
}
}
Loops
- may be a conventional loop with a condition and a block statement for a body
- need not have parentheses around the condition
- may have a
*
for the condition (the loop is then non-deterministic) - binding guards are not allowed
- may have a case-based structure
- may have no body — a bodyless loop is not compilable, but can be reaosnaed about
Importantly, loops need loop specifications in order for Dafny to prove that they obey expected behavior. In some cases Dafny can infer the loop specifications by analyzing the code, so the loop specifications need not always be explicit. These specifications are described in Section 7.6 and Section 8.15.
The general loop statement in Dafny is the familiar while
statement.
It has two general forms.
The first form is similar to a while loop in a C-like language. For example:
method m(){
var i := 0;
while i < 5 {
i := i + 1;
}
}
In this form, the condition following the while
is one of these:
- A boolean expression. If true it means execute one more iteration of the loop. If false then terminate the loop.
- An asterisk (
*
), meaning non-deterministically yield eithertrue
orfalse
as the value of the condition
The body of the loop is usually a block statement, but it can also be missing altogether. A loop with a missing body may still pass verification, but any attempt to compile the containing program will result in an error message. When verifying a loop with a missing body, the verifier will skip attempts to prove loop invariants and decreases assertions that would normally be asserted at the end of the loop body. There is more discussion about bodyless loops in Section 8.15.4.
The second form uses a case-based block. It is similar to the
do ... od
construct used in the book “A Discipline of Programming” by
Edsger W. Dijkstra. For example:
method m(n: int){
var r := n;
while
decreases if 0 <= r then r else -r
{
case r < 0 =>
r := r + 1;
case 0 < r =>
r := r - 1;
}
}
For this form, the guards are evaluated in some undetermined order until one is found that is true, in which case the corresponding statements are executed and the while statement is repeated. If none of the guards evaluates to true, then the loop execution is terminated.
The form that used ...
(a refinement feature) as the guard is deprecated.
8.13. For Loops (grammar)
Examples:
method m() decreases * {
for i := 0 to 10 {}
for _ := 0 to 10 {}
for i := 0 to * invariant i >= 0 decreases * {}
for i: int := 10 downto 0 {}
for i: int := 10 downto 0
}
The for
statement provides a convenient way to write some common loops.
The statement introduces a local variable with optional type, which is called
the loop index. The loop index is in scope in the specification and the body,
but not after the for
loop. Assignments to the loop index are not allowed.
The type of the loop index can typically be inferred; if so, it need not be given
explicitly. If the identifier is not used, it can be written as _
, as illustrated
in this repeat-20-times loop:
for _ := 0 to 20 {
Body
}
There are four basic variations of the for
loop:
for i: T := lo to hi
LoopSpec
{ Body }
for i: T := hi downto lo
LoopSpec
{ Body }
for i: T := lo to *
LoopSpec
{ Body }
for i: T := hi downto *
LoopSpec
{ Body }
Semantically, they are defined as the following respective while
loops:
{
var _lo, _hi := lo, hi;
assert _lo <= _hi && forall _i: int :: _lo <= _i <= _hi ==> _i is T;
var i := _lo;
while i != _hi
invariant _lo <= i <= _hi
LoopSpec
decreases _hi - i
{
Body
i := i + 1;
}
}
{
var _lo, _hi := lo, hi;
assert _lo <= _hi && forall _i: int :: _lo <= _i <= _hi ==> _i is T;
var i := _hi;
while i != lo
invariant _lo <= i <= _hi
LoopSpec
decreases i - _lo
{
i := i - 1;
Body
}
}
{
var _lo := lo;
assert forall _i: int :: _lo <= _i ==> _i is T;
var i := _lo;
while true
invariant _lo <= i
LoopSpec
{
Body
i := i + 1;
}
}
{
var _hi := hi;
assert forall _i: int :: _i <= _hi ==> _i is T;
var i := _hi;
while true
invariant i <= _hi
LoopSpec
{
i := i - 1;
Body
}
}
The expressions lo
and hi
are evaluated just once, before the loop
iterations start.
Also, in all variations the values of i
in the body are the values
from lo
to, but not including, hi
. This makes it convenient to
write common loops, including these:
for i := 0 to a.Length {
Process(a[i]);
}
for i := a.Length downto 0 {
Process(a[i]);
}
Nevertheless, hi
must be a legal value for the type of the index variable,
since that is how the index variable is used in the invariant.
If the end-expression is not *
, then no explicit decreases
is
allowed, since such a loop is already known to terminate.
If the end-expression is *
, then the absence of an explicit decreases
clause makes it default to decreases *
. So, if the end-expression is *
and no
explicit decreases
clause is given, the loop is allowed only in methods
that are declared with decreases *
.
The directions to
or downto
are contextual keywords. That is, these two
words are part of the syntax of the for
loop, but they are not reserved
keywords elsewhere.
Just like for while loops, the body of a for-loop may be omitted during verification. This suppresses attempts to check assertions (like invariants) that would occur at the end of the loop. Eventually, however a body must be provided; the compiler will not compile a method containing a body-less for-loop. There is more discussion about bodyless loops in Section 8.15.4.
8.14. Break and Continue Statements (grammar)
Examples:
class A { var f: int }
method m(a: A) {
label x:
while true {
if (*) { break; }
}
label y: {
var z := 1;
if * { break y; }
z := 2;
}
}
Break and continue statements provide a means to transfer control in a way different than the usual nested control structures. There are two forms of each of these statements: with and without a label.
If a label is used, the break or continue statement must be enclosed in a statement with that label. The enclosing statement is called the target of the break or continue.
A break
statement transfers control to the point immediately
following the target statement. For example, such a break statement can be
used to exit a sequence of statements in a block statement before
reaching the end of the block.
For example,
label L: {
var n := ReadNext();
if n < 0 {
break L;
}
DoSomething(n);
}
is equivalent to
{
var n := ReadNext();
if 0 <= n {
DoSomething(n);
}
}
If no label is specified and the statement lists n
occurrences of break
, then the statement must be enclosed in
at least n
levels of loop statements. Control continues after exiting n
enclosing loops. For example,
method m() {
for i := 0 to 10 {
for j := 0 to 10 {
label X: {
for k := 0 to 10 {
if j + k == 15 {
break break;
}
}
}
}
// control continues here after the "break break", exiting two loops
}
}
Note that a non-labeled break
pays attention only to loops, not to labeled
statements. For example, the labeled block X
in the previous example
does not play a role in determining the target statement of the break break;
.
For a continue
statement, the target statement must be a loop statement.
The continue statement transfers control to the point immediately
before the closing curly-brace of the loop body.
For example,
method m() {
for i := 0 to 100 {
if i == 17 {
continue;
}
DoSomething(i);
}
}
method DoSomething(i:int){}
is equivalent to
method m() {
for i := 0 to 100 {
if i != 17 {
DoSomething(i);
}
}
}
method DoSomething(i:int){}
The same effect can also be obtained by wrapping the loop body in a labeled
block statement and then using break
with a label, but that usually makes
for a more cluttered program:
method m() {
for i := 0 to 100 {
label LoopBody: {
if i == 17 {
break LoopBody;
}
DoSomething(i);
}
}
}
method DoSomething(i:int){}
Stated differently, continue
has the effect of ending the current loop iteration,
after which control continues with any remaining iterations. This is most natural
for for
loops. For a while
loop, be careful to make progress toward termination
before a continue
statement. For example, the following program snippet shows
an easy mistake to make (the verifier will complain that the loop may not terminate):
method m() {
var i := 0;
while i < 100 {
if i == 17 {
continue; // error: this would cause an infinite loop
}
DoSomething(i);
i := i + 1;
}
}
method DoSomething(i:int){}
The continue
statement can give a label, provided the label is a label of a loop.
For example,
method m() {
label Outer:
for i := 0 to 100 {
for j := 0 to 100 {
if i + j == 19 {
continue Outer;
}
WorkIt(i, j);
}
PostProcess(i);
// the "continue Outer" statement above transfers control to here
}
}
method WorkIt(i:int, j:int){}
method PostProcess(i:int){}
If a non-labeled continue statement lists n
occurrences of break
before the
continue
keyword, then the statement must be enclosed in at least n + 1
levels
of loop statements. The effect is to break
out of the n
most closely enclosing
loops and then continue
the iterations of the next loop. That is, n
occurrences
of break
followed by one more break;
will break out of n
levels of loops
and then do a break
, whereas n
occurrences of break
followed by continue;
will break out of n
levels of loops and then do a continue
.
For example, the WorkIt
example above can equivalently be written without labels
as
method m() {
for i := 0 to 100 {
for j := 0 to 100 {
if i + j == 19 {
break continue;
}
WorkIt(i, j);
}
PostProcess(i);
// the "break continue" statement above transfers control to here
}
}
method WorkIt(i:int, j:int){}
method PostProcess(i:int){}
Note that a loop invariant is checked on entry to a loop and at the closing curly-brace
of the loop body. It is not checked at break statements. For continue statements,
the loop invariant is checked as usual at the closing curly-brace
that the continue statement jumps to.
This checking ensures that the loop invariant holds at the very top of
every iteration. Commonly, the only exit out of a loop happens when the loop guard evaluates
to false
. Since no state is changed between the top of an iteration (where the loop
invariant is known to hold) and the evaluation of the loop guard, one can also rely on
the loop invariant to hold immediately following the loop. But the loop invariant may
not hold immediately following a loop if a loop iteration changes the program state and
then exits the loop with a break statement.
For example, the following program verifies:
method m() {
var i := 0;
while i < 10
invariant 0 <= i <= 10
{
if P(i) {
i := i + 200;
break;
}
i := i + 1;
}
assert i == 10 || 200 <= i < 210;
}
predicate P(i:int)
To explain the example, the loop invariant 0 <= i <= 10
is known to hold at the very top
of each iteration,
that is, just before the loop guard i < 10
is evaluated. If the loop guard evaluates
to false
, then the negated guard condition (10 <= i
) and the invariant hold, so
i == 10
will hold immediately after the loop. If the loop guard evaluates to true
(that is, i < 10
holds), then the loop body is entered. If the test P(i)
then evaluates
to true
, the loop adds 200
to i
and breaks out of the loop, so on such a
path, 200 <= i < 210
is known to hold immediately after the loop. This is summarized
in the assert statement in the example.
So, remember, a loop invariant holds at the very top of every iteration, not necessarily
immediately after the loop.
8.15. Loop Specifications
For some simple loops, such as those mentioned previously, Dafny can figure out what the loop is doing without more help. However, in general the user must provide more information in order to help Dafny prove the effect of the loop. This information is provided by a loop specification. A loop specification provides information about invariants, termination, and what the loop modifies. For additional tutorial information see [@KoenigLeino:MOD2011] or the online Dafny tutorial.
8.15.1. Loop invariants
Loops present a problem for specification-based reasoning. There is no way to know in advance how many times the code will go around the loop and a tool cannot reason about every one of a possibly unbounded sequence of unrollings. In order to consider all paths through a program, specification-based program verification tools require loop invariants, which are another kind of annotation.
A loop invariant is an expression that holds just prior to the loop test, that is, upon entering a loop and after every execution of the loop body. It captures something that is invariant, i.e. does not change, about every step of the loop. Now, obviously we are going to want to change variables, etc. each time around the loop, or we wouldn’t need the loop. Like pre- and postconditions, an invariant is a property that is preserved for each execution of the loop, expressed using the same boolean expressions we have seen. For example,
var i := 0;
while i < n
invariant 0 <= i
{
i := i + 1;
}
When you specify an invariant, Dafny proves two things: the invariant holds upon entering the loop, and it is preserved by the loop. By preserved, we mean that assuming that the invariant holds at the beginning of the loop (just prior to the loop test), we must show that executing the loop body once makes the invariant hold again. Dafny can only know upon analyzing the loop body what the invariants say, in addition to the loop guard (the loop condition). Just as Dafny will not discover properties of a method on its own, it will not know that any but the most basic properties of a loop are preserved unless it is told via an invariant.
8.15.2. Loop termination
Dafny proves that code terminates, i.e. does not loop forever, by using
decreases
annotations. For many things, Dafny is able to guess the right
annotations, but sometimes it needs to be made explicit.
There are two places Dafny proves termination: loops and recursion.
Both of these situations require either an explicit annotation or a
correct guess by Dafny.
A decreases
annotation, as its name suggests, gives Dafny an expression
that decreases with every loop iteration or recursive call. There are two
conditions that Dafny needs to verify when using a decreases
expression:
- that the expression actually gets smaller, and
- that it is bounded.
That is, the expression must strictly decrease in a well-founded ordering (cf. Section 12.7).
Many times, an integral value (natural or plain integer) is the quantity
that decreases, but other values can be used as well. In the case of
integers, the bound is assumed to be zero.
For each loop iteration the decreases
expression at the end of the loop
body must be strictly smaller than its value at the beginning of the loop
body (after the loop test). For integers, the well-founded relation between
x
and X
is x < X && 0 <= X
.
Thus if the decreases
value (X
) is negative at the
loop test, it must exit the loop, since there is no permitted value for
x
to have at the end of the loop body.
For example, the following is
a proper use of decreases
on a loop:
method m(n: nat){
var i := n;
while 0 < i
invariant 0 <= i
decreases i
{
i := i - 1;
}
}
Here Dafny has all the ingredients it needs to prove termination. The
variable i
becomes smaller each loop iteration, and is bounded below by
zero. When i
becomes 0, the lower bound of the well-founded order, control
flow exits the loop.
This is fine, except the loop is backwards from most loops, which tend to count up instead of down. In this case, what decreases is not the counter itself, but rather the distance between the counter and the upper bound. A simple trick for dealing with this situation is given below:
method m(m: nat, n: int) {
assume m <= n;
var i := m;
while i < n
invariant 0 <= i <= n
decreases n - i
{
i := i + 1;
}
}
This is actually Dafny’s guess for this situation, as it sees i < n
and
assumes that n - i
is the quantity that decreases. The upper bound of the
loop invariant implies that 0 <= n – i
, and gives Dafny a lower bound on
the quantity. This also works when the bound n
is not constant, such as
in the binary search algorithm, where two quantities approach each other,
and neither is fixed.
If the decreases
clause of a loop specifies *
, then no
termination check will be performed. Use of this feature is sound only with
respect to partial correctness.
8.15.3. Loop framing
The specification of a loop also includes framing, which says what the loop modifies. The loop frame includes both local variables and locations in the heap.
For local variables, the Dafny verifier performs a syntactic scan of the loop body to find every local variable or out-parameter that occurs as a left-hand side of an assignment. These variables are called syntactic assignment targets of the loop, or syntactic loop targets for short. Any local variable or out-parameter that is not a syntactic assignment target is known by the verifier to remain unchanged by the loop.
The heap may or may not be a syntactic loop target. It is when the loop body
syntactically contains a statement that can modify a heap location. This
includes calls to compiled methods, even if such a method has an empty
modifies
clause, since a compiled method is always allowed to allocate
new objects and change their values in the heap.
If the heap is not a syntactic loop target, then the verifier knows the heap
remains unchanged by the loop. If the heap is a syntactic loop target,
then the loop’s effective modifies
clause determines what is allowed to be
modified by iterations of the loop body.
A loop can use modifies
clauses to declare the effective modifies
clause
of the loop. If a loop does not explicitly declare any modifies
clause, then
the effective modifies
clause of the loop is the effective modifies
clause
of the most tightly enclosing loop or, if there is no enclosing loop, the
modifies
clause of the enclosing method.
In most cases, there is no need to give an explicit modifies
clause for a
loop. The one case where it is sometimes needed is if a loop modifies less
than is allowed by the enclosing method. Here are two simple methods that
illustrate this case:
class Cell {
var data: int
}
method M0(c: Cell, d: Cell)
requires c != d
modifies c, d
ensures c.data == d.data == 100
{
c.data, d.data := 100, 0;
var i := 0;
while i < 100
invariant d.data == i
// Needs "invariant c.data == 100" or "modifies d" to verify
{
d.data := d.data + 1;
i := i + 1;
}
}
method M1(c: Cell)
modifies c
ensures c.data == 100
{
c.data := 100;
var i := 0;
while i < 100
// Needs "invariant c.data == 100" or "modifies {}" to verify
{
var tmp := new Cell;
tmp.data := i;
i := i + 1;
}
}
In M0
, the effective modifies
clause of the loop is modifies c, d
. Therefore,
the method’s postcondition c.data == 100
is not provable. To remedy the situation,
the loop needs to be declared either with invariant c.data == 100
or with
modifies d
.
Similarly, the effective modifies
clause of the loop in M1
is modifies c
. Therefore,
the method’s postcondition c.data == 100
is not provable. To remedy the situation,
the loop needs to be declared either with invariant c.data == 100
or with
modifies {}
.
When a loop has an explicit modifies
clause, there is, at the top of
every iteration, a proof obligation that
- the expressions given in the
modifies
clause are well-formed, and - everything indicated in the loop
modifies
clause is allowed to be modified by the (effectivemodifies
clause of the) enclosing loop or method.
8.15.4. Body-less methods, functions, loops, and aggregate statements
Methods (including lemmas), functions, loops, and forall
statements are ordinarily
declared with a body, that is, a curly-braces pair that contains (for methods, loops, and forall
)
a list of zero-or-more statements or (for a function) an expression. In each case, Dafny syntactically
allows these constructs to be given without a body (no braces at all). This is to allow programmers to
temporarily postpone the development of the implementation of the method, function, loop, or
aggregate statement.
If a method has no body, there is no difference for callers of the method. Callers still reason
about the call in terms of the method’s specification. But without a body, the verifier has
no method implementation to check against the specification, so the verifier is silently happy.
The compiler, on the other hand, will complain if it encounters a body-less method, because the
compiler is supposed to generate code for the method, but it isn’t clever enough to do that by
itself without a given method body. If the method implementation is provided by code written
outside of Dafny, the method can be marked with an {:extern}
annotation, in which case the
compiler will no longer complain about the absence of a method body; the verifier will not
object either, even though there is now no proof that the Dafny specifications are satisfied
by the external implementation.
A lemma is a special kind of (ghost) method. Callers are therefore unaffected by the absence of a body,
and the verifier is silently happy with not having a proof to check against the lemma specification.
Despite a lemma being ghost, it is still the compiler that checks for, and complains about,
body-less lemmas. A body-less lemma is an unproven lemma, which is often known as an axiom.
If you intend to use a lemma as an axiom, omit its body and add the attribute {:axiom}
, which
causes the compiler to suppress its complaint about the lack of a body.
Similarly, calls to a body-less function use only the specification of the function. The
verifier is silently happy, but the compiler complains (whether or not the function is ghost).
As for methods and lemmas, the {:extern}
and {:axiom}
attributes can be used to suppress the
compiler’s complaint.
By supplying a body for a method or function, the verifier will in effect show the feasibility of
the specification of the method or function. By supplying an {:extern}
or {:axiom}
attribute,
you are taking that responsibility into your own hands. Common mistakes include forgetting to
provide an appropriate modifies
or reads
clause in the specification, or forgetting that
the results of functions in Dafny (unlike in most other languages) must be deterministic.
Just like methods and functions have two sides, callers and implementations, loops also have two sides. One side (analogous to callers) is the context that uses the loop. That context treats the loop in the same way, using its specifications, regardless of whether or not the loop has a body. The other side is the loop body, that is, the implementation of each loop iteration. The verifier checks that the loop body maintains the loop invariant and that the iterations will eventually terminate, but if there is no loop body, the verifier is silently happy. This allows you to temporarily postpone the authoring of the loop body until after you’ve made sure that the loop specification is what you need in the context of the loop.
There is one thing that works differently for body-less loops than for loops with bodies.
It is the computation of syntactic loop targets, which become part of the loop frame
(see Section 8.15.3). For a body-less loop, the local variables
computed as part of the loop frame are the mutable variables that occur free in the
loop specification. The heap is considered a part of the loop frame if it is used
for mutable fields in the loop specification or if the loop has an explicit modifies
clause.
The IDE will display the computed loop frame in hover text.
For example, consider
class Cell {
var data: int
const K: int
}
method BodylessLoop(n: nat, c: Cell)
requires c.K == 8
modifies c
{
c.data := 5;
var a, b := n, n;
for i := 0 to n
invariant c.K < 10
invariant a <= n
invariant c.data < 10
assert a == n;
assert b == n;
assert c.data == 5;
}
The loop specification mentions local variable a
, and thus a
is considered part of
the loop frame. Since what the loop invariant says about a
is not strong enough to
prove the assertion a == n
that follows the loop, the verifier complains about that
assertion.
Local variable b
is not mentioned in the loop specification, and thus b
is not
included in the loop frame. Since in-parameter n
is immutable, it is not included
in the loop frame, either, despite being mentioned in the loop specification. For
these reasons, the assertion b == n
is provable after the loop.
Because the loop specification mentions the mutable field data
, the heap becomes
part of the loop frame. Since the loop invariant is not strong enough to prove the
assertion c.data == 5
that follows the loop, the verifier complains about that
assertion. On the other hand, had c.data < 10
not been mentioned in the loop
specification, the assertion would be verified, since field K
is then the only
field mentioned in the loop specification and K
is immutable.
Finally, the aggregate statement (forall
) can also be given without a body. Such
a statement claims that the given ensures
clause holds true for all values of
the bound variables that satisfy the given range constraint. If the statement has
no body, the program is in effect omitting the proof, much like a body-less lemma
is omitting the proof of the claim made by the lemma specification. As with the
other body-less constructs above, the verifier is silently happy with a body-less
forall
statement, but the compiler will complain.
8.16. Print Statement (grammar)
Examples:
print 0, x, list, array;
The print
statement is used to print the values of a comma-separated
list of expressions to the console (standard-out). The generated code uses
target-language-specific idioms to perform this printing.
The expressions may of course include strings that are used
for captions. There is no implicit new line added, so to add a new
line you must include "\n"
as part of one of the expressions.
Dafny automatically creates implementations of methods that convert values to strings
for all Dafny data types. For example,
datatype Tree = Empty | Node(left: Tree, data: int, right: Tree)
method Main()
{
var x : Tree := Node(Node(Empty, 1, Empty), 2, Empty);
print "x=", x, "\n";
}
produces this output:
x=Tree.Node(Tree.Node(Tree.Empty, 1, Tree.Empty), 2, Tree.Empty)
Note that Dafny does not have method overriding and there is no mechanism to
override the built-in value->string conversion. Nor is there a way to
explicitly invoke this conversion.
One can always write an explicit function to convert a data value to a string
and then call it explicitly in a print
statement or elsewhere.
By default, Dafny does not keep track of print effects, but this can be changed
using the --track-print-effects
command line flag. print
statements are allowed
only in non-ghost contexts and not in expressions, with one exception.
The exception is that a function-by-method may contain print
statements,
whose effect may be observed as part of the run-time evaluation of such functions
(unless --track-print-effects
is enabled).
The verifier checks that each expression is well-defined, but otherwise
ignores the print
statement.
Note: print
writes to standard output. To improve compatibility with
native code and external libraries, the process of encoding Dafny strings passed
to print
into standard-output byte strings is left to the runtime of the
language that the Dafny code is compiled to (some language runtimes use UTF-8 in
all cases; others obey the current locale or console encoding).
In most cases, the standard-output encoding can be set before running the
compiled program using language-specific flags or environment variables
(e.g. -Dfile.encoding=
for Java). This is in fact how dafny run
operates:
it uses language-specific flags and variables to enforce UTF-8 output regardless
of the target language (but note that the C++ and Go backends currently have
limited support for UTF-16 surrogates).
8.17. Assert statement (grammar)
Examples:
assert i > 0;
assert IsPositive: i > 0;
assert i > 0 by {
...
}
Assert
statements are used to express logical propositions that are
expected to be true. Dafny will attempt to prove that the assertion
is true and give an error if the assertion cannot be proven.
Once the assertion is proved,
its truth may aid in proving subsequent deductions.
Thus if Dafny is having a difficult time verifying a method,
the user may help by inserting assertions that Dafny can prove,
and whose truth may aid in the larger verification effort,
much as lemmas might be used in mathematical proofs.
Assert
statements are ignored by the compiler.
In the by
form of the assert
statement, there is an additional block of statements that provide the Dafny verifier with additional proof steps.
Those statements are often a sequence of lemmas, calc
statements, reveal
statements or other assert
statements,
combined with ghost control flow, ghost variable declarations and ghost update statements of variables declared in the by
block.
The intent is that those statements be evaluated in support of proving the assert
statement.
For that purpose, they could be simply inserted before the assert
statement.
But by using the by
block, the statements in the block are discarded after the assertion is proved.
As a result, the statements in the block do not clutter or confuse the solver in performing subsequent
proofs of assertions later in the program. Furthermore, by isolating the statements in the by
block,
their purpose – to assist in proving the given assertion – is manifest in the structure of the code.
Examples of this form of assert are given in the section of the reveal
statement and in Different Styles of Proof
An assert statement may have a label, whose use is explained in Section 8.20.1.
The attributes recognized for assert statements are discussed in Section 11.3.
Using ...
as the argument of the statement is deprecated.
An assert statement can have custom error and success messages.
8.18. Assume Statement (grammar)
Examples:
assume i > 0;
assume {:axiom} i > 0 ==> -i < 0;
The assume
statement lets the user specify a logical proposition
that Dafny may assume to be true without proof. If in fact the
proposition is not true this may lead to invalid conclusions.
An assume
statement would ordinarily be used as part of a larger
verification effort where verification of some other part of
the program required the proposition. By using the assume
statement
the other verification can proceed. Then when that is completed the
user would come back and replace the assume
with assert
.
An assume
statement cannot be compiled. In fact, the compiler
will complain if it finds an assume
anywhere.
Using an {:axiom}
attribute makes the claim that the assume statement is
OK because it is known outside the Dafny program to be true.
The verifier will not complain about it, but it is the user’s
responsibility to be absolutely sure that the proposition is
indeed true.
Using ...
as the argument of the statement is deprecated.
8.19. Expect Statement (grammar)
Examples:
expect i > 0;
expect i > 0, "i is positive";
The expect
statement states a boolean expression that is
(a) assumed to be true by the verifier
and (b) checked to be true
at run-time. That is, the compiler inserts into the run-time executable a
check that the given expression is true; if the expression is false, then
the execution of the program halts immediately. If a second argument is
given, it may be a value of any type.
That value is converted to a string (just like the print
statement)
and the string is included
in the message emitted by the program
when it halts; otherwise a default message is emitted.
Because the expect expression and optional second argument are compiled, they cannot be ghost expressions.
The expect
statement behaves like
assume
for the verifier, but also inserts a run-time check that the
assumption is indeed correct (for the test cases used at run-time).
Here are a few use-cases for the expect
statement.
A) To check the specifications of external methods.
Consider an external method Random
that takes a nat
as input
and returns a nat
value that is less than the input.
Such a method could be specified as
method {:extern} Random(n: nat) returns (r: nat)
ensures r < n
But because there is no body for Random
(only the external non-dafny implementation),
it cannot be verified that Random
actually satisfies this specification.
To mitigate this situation somewhat, we can define a wrapper function, Random'
,
that calls Random
but in which we can put some run-time checks:
method {:extern} Random(n: nat) returns (r: nat)
method Random'(n: nat) returns (r: nat)
ensures r < n
{
r := Random(n);
expect r < n;
}
Here we can verify that Random'
satisfies its own specification,
relying on the unverified specification of Random
.
But we are also checking at run-time that any input-output pairs for Random
encountered during execution
do satisfy the specification,
as they are checked by the expect
statement.
Note, in this example, two problems still remain.
One problem is that the out-parameter of the extern Random
has type nat
,
but there is no check that the value returned really is non-negative.
It would be better to declare the out-parameter of Random
to be int
and
to include 0 <= r
in the condition checked by the expect
statement in Random'
.
The other problem is that Random
surely will need n
to be strictly positive.
This can be fixed by adding requires n != 0
to Random'
and Random
.
B) Run-time testing
Verification and run-time testing are complementary
and both have their role in assuring that software does what is intended.
Dafny can produce executables
and these can be instrumented with unit tests.
Annotating a method with the {:test}
attribute
indicates to the compiler
that it should produce target code
that is correspondingly annotated to mark the method
as a unit test (e.g., an XUnit test) in the target language.
Alternatively, the dafny test
command will produce a main method
that invokes all methods with the {:test}
attribute, and hence does not
depend on any testing framework in the target language.
Within such methods one might use expect
statements (as well as print
statements)
to insert checks that the target program is behaving as expected.
C) Debugging
While developing a new program, one work style uses proof attempts and runtime tests in combination. If an assert statement does not prove, one might run the program with a corresponding expect statement to see if there are some conditions when the assert is not actually true. So one might have paired assert/expect statements:
assert _P_;
expect _P_;
Once the program is debugged, both statements can be removed.
Note that it is important that the assert
come before the expect
, because
by the verifier, the expect
is interpreted as an assume
, which would automatically make
a subsequent assert
succeed.
D) Compiler tests
The same approach might be taken to assure that compiled code is behaving at run-time consistently with the statically verified code, one can again use paired assert/expect statements with the same expression:
assert _P_;
expect _P_;
The verifier will check that P is always true at the given point in a program
(at the assert
statement).
At run-time, the compiler will insert checks that the same predicate,
in the expect
statement, is true.
Any difference identifies a compiler bug.
Again the expect
must be after the assert
:
if the expect
is first,
then the verifier will interpret the expect
like an assume
,
in which case the assert
will be proved trivially
and potential unsoundness will be hidden.
Using ...
as the argument of the statement is deprecated.
8.20. Reveal Statement (grammar)
Examples:
reveal f(), L;
The reveal
statement makes available to the solver information that is otherwise not visible, as described in the following subsections.
8.20.1. Revealing assertions
If an assert statement has an expression label, then a proof of that assertion is attempted, but the assertion itself is not used subsequently. For example, consider
method m(i: int) {
assert x: i == 0; // Fails
assert i == 0; // Fails also because the x: makes the first assertion opaque
}
The first assertion fails. Without the label x:
, the second would succeed because after a failing assertion, the
assertion is assumed in the context of the rest of the program. But with the label, the first assertion is hidden from
the rest of the program. That assertion can be revealed by adding a reveal
statement:
method m(i: int) {
assert x: i == 0; // Fails
reveal x;
assert i == 0; // Now succeeds
}
or
method m(i: int) {
assert x: i == 0; // Fails
assert i == 0 by { reveal x; } // Now succeeds
}
At the point of the reveal
statement, the labeled assertion is made visible and can be used in proving the second assertion.
In this example there is no point to labeling an assertion and then immediately revealing it. More useful are the cases where
the reveal is in an assert-by block or much later in the method body.
8.20.2. Revealing preconditions
In the same way as assertions, preconditions can be labeled.
Within the body of a method, a precondition is an assumption; if the precondition is labeled then that assumption is not visible in the body of the method.
A reveal
statement naming the label of the precondition then makes the assumption visible.
Here is a toy example:
method m(x: int, y: int) returns (z: int)
requires L: 0 < y
ensures z == x+y
ensures x < z
{
z := x + y;
}
The above method will not verify. In particular, the second postcondition cannot be proved.
However, if we add a reveal L;
statement in the body of the method, then the precondition is visible
and both postconditions can be proved.
One could also use this style:
method m(x: int, y: int) returns (z: int)
requires L: 0 < y
ensures z == x+y
ensures x < z
{
z := x + y;
assert x < z by { reveal L; }
}
The reason to possibly hide a precondition is the same as the reason to hide assertions: sometimes less information is better for the solver as it helps the solver focus attention on relevant information.
Section 7 of http://leino.science/papers/krml276.html provides
an extended illustration of this technique to make all the dependencies of an assert
explicit.
8.20.3. Revealing function bodies
Normally function bodies are transparent and available for constructing proofs of assertions that use those functions.
However, sometimes it is helpful to mark a function {:opaque}
and treat it as an uninterpreted function, whose properties are
just its specifications. This action limits the information available to the logical reasoning engine and may make a proof
possible where there might be information overload otherwise.
But then there may be specific instances where the definition of that opaque function is needed. In that situation, the body of the function can be revealed using the reveal statement. Here is an example:
opaque function f(i: int): int { i + 1 }
method m(i: int) {
assert f(i) == i + 1;
}
Without the [opaque
] modifier, the assertion is valid; with the modifier it cannot be proved because the body of the
function is not visible. However if a reveal f();
statement is inserted before the assertion, the proof succeeds.
Note that the pseudo-function-call in the reveal
statement is written without arguments and serves to mark f
as a function name
instead of a label.
8.20.4. Revealing constants
A const
declaration can be opaque
. If so the value of the constant is not known in reasoning about its uses, just its type and the
fact that the value does not change. The constant’s identifier can be listed in a reveal statement. In that case, like other revealed items,
the value of the constant will be known to the reasoning engine until the end of the block containing the reveal statement.
A label or locally declared name in a method body will shadow an opaque constant with the same name outside the method body, making it unable to be revealed without using a qualified name.
8.21. Forall Statement (grammar)
Examples:
forall i | 0 <= i < a.Length {
a[i] := 0;
}
forall i | 0 <= i < 100 {
P(i); // P a lemma
}
forall i | 0 <= i < 100
ensures i < 1000 {
}
The forall
statement executes the body
simultaneously for all quantified values in the specified quantifier domain.
You can find more details about quantifier domains here.
There are several variant uses of the forall
statement and there are a number of restrictions.
A forall
statement can be classified as one of the following:
- Assign - the
forall
statement is used for simultaneous assignment. The target must be an array element or an object field. - Call - The body consists of a single call to a ghost method without side effects
- Proof - The
forall
hasensure
expressions which are effectively quantified or proved by the body (if present).
An assign forall
statement performs simultaneous assignment.
The left-hand sides must denote different l-values, unless the
corresponding right-hand sides also coincide.
The following is an excerpt of an example given by Leino in
Developing Verified Programs with Dafny.
When the buffer holding the queue needs to be resized,
the forall
statement is used to simultaneously copy the old contents
into the new buffer.
class SimpleQueue<Data(0)>
{
ghost var Contents: seq<Data>
var a: array<Data> // Buffer holding contents of queue.
var m: int // Index head of queue.
var n: int // Index just past end of queue
method Enqueue(d: Data)
requires a.Length > 0
requires 0 <= m <= n <= a.Length
modifies this, this.a
ensures Contents == old(Contents) + [d]
{
if n == a.Length {
var b := a;
if m == 0 { b := new Data[2 * a.Length]; }
forall i | 0 <= i < n - m {
b[i] := a[m + i];
}
a, m, n := b, 0, n - m;
}
a[n], n, Contents := d, n + 1, Contents + [d];
}
}
Here is an example of a call forall
statement and the
callee. This is contained in the CloudMake-ConsistentBuilds.dfy
test in the Dafny repository.
method m() {
forall cmd', deps', e' |
Hash(Loc(cmd', deps', e')) == Hash(Loc(cmd, deps, e)) {
HashProperty(cmd', deps', e', cmd, deps, e);
}
}
lemma HashProperty(cmd: Expression, deps: Expression, ext: string,
cmd': Expression, deps': Expression, ext': string)
requires Hash(Loc(cmd, deps, ext)) == Hash(Loc(cmd', deps', ext'))
ensures cmd == cmd' && deps == deps' && ext == ext'
The following example of a proof forall
statement comes from the same file:
forall p | p in DomSt(stCombinedC.st) && p in DomSt(stExecC.st)
ensures GetSt(p, stCombinedC.st) == GetSt(p, stExecC.st)
{
assert DomSt(stCombinedC.st) <= DomSt(stExecC.st);
assert stCombinedC.st == Restrict(DomSt(stCombinedC.st),
stExecC.st);
}
More generally, the statement
forall x | P(x) { Lemma(x); }
is used to invoke Lemma(x)
on all x
for which P(x)
holds. If
Lemma(x)
ensures Q(x)
, then the forall statement establishes
forall x :: P(x) ==> Q(x).
The forall
statement is also used extensively in the de-sugared forms of
co-predicates and co-lemmas. See datatypes.
8.22. Modify Statement (grammar)
The effect of the modify
statement
is to say that some undetermined
modifications have been made to any or all of the memory
locations specified by the given frame expressions.
In the following example, a value is assigned to field x
followed by a modify
statement that may modify any field
in the object. After that we can no longer prove that the field
x
still has the value we assigned to it. The now unknown values
still are values of their type (e.g. of the subset type or newtype).
class MyClass {
var x: int
method N()
modifies this
{
x := 18;
modify this;
assert x == 18; // error: cannot conclude this here
}
}
Using ...
as the argument of the statement is deprecated.
The form of the modify
statement which includes a block
statement is also deprecated.
The havoc assignment also sets a variable or field to some arbitrary (but type-consistent) value. The difference is that the havoc assignment acts on one LHS variable or memory location; the modify statement acts on all the fields of an object.
8.23. Calc Statement (grammar)
See also: Verified Calculations.
The calc
statement supports calculational proofs using a language
feature called program-oriented calculations (poC). This feature was
introduced and explained in the [Verified Calculations] paper by Leino
and Polikarpova[@LEINO:Dafny:Calc]. Please see that paper for a more
complete explanation of the calc
statement. We here mention only the
highlights.
Calculational proofs are proofs by stepwise formula manipulation as is taught in elementary algebra. The typical example is to prove an equality by starting with a left-hand-side and through a series of transformations morph it into the desired right-hand-side.
Non-syntactic rules further restrict hints to only ghost and side-effect free statements, as well as imposing a constraint that only chain-compatible operators can be used together in a calculation. The notion of chain-compatibility is quite intuitive for the operators supported by poC; for example, it is clear that “<” and “>” cannot be used within the same calculation, as there would be no relation to conclude between the first and the last line. See the [paper][Verified Calculations] for a more formal treatment of chain-compatibility.
Note that we allow a single occurrence of the intransitive operator “!=” to
appear in a chain of equalities (that is, “!=” is chain-compatible with
equality but not with any other operator, including itself). Calculations
with fewer than two lines are allowed, but have no effect. If a step
operator is omitted, it defaults to the calculation-wide operator,
defined after the calc
keyword. If that operator is omitted, it defaults
to equality.
Here is an example using calc
statements to prove an elementary
algebraic identity. As it turns out, Dafny is able to prove this without
the calc
statements, but the example illustrates the syntax.
lemma docalc(x : int, y: int)
ensures (x + y) * (x + y) == x * x + 2 * x * y + y * y
{
calc {
(x + y) * (x + y);
==
// distributive law: (a + b) * c == a * c + b * c
x * (x + y) + y * (x + y);
==
// distributive law: a * (b + c) == a * b + a * c
x * x + x * y + y * x + y * y;
==
calc {
y * x;
==
x * y;
}
x * x + x * y + x * y + y * y;
==
calc {
x * y + x * y;
==
// a = 1 * a
1 * x * y + 1 * x * y;
==
// Distributive law
(1 + 1) * x * y;
==
2 * x * y;
}
x * x + 2 * x * y + y * y;
}
}
Here we started with (x + y) * (x + y)
as the left-hand-side
expressions and gradually transformed it using distributive,
commutative and other laws into the desired right-hand-side.
The justification for the steps are given as comments or as
nested calc
statements that prove equality of some sub-parts
of the expression.
The ==
operators show the relation between
the previous expression and the next. Because of the transitivity of
equality we can then conclude that the original left-hand-side is
equal to the final expression.
We can avoid having to supply the relational operator between
every pair of expressions by giving a default operator between
the calc
keyword and the opening brace as shown in this abbreviated
version of the above calc statement:
lemma docalc(x : int, y: int)
ensures (x + y) * (x + y) == x * x + 2 * x * y + y * y
{
calc == {
(x + y) * (x + y);
x * (x + y) + y * (x + y);
x * x + x * y + y * x + y * y;
x * x + x * y + x * y + y * y;
x * x + 2 * x * y + y * y;
}
}
And since equality is the default operator, we could have omitted
it after the calc
keyword.
The purpose of the block statements or the calc
statements between
the expressions is to provide hints to aid Dafny in proving that
step. As shown in the example, comments can also be used to aid
the human reader in cases where Dafny can prove the step automatically.
9. Expressions
Dafny expressions come in three flavors.
- The bulk of expressions have no side-effects and can be used within methods, functions, and specifications, and in either compiled or ghost code.
- Some expressions, called right-hand-side expressions, do have side-effects and may only be used in specific syntactic locations, such as the right-hand-side of update (assignment) statements; object allocation and method calls are two typical examples of right-hand-side expressions. Note that method calls are syntactically indistinguishable from function calls; both are Expressions (PrimaryExpressions with an ArgumentList suffix). However, method calls are semantically permitted only in right-hand-side expression locations.
- Some expressions are allowed only in specifications and other ghost code, as listed here.
The grammar of Dafny expressions follows a hierarchy that reflects the precedence of Dafny operators. The following table shows the Dafny operators and their precedence in order of increasing binding power.
operator | precedence | description |
---|---|---|
; |
0 | That is LemmaCall; Expression |
<==> |
1 | equivalence (if and only if) |
==> |
2 | implication (implies) |
<== |
2 | reverse implication (follows from) |
&& , & |
3 | conjunction (and) |
|| , | |
3 | disjunction (or) |
== |
4 | equality |
==#[k] |
4 | prefix equality (coinductive) |
!= |
4 | disequality |
!=#[k] |
4 | prefix disequality (coinductive) |
< |
4 | less than |
<= |
4 | at most |
>= |
4 | at least |
> |
4 | greater than |
in |
4 | collection membership |
!in |
4 | collection non-membership |
!! |
4 | disjointness |
<< |
5 | left-shift |
>> |
5 | right-shift |
+ |
6 | addition (plus) |
- |
6 | subtraction (minus) |
* |
7 | multiplication (times) |
/ |
7 | division (divided by) |
% |
7 | modulus (mod) |
| |
8 | bit-wise or |
& |
8 | bit-wise and |
^ |
8 | bit-wise exclusive-or (not equal) |
as operation |
9 | type conversion |
is operation |
9 | type test |
- |
10 | arithmetic negation (unary minus) |
! |
10 | logical negation, bit-wise complement |
Primary Expressions | 11 |
9.1. Lemma-call expressions (grammar)
Examples:
var a := L(a,b); a*b
This expression has the form S; E
.
The type of the expression is the type of E
.
S
must be a lemma call (though the grammar appears more lenient).
The lemma introduces a fact necessary to establish properties of E
.
Sometimes an expression will fail unless some relevant fact is known.
In the following example the F_Fails
function fails to verify
because the Fact(n)
divisor may be zero. But preceding
the expression by a lemma that ensures that the denominator
is not zero allows function F_Succeeds
to succeed.
function Fact(n: nat): nat
{
if n == 0 then 1 else n * Fact(n-1)
}
lemma L(n: nat)
ensures 1 <= Fact(n)
{
}
function F_Fails(n: nat): int
{
50 / Fact(n) // error: possible division by zero
}
function F_Succeeds(n: nat): int
{
L(n); // note, this is a lemma call in an expression
50 / Fact(n)
}
One restriction is that a lemma call in this form is permitted only in situations in which the expression itself is not terminated by a semicolon.
A second restriction is that E
is not always permitted to contain lambda expressions, such
as in the expressions that are the body of a lambda expression itself, function, method and iterator specifications,
and if and while statements with guarded alternatives.
A third restriction is that E
is not always permitted to contain a bit-wise or (|
) operator,
because it would be ambiguous with the vertical bar used in comprehension expressions.
Note that the effect of the lemma call only extends to the succeeding expression E
(which may be another ;
expression).
9.2. Equivalence Expressions (grammar)
Examples:
A
A <==> B
A <==> C ==> D <==> B
An Equivalence Expression that contains one or more <==>
s is
a boolean expression and all the operands
must also be boolean expressions. In that case each <==>
operator tests for logical equality which is the same as
ordinary equality (but with a different precedence).
See Section 5.2.1.1 for an explanation of the
<==>
operator as compared with the ==
operator.
The <==>
operator is commutative and associative: A <==> B <==> C
and (A <==> B) <==> C
and A <==> (B <==> C)
and C <==> B <==> A
are all equivalent and are all true iff an even number of operands are false.
9.3. Implies or Explies Expressions (grammar)
Examples:
A ==> B
A ==> B ==> C ==> D
B <== A
See Section 5.2.1.3 for an explanation
of the ==>
and <==
operators.
9.4. Logical Expressions (grammar)
Examples:
A && B
A || B
&& A && B && C
Note that the Dafny grammar allows a conjunction or disjunction to be
prefixed with &&
or ||
respectively. This form simply allows a
parallel structure to be written:
method m(x: object?, y:object?, z: object?) {
var b: bool :=
&& x != null
&& y != null
&& z != null
;
}
This is purely a syntactic convenience allowing easy edits such as reordering lines or commenting out lines without having to check that the infix operators are always where they should be.
Note also that &&
and ||
cannot be mixed without using parentheses:
A && B || C
is not permitted. Write (A && B) || C
or A && (B || C)
instead.
See Section 5.2.1.2 for an explanation
of the &&
and ||
operators.
9.5. Relational Expressions (grammar)
Examples:
x == y
x != y
x < y
x >= y
x in y
x ! in y
x !! y
x ==#[k] y
The relation expressions compare two or more terms.
As explained in the section about basic types, ==
, !=
, <
, >
, <=
, and >=
are chaining.
The in
and !in
operators apply to collection types as explained in
Section 5.5 and represent membership or non-membership
respectively.
The !!
represents disjointness for sets and multisets as explained in
Section 5.5.1 and Section 5.5.2.
x ==#[k] y
is the prefix equality operator that compares
coinductive values for equality to a nesting level of k, as
explained in the section about co-equality.
9.6. Bit Shifts (grammar)
Examples:
k << 5
j >> i
These operators are the left and right shift operators for bit-vector values.
They take a bit-vector value and an int
, shifting the bits by the given
amount; the result has the same bit-vector type as the LHS.
For the expression to be well-defined, the RHS value must be in the range 0 to the number of
bits in the bit-vector type, inclusive.
The operations are left-associative: a << i >> j
is (a << i) >> j
.
9.7. Terms (grammar)
Examples:
x + y - z
Terms
combine Factors
by adding or subtracting.
Addition has these meanings for different types:
- arithmetic addition for numeric types (Section 5.2.2])
- union for sets and multisets (Section 5.5.1 and Section 5.5.2)
- concatenation for sequences (Section 5.5.3)
- map merging for maps (Section 5.5.4)
Subtraction is
- arithmetic subtraction for numeric types
- set or multiset subtraction for sets and multisets
- domain subtraction for maps.
All addition operations are associative. Arithmetic addition and union are commutative. Subtraction is neither; it groups to the left as expected:
x - y -z
is (x - y) -z
.
9.8. Factors (grammar)
Examples:
x * y
x / y
x % y
A Factor
combines expressions using multiplication,
division, or modulus. For numeric types these are explained in
Section 5.2.2.
As explained there, /
and %
on int
values represent Euclidean
integer division and modulus and not the typical C-like programming
language operations.
Only *
has a non-numeric application. It represents set or multiset
intersection as explained in Section 5.5.1 and Section 5.5.2.
*
is commutative and associative; /
and %
are neither but do group to the left.
9.9. Bit-vector Operations (grammar)
Examples:
x | y
x & y
x ^ y
These operations take two bit-vector values of the same type, returning
a value of the same type. The operations perform bit-wise or (|
),
and (&
), and exclusive-or (^
). To perform bit-wise equality, use
^
and !
(unary complement) together. (==
is boolean equality of the whole bit-vector.)
These operations are associative and commutative but do not associate with each other.
Use parentheses: a & b | c
is illegal; use (a & b) | c
or a & (b | c)
instead.
Bit-vector operations are not allowed in some contexts.
The |
symbol is used both for bit-wise or and as the delimiter in a
cardinality expression: an ambiguity arises if
the expression E in | E |
contains a |
. This situation is easily
remedied: just enclose E in parentheses, as in |(E)|
.
The only type-correct way this can happen is if the expression is
a comprehension, as in | set x: int :: x | 0x101 |
.
9.10. As (Conversion) and Is (type test) Expressions (grammar)
Examples:
e as MyClass
i as bv8
e is MyClass
The as
expression converts the given LHS to the type stated on the RHS,
with the result being of the given type. The following combinations
of conversions are permitted:
- Any type to itself
- Any int or real based numeric type or bit-vector type to another int or real based numeric type or bit-vector type
- Any base type to a subset or newtype with that base
- Any subset or newtype to its base type or a subset or newtype of the same base
- Any type to a subset or newtype that has the type as its base
- Any trait to a class or trait that extends (perhaps recursively) that trait
- Any class or trait to a trait extended by that class or trait
Some of the conversions above are already implicitly allowed, without the
as
operation, such as from a subset type to its base. In any case, it
must be able to be proved that the value of the given expression is a
legal value of the given type. For example, 5 as MyType
is permitted (by the verifier) only if 5
is a legitimate value ofMyType
(which must be a numeric type).
The as
operation is like a grammatical suffix or postfix operation.
However, note that the unary operations bind more tightly than does as
.
That is - 5 as nat
is (- 5) as nat
(which fails), whereas a * b as nat
is a * (b as nat)
. On the other hand, - a[4]
is - (a[4])
.
The is
expression is grammatically similar to the as
expression, with the
same binding power. The is
expression is a type test that
returns a bool
value indicating whether the LHS expression is a legal
value of the RHS type. The expression can be used to check
whether a trait value is of a particular class type. That is, the expression
in effect checks the allocated type of a trait.
The RHS type of an is
expression can always be a supertype of the type of the LHS
expression, in which case the result is trivally true.
Other than that, the RHS must be based on a reference type and the
LHS expression must be assignable to the RHS type. Furthermore, in order to be
compilable, the RHS type must not be a subset type other than a non-null reference
type, and the type parameters of the RHS must be uniquely determined from the
type parameters of the LHS type. The last restriction is designed to make it
possible to perform type tests without inspecting type parameters at run time.
For example, consider the following types:
trait A { }
trait B<X> { }
class C<Y> extends B<Y> { }
class D<Y(==)> extends B<set<Y>> { }
class E extends B<int> { }
class F<Z> extends A { }
A LHS expression of type B<set<int>>
can be used in a type test where the RHS is
B<set<int>>
, C<set<int>>
, or D<int>
, and a LHS expression of type B<int>
can be used in a type test where the RHS is B<int>
, C<int>
, or E
. Those
are always allowed in compiled (and ghost) contexts.
For an expression a
of type A
, the expression a is F<int>
is a ghost expression;
it can be used in ghost contexts, but not in compiled contexts.
For an expression e
and type t
, e is t
is the condition determining whether
e as t
is well-defined (but, as noted above, is not always a legal expression).
The repertoire of types allowed in is
tests may be expanded in the future.
9.11. Unary Expressions (grammar)
Examples:
-x
- - x
! x
A unary expression applies
- logical complement (
!
– Section 5.2.1), - bit-wise complement (
!
– Section 5.2.3), - numeric negation (
-
– Section 5.2.2), or - bit-vector negation (
-
– Section 5.2.3)
to its operand.
9.12. Primary Expressions (grammar)
Examples:
true
34
M(i,j)
b.c.d
[1,2,3]
{2,3,4}
map[1 => 2, 3 => 4]
(i:int,j:int)=>i+j
if b then 4 else 5
After descending through all the binary and unary operators we arrive at the primary expressions, which are explained in subsequent sections. A number of these can be followed by 0 or more suffixes to select a component of the value.
9.13. Lambda expressions (grammar)
Examples:
x => -x
_ => true
(x,y) => x*y
(x:int, b:bool) => if b then x else -x
x requires x > 0 => x-1
See Section 7.4 for a description of specifications for lambda expressions.
In addition to named functions, Dafny supports expressions that define functions. These are called lambda (expression)s (some languages know them as anonymous functions). A lambda expression has the form:
( _params_ ) _specification_ => _body_
where params is a comma-delimited list of parameter
declarations, each of which has the form x
or x: T
. The type T
of a parameter can be omitted when it can be inferred. If the
identifier x
is not needed, it can be replaced by _
. If
params consists of a single parameter x
(or _
) without an
explicit type, then the parentheses can be dropped; for example, the
function that returns the successor of a given integer can be written
as the following lambda expression:
x => x + 1
The specification is a list of clauses requires E
or
reads W
, where E
is a boolean expression and W
is a frame
expression.
body is an expression that defines the function’s return
value. The body must be well-formed for all possible values of the
parameters that satisfy the precondition (just like the bodies of
named functions and methods). In some cases, this means it is
necessary to write explicit requires
and reads
clauses. For
example, the lambda expression
x requires x != 0 => 100 / x
would not be well-formed if the requires
clause were omitted,
because of the possibility of division-by-zero.
In settings where functions cannot be partial and there are no
restrictions on reading the heap, the eta expansion of a function
F: T -> U
(that is, the wrapping of F
inside a lambda expression
in such a way that the lambda expression is equivalent to F
) would
be written x => F(x)
. In Dafny, eta expansion must also account for
the precondition and reads set of the function, so the eta expansion
of F
looks like:
x requires F.requires(x) reads F.reads(x) => F(x)
9.14. Left-Hand-Side Expressions (grammar)
Examples:
x
a[k]
LibraryModule.F().x
old(o.f).x
A left-hand-side expression is only used on the left hand side of an Update statement or an Update with Failure Statement.
An LHS can be
- a simple identifier:
k
- an expression with a dot suffix:
this.x
,f(k).y
- an expression with an array selection:
a[k]
,f(a8)[6]
9.15. Right-Hand-Side Expressions (grammar)
Examples:
new int[6]
new MyClass
new MyClass(x,y,z)
x+y+z
*
A Right-Hand-Side expression is an expression-like construct that may have side-effects. Consequently such expressions can only be used within certain statements within methods, and not as general expressions or within functions or specifications.
An RHS is either an array allocation, an object allocation, a havoc right-hand-side, a method call, or a simple expression, optionally followed by one or more attributes.
Right-hand-side expressions (that are not just regular expressions) appear in the following constructs:
- return statements,
- yield statements,
- update statements,
- update-with-failure statements, or
- variable declaration statements.
These are the only contexts in which arrays or objects may be allocated or in which havoc may be stipulated.
9.16. Array Allocation (grammar)
Examples:
new int[5,6]
new int[5][2,3,5,7,11]
new int[][2,3,5,7,11]
new int[5](i => i*i)
new int[2,3]((i,j) => i*j)
This right-hand-side expression allocates a new single or multi-dimensional array (cf. Section 5.10). The initialization portion is optional. One form is an explicit list of values, in which case the dimension is optional:
var a := new int[5];
var b := new int[5][2,3,5,7,11];
var c := new int[][2,3,5,7,11];
var d := new int[3][4,5,6,7]; // error
The comprehension form requires a dimension and uses a function of
type nat -> T
where T
is the array element type:
var a := new int[5](i => i*i);
To allocate a multi-dimensional array, simply give the sizes of each dimension. For example,
var m := new real[640, 480];
allocates a 640-by-480 two-dimensional array of real
s. The initialization
portion cannot give a display of elements like in the one-dimensional
case, but it can use an initialization function. A function used to initialize
a n-dimensional array requires a function from n nat
s to a T
, where T
is the element type of the array. Here is an example:
var diag := new int[30, 30]((i, j) => if i == j then 1 else 0);
Array allocation is permitted in ghost contexts. If any expression
used to specify a dimension or initialization value is ghost, then the
new
allocation can only be used in ghost contexts. Because the
elements of an array are non-ghost, an array allocated in a ghost
context in effect cannot be changed after initialization.
9.17. Object Allocation (grammar)
Examples:
new MyClass
new MyClass.Init
new MyClass.Init(1,2,3)
This right-hand-side expression allocates a new object of a class type as explained in section Class Types.
9.18. Havoc Right-Hand-Side (grammar)
Examples:
*
A havoc right-hand-side is just a *
character.
It produces an arbitrary value of its associated
type. The “assign-such-that”
operator (:|
) can be used to obtain a more constrained arbitrary value.
See Section 8.5.
9.19. Constant Or Atomic Expressions (grammar)
Examples:
this
null
5
5.5
true
'a'
"dafny"
( e )
| s |
old(x)
allocated(x)
unchanged(x)
fresh(e)
These expressions are never l-values. They include
- literal expressions
- parenthesized expressions
this
expressions- fresh expressions
- allocated expressions
- unchanged expressions
- old expressions
- cardinality expressions
9.20. Literal Expressions (grammar}
Examples:
5
5.5
true
'a'
"dafny"
A literal expression is a null object reference or a boolean, integer, real, character or string literal.
9.21. this
Expression (grammar)
Examples:
this
The this
token denotes the current object in the context of
a constructor, instance method, or instance function.
9.22. Old and Old@ Expressions (grammar)
Examples:
old(c)
old@L(c)
An old expression is used in postconditions or in the body of a method or in the body or specification of any two-state function or two-state lemma; an old expression with a label is used only in the body of a method at a point where the label dominates its use in the expression.
old(e)
evaluates
the argument using the value of the heap on entry to the method;
old@ident(e)
evaluates the argument using the value of the heap at the
given statement label.
Note that old and old@ only affect heap dereferences,
like o.f
and a[i]
.
In particular, neither form has any effect on the value returned for local
variables or out-parameters (as they are not on the heap).10
If the value of an entire expression at a
particular point in the method body is needed later on in the method body,
the clearest means is to declare a ghost variable, initializing it to the
expression in question.
If the argument of old
is a local variable or out-parameter. Dafny issues a warning.
The argument of an old
expression may not contain nested old
,
fresh
,
or unchanged
expressions,
nor two-state functions or two-state lemmas.
Here are some explanatory examples. All assert
statements verify to be true.
class A {
var value: int
method m(i: int)
requires i == 6
requires value == 42
modifies this
{
var j: int := 17;
value := 43;
label L:
j := 18;
value := 44;
label M:
assert old(i) == 6; // i is local, but can't be changed anyway
assert old(j) == 18; // j is local and not affected by old
assert old@L(j) == 18; // j is local and not affected by old
assert old(value) == 42;
assert old@L(value) == 43;
assert old@M(value) == 44 && this.value == 44;
// value is this.value; 'this' is the same
// same reference in current and pre state but the
// values stored in the heap as its fields are different;
// '.value' evaluates to 42 in the pre-state, 43 at L,
// and 44 in the current state
}
}
class A {
var value: int
constructor ()
ensures value == 10
{
value := 10;
}
}
class B {
var a: A
constructor () { a := new A(); }
method m()
requires a.value == 11
modifies this, this.a
{
label L:
a.value := 12;
label M:
a := new A(); // Line X
label N:
a.value := 20;
label P:
assert old(a.value) == 11;
assert old(a).value == 12; // this.a is from pre-state,
// but .value in current state
assert old@L(a.value) == 11;
assert old@L(a).value == 12; // same as above
assert old@M(a.value) == 12; // .value in M state is 12
assert old@M(a).value == 12;
assert old@N(a.value) == 10; // this.a in N is the heap
// reference at Line X
assert old@N(a).value == 20; // .value in current state is 20
assert old@P(a.value) == 20;
assert old@P(a).value == 20;
}
}
class A {
var value: int
constructor ()
ensures value == 10
{
value := 10;
}
}
class B {
var a: A
constructor () { a := new A(); }
method m()
requires a.value == 11
modifies this, this.a
{
label L:
a.value := 12;
label M:
a := new A(); // Line X
label N:
a.value := 20;
label P:
assert old(a.value) == 11;
assert old(a).value == 12; // this.a is from pre-state,
// but .value in current state
assert old@L(a.value) == 11;
assert old@L(a).value == 12; // same as above
assert old@M(a.value) == 12; // .value in M state is 12
assert old@M(a).value == 12;
assert old@N(a.value) == 10; // this.a in N is the heap
// reference at Line X
assert old@N(a).value == 20; // .value in current state is 20
assert old@P(a.value) == 20;
assert old@P(a).value == 20;
}
}
The next example demonstrates the interaction between old
and array elements.
class A {
var z1: array<nat>
var z2: array<nat>
method mm()
requires z1.Length > 10 && z1[0] == 7
requires z2.Length > 10 && z2[0] == 17
modifies z2
{
var a: array<nat> := z1;
assert a[0] == 7;
a := z2;
assert a[0] == 17;
assert old(a[0]) == 17; // a is local with value z2
z2[0] := 27;
assert old(a[0]) == 17; // a is local, with current value of
// z2; in pre-state z2[0] == 17
assert old(a)[0] == 27; // a is local, so old(a) has no effect
}
}
9.23. Fresh Expressions (grammar)
Examples:
fresh(e)
fresh@L(e)
fresh(e)
returns a boolean value that is true if
the objects denoted by expression e
were all
freshly allocated since the time of entry to the enclosing method,
or since label L:
in the variant fresh@L(e)
.
The argument is an object or set of objects.
For example, consider this valid program:
class C { constructor() {} }
method f(c1: C) returns (r: C)
ensures fresh(r)
{
assert !fresh(c1);
var c2 := new C();
label AfterC2:
var c3 := new C();
assert fresh(c2) && fresh(c3);
assert fresh({c2, c3});
assert !fresh@AfterC2(c2) && fresh@AfterC2(c3);
r := c2;
}
The L
in the variant fresh@L(e)
must denote a label that, in the
enclosing method’s control flow, dominates the expression. In this
case, fresh@L(e)
returns true
if the objects denoted by e
were all
freshly allocated since control flow reached label L
.
The argument of fresh
must be either an object
reference
or a set or sequence of object references.
In this case, fresh(e)
(respectively fresh@L(e)
with a label)
is a synonym of old(!allocated(e))
(respectively old@L(!allocated(e))
)
9.24. Allocated Expressions (grammar)
Examples:
allocated(c)
allocated({c1,c2})
allocated@L(c)
For any expression e
, the expression allocated(e)
evaluates to true
in a state if the value of e
is available in that state, meaning that
it could in principle have been the value of a variable in that state.
For example, consider this valid program:
class C { constructor() {} }
datatype D = Nil | Cons(C, D)
method f() {
var d1, d2 := Nil, Nil;
var c1 := new C();
label L1:
var c2 := new C();
label L2:
assert old(allocated(d1) && allocated(d2));
d1 := Cons(c1, Nil);
assert old(!allocated(d1) && allocated(d2));
d2 := Cons(c2, Nil);
assert old(!allocated(d1) && !allocated(d2));
assert allocated(d1) && allocated(d2);
assert old@L1(allocated(d1) && !allocated(d2));
assert old@L2(allocated(d1) && allocated(d2));
d1 := Nil;
assert old(allocated(d1) && !allocated(d2));
}
This can be useful when, for example, allocated(e)
is evaluated in an
old
state. Like in the example, where d1
is a local variable holding a datatype value
Cons(c1, Nil)
where c1
is an object that was allocated in the enclosing
method, then old(allocated(d))
is false
.
If the expression e
is of a reference type, then !old(allocated(e))
is the same as fresh(e)
.
9.25. Unchanged Expressions (grammar)
Examples:
unchanged(c)
unchanged([c1,c2])
unchanged@L(c)
The unchanged
expression returns true
if and only if every reference
denoted by its arguments has the same value for all its fields in the
old and current state. For example, if c
is an object with two
fields, x
and y
, then unchanged(c)
is equivalent to
c.x == old(c.x) && c.y == old(c.y)
Each argument to unchanged
can be a reference, a set of references, or
a sequence of references, each optionally followed by a back-tick and field name.
This form with a frame field expresses that just the field f
,
not necessarily all fields, has the same value in the old and current
state.
If there is such a frame field, all the references must have the same type,
which must have a field of that name.
The optional @
-label says to use the state at that label as the old-state instead of using
the old
state (the pre-state of the method). That is, using the example c
from above, the expression
unchanged@Lbl(c)
is equivalent to
c.x == old@Lbl(c.x) && c.y == old@Lbl(c.y)
Each reference denoted by the arguments of unchanged
must be non-null and
must be allocated in the old-state of the expression.
9.26. Cardinality Expressions (grammar)
Examples:
|s|
|s[1..i]|
For a finite-collection expression c
, |c|
is the cardinality of c
. For a
finite set or sequence, the cardinality is the number of elements. For
a multiset, the cardinality is the sum of the multiplicities of the
elements. For a finite map, the cardinality is the cardinality of the
domain of the map. Cardinality is not defined for infinite sets or infinite maps.
For more information, see Section 5.5.
9.27. Parenthesized Expressions (grammar)
A parenthesized expression is a list of zero or more expressions enclosed in parentheses.
If there is exactly one expression enclosed then the value is just the value of that expression.
If there are zero or more than one, the result is a tuple
value.
See Section 5.13.
9.28. Sequence Display Expression (grammar)
Examples:
[1, 2, 3]
[1]
[]
seq(k, n => n+1)
A sequence display expression provides a way to construct a sequence with given values. For example
[1, 2, 3]
is a sequence with three elements in it.
seq(k, n => n+1)
is a sequence of k elements whose values are obtained by evaluating the second argument (a function, in this case a lambda expression) on the indices 0 up to k.
See this section for more information on sequences.
9.29. Set Display Expression (grammar)
Examples:
{}
{1,2,3}
iset{1,2,3,4}
multiset{1,2,2,3,3,3}
multiset(s)
A set display expression provides a way of constructing a set with given
elements. If the keyword iset
is present, then a potentially infinite
set (with the finite set of given elements) is constructed.
For example
{1, 2, 3}
is a set with three elements in it. See Section 5.5.1 for more information on sets.
A multiset display expression provides a way of constructing a multiset with given elements and multiplicities. For example
multiset{1, 1, 2, 3}
is a multiset with three elements in it. The number 1 has a multiplicity of 2, and the numbers 2 and 3 each have a multiplicity of 1.
A multiset cast expression converts a set or a sequence into a multiset as shown here:
var s : set<int> := {1, 2, 3};
var ms : multiset<int> := multiset(s);
ms := ms + multiset{1};
var sq : seq<int> := [1, 1, 2, 3];
var ms2 : multiset<int> := multiset(sq);
assert ms == ms2;
Note that multiset{1, 1}
is a multiset holding the value 1
with multiplicity 2,
but in multiset({1,1})
the multiplicity is 1, because the expression {1,1}
is the set {1}
,
which is then converted to a multiset.
See Section 5.5.2 for more information on multisets.
9.30. Map Display Expression (grammar)
Examples:
map[]
map[1 := "a", 2 := "b"]
imap[1 := "a", 2 := "b"]
A map display expression builds a finite or potentially infinite map from explicit mappings. For example:
const m := map[1 := "a", 2 := "b"]
ghost const im := imap[1 := "a", 2 := "b"]
See Section 5.5.4 for more details on maps and imaps.
9.31. Endless Expression (grammar)
Endless expression gets it name from the fact that all its alternate productions have no terminating symbol to end them, but rather they all end with an arbitrary expression at the end. The various endless expression alternatives are described in the following subsections.
9.31.1. If Expression (grammar)
Examples:
if c then e1 else e2
if x: int :| P(x) then x else 0
An if expression is a conditional (ternary) expression. It first evaluates
the condition expression that follows the if
. If the condition evaluates to true
then
the expression following the then
is evaluated and its value is the
result of the expression. If the condition evaluates to false
then the
expression following the else
is evaluated and that value is the result
of the expression. It is important that only the selected expression
is evaluated as the following example shows.
var k := 10 / x; // error, may divide by 0.
var m := if x != 0 then 10 / x else 1; // ok, guarded
The if
expression also permits a binding form.
In this case the condition of the if
is an existential asking
“does there exist a value satisfying the given predicate?”.
If not, the else branch is evaluated. But if so, then an
(arbitrary) value that does satisfy the given predicate is
bound to the given variable and that variable is in scope in
the then-branch of the expression.
For example, in the code
predicate P(x: int) {
x == 5 || x == -5
}
method main() {
assert P(5);
var y := if x: int :| P(x) then x else 0;
assert y == 5 || y == -5;
}
x
is given some value that satisfies P(x)
, namely either 5
or -5
.
That value of x
is the value of the expression in the then
branch above; if there is no value satisfying P(x)
,
then 0
is returned. Note that if x
is declared to be a nat
in this example, then only
the value 5
would be permissible.
This binding form of the if
expression acts in the same way as the binding form of the if
statement.
In the example given, the binder for x
has no constraining range, so the expression is ghost
;
if a range is given, such as var y := if x: int :| 0 <= x < 10 && P(x) then x else 0;
,
then the if
and y
are no longer ghost, and y
could be used, for example, in a print
statement.
9.31.2. Case and Extended Patterns (grammar)
Patterns are used for (possibly nested) pattern matching on inductive, coinductive or base type values. They are used in match statements, match expressions, let expressions, and variable declarations. The match expressions and statements allow literals, symbolic constants, and disjunctive (“or”) patterns.
When matching an inductive or coinductive value in a match statement or expression, the pattern must correspond to one of the following:
- (0) a case disjunction (“or-pattern”)
- (1) bound variable (a simple identifier),
- (2) a constructor of the type of the value,
- (3) a literal of the correct type, or
- (4) a symbolic constant.
If the extended pattern is
- a sequence of
|
-separated sub-patterns, then the pattern matches values matched by any of the sub-patterns. - a parentheses-enclosed possibly-empty list of patterns, then the pattern matches a tuple.
- an identifier followed by a parentheses-enclosed possibly-empty list of patterns, then the pattern matches a constructor.
- a literal, then the pattern matches exactly that literal.
- a simple identifier, then the pattern matches
- a parameter-less constructor if there is one defined with the correct type and the given name, else
- the value of a symbolic constant, if a name lookup finds a declaration for a constant with the given name (if the name is declared but with a non-matching type, a type resolution error will occur),
- otherwise, the identifier is a new bound variable
Disjunctive patterns may not bind variables, and may not be nested inside other patterns.
Any patterns inside the parentheses of a constructor (or tuple) pattern are then matched against the arguments that were given to the constructor when the value was constructed. The number of patterns must match the number of parameters to the constructor (or the arity of the tuple).
When matching a value of base type, the pattern should either be a literal expression of the same type as the value, or a single identifier matching all values of this type.
Patterns may be nested. The bound variable identifiers contained in all the patterns must be distinct. They are bound to the corresponding values in the value being matched. (Thus, for example, one cannot repeat a bound variable to attempt to match a constructor that has two identical arguments.)
9.31.3. Match Expression (grammar)
A match expression is used to conditionally evaluate and select an expression depending on the value of an algebraic type, i.e. an inductive type, a coinductive type, or a base type.
All of the variables in the patterns must be distinct. If types for the identifiers are not given then types are inferred from the types of the constructor’s parameters. If types are given then they must agree with the types of the corresponding parameters.
The expression following the match
keyword is called the
selector. A match expression is evaluated by first evaluating the selector.
The patterns of each match alternative are then compared, in order,
with the resulting value until a matching pattern is found, as described in
the section on case bindings.
If the constructor had
parameters, then the actual values used to construct the selector
value are bound to the identifiers in the identifier list.
The expression to the right of the =>
in the matched alternative is then
evaluated in the environment enriched by this binding. The result
of that evaluation is the result of the match expression.
Note that the braces enclosing the sequence of match alternatives may be omitted. Those braces are required if lemma or lambda expressions are used in the body of any match alternative; they may also be needed for disambiguation if there are nested match expressions.
9.31.4. Quantifier Expression (grammar)
Examples:
forall x: int :: x > 0
forall x: nat | x < 10 :: x*x < 100
exists x: int :: x * x == 25
A quantifier expression is a boolean expression that specifies that a
given expression (the one following the ::
) is true for all (for
forall) or some (for exists) combination of values of the
quantified variables, namely those in the given quantifier domain.
See Section 2.7.4 for more details on quantifier domains.
Here are some examples:
assert forall x : nat | x <= 5 :: x * x <= 25;
(forall n :: 2 <= n ==> (exists d :: n < d < 2*n))
assert forall x: nat | 0 <= x < |s|, y <- s[x] :: y < x;
The quantifier identifiers are bound within the scope of the expressions in the quantifier expression.
If types are not given for the quantified identifiers, then Dafny attempts to infer their types from the context of the expressions. It this is not possible, the program is in error.
9.31.5. Set Comprehension Expressions (grammar)
Examples:
const c1 := set x: nat | x < 100
const c2 := set x: nat | x < 100 :: x * x
const c3 := set x: nat, y: nat | x < y < 100 :: x * y
ghost const c4 := iset x: nat | x > 100
ghost const c5: iset<int> := iset s
const c6 := set x <- c3 :: x + 1
A set comprehension expression is an expression that yields a set
(possibly infinite only if iset
is used) that
satisfies specified conditions. There are two basic forms.
If there is only one quantified variable, the optional "::" Expression
need not be supplied, in which case it is as if it had been supplied
and the expression consists solely of the quantified variable.
That is,
set x : T | P(x)
is equivalent to
set x : T | P(x) :: x
For the full form
var S := set x1: T1 <- C1 | P1(x1),
x2: T2 <- C2 | P2(x1, x2),
...
:: Q(x1, x2, ...)
the elements of S
will be all values resulting from evaluation of Q(x1, x2, ...)
for all combinations of quantified variables x1, x2, ...
(from their respective C1, C2, ...
domains) such that all predicates P1(x1), P2(x1, x2), ...
hold.
For example,
var S := set x:nat, y:nat | x < y < 3 :: (x, y)
yields S == {(0, 1), (0, 2), (1, 2) }
The types on the quantified variables are optional and if not given Dafny
will attempt to infer them from the contexts in which they are used in the
various expressions. The <- C
domain expressions are also optional and default to
iset x: T
(i.e. all values of the variable’s type), as are the | P
expressions which
default to true
. See also Section 2.7.4 for more details on quantifier domains.
If a finite set was specified (“set” keyword used), Dafny must be able to prove that the result is finite otherwise the set comprehension expression will not be accepted.
Set comprehensions involving reference types such as
set o: object
are allowed in ghost expressions within methods, but not in ghost functions11. In particular, in ghost contexts, the check that the result is finite should allow any set comprehension where the bound variable is of a reference type. In non-ghost contexts, it is not allowed, because–even though the resulting set would be finite–it is not pleasant or practical to compute at run time.
The universe in which set comprehensions are evaluated is the set of all allocated objects, of the appropriate type and satisfying the given predicate. For example, given
class I {
var i: int
}
method test() {
ghost var m := set x: I :: 0 <= x.i <= 10;
}
the set m
contains only those instances of I
that have been allocated
at the point in program execution that test
is evaluated. This could be
no instances, one per value of x.i
in the stated range, multiple instances
of I
for each value of x.i
, or any other combination.
9.31.6. Statements in an Expression (grammar)
Examples:
assert x != 0; 10/x
assert x != 0; assert y > 0; y/x
assume x != 0; 10/x
expect x != 0; 10/x
reveal M.f; M.f(x)
calc { x * 0; == 0; } x/1;
A StmtInExpr
is a kind of statement that is allowed to
precede an expression in order to ensure that the expression
can be evaluated without error. For example:
assume x != 0; 10/x
Assert
, assume
, expect
, reveal
and calc
statements can be used in this way.
9.31.7. Let and Let or Fail Expression (grammar)
Examples:
var x := f(y); x*x
var x :- f(y); x*x
var x :| P(x); x*x
var (x, y) := T(); x + y // T returns a tuple
var R(x,y) := T(); x + y // T returns a datatype value R
A let
expression allows binding of intermediate values to identifiers
for use in an expression. The start of the let
expression is
signaled by the var
keyword. They look much like a local variable
declaration except the scope of the variable only extends to the
enclosed expression.
For example:
var sum := x + y; sum * sum
In the simple case, the pattern is just an identifier with optional type (which if missing is inferred from the rhs).
The more complex case allows destructuring of constructor expressions. For example:
datatype Stuff = SCons(x: int, y: int) | Other
function GhostF(z: Stuff): int
requires z.SCons?
{
var SCons(u, v) := z; var sum := u + v; sum * sum
}
The Let expression has a failure variant
that simply uses :-
instead of :=
. This Let-or-Fail expression also permits propagating
failure results. However, in statements (Section 8.6), failure results in
immediate return from the method; expressions do not have side effects or immediate return
mechanisms. Rather, if the expression to the right of :-
results in a failure value V
,
the overall expression returns V.PropagateFailure()
; if there is no failure, the expression following the
semicolon is returned. Note that these two possible return values must have the same type (or be
implicitly convertible to the same type). Typically that means that tmp.PropagateFailure()
is a failure value and
E
is a value-carrying success value, both of the same failure-compatible type,
as described in Section 8.6.
The expression :- V; E
is desugared into the expression
var tmp := V;
if tmp.IsFailure()
then tmp.PropagateFailure()
else E
The expression var v :- V; E
is desugared into the expression
var tmp := V;
if tmp.IsFailure()
then tmp.PropagateFailure()
else var v := tmp.Extract(); E
If the RHS is a list of expressions then the desugaring is similar. var v, v1 :- V, V1; E
becomes
var tmp := V;
if tmp.IsFailure()
then tmp.PropagateFailure()
else var v, v1 := tmp.Extract(), V1; E
So, if tmp is a failure value, then a corresponding failure value is propagated along; otherwise, the expression is evaluated as normal.
9.31.8. Map Comprehension Expression (grammar)
Examples:
map x : int | 0 <= x <= 10 :: x * x;
map x : int | 0 <= x <= 10 :: -x := x * x;
imap x : int | 10 < x :: x * x;
A map comprehension expression defines a finite or infinite map value by defining a domain and for each value in the domain, giving the mapped value using the expression following the “::”. See Section 2.7.4 for more details on quantifier domains.
For example:
function square(x : int) : int { x * x }
method test()
{
var m := map x : int | 0 <= x <= 10 :: x * x;
ghost var im := imap x : int :: x * x;
ghost var im2 := imap x : int :: square(x);
}
Dafny finite maps must be finite, so the domain must be constrained to be finite. But imaps may be infinite as the examples show. The last example shows creation of an infinite map that gives the same results as a function.
If the expression includes the :=
token, that token separates
domain values from range values. For example, in the following code
method test()
{
var m := map x : int | 1 <= x <= 10 :: 2*x := 3*x;
}
m
maps 2
to 3
, 4
to 6
, and so on.
9.32. Name Segment (grammar)
Examples:
I
I<int,C>
I#[k]
I#<int>[k]
A name segment names a Dafny entity by giving its declared name optionally followed by information to make the name more complete. For the simple case, it is just an identifier. Note that a name segment may be followed by suffixes, including the common ‘.’ and further name segments.
If the identifier is for a generic entity, it is followed by
a GenericInstantiation
which provides actual types for
the type parameters.
To reference a prefix predicate (see Section 5.14.3.5) or prefix lemma (see Section 5.14.3.6.3), the identifier must be the name of the greatest predicate or greatest lemma and it must be followed by a hash call.
9.33. Hash call (grammar)
A hash call is used to call the prefix for a greatest predicate or greatest lemma.
In the non-generic case, just insert "#[k]"
before the call argument
list where k is the number of recursion levels.
In the case where the greatest lemma
is generic, the generic type
argument is given before. Here is an example:
codatatype Stream<T> = Nil | Cons(head: int, stuff: T,
tail: Stream<T>)
function append(M: Stream, N: Stream): Stream
{
match M
case Nil => N
case Cons(t, s, M') => Cons(t, s, append(M', N))
}
function zeros<T>(s : T): Stream<T>
{
Cons(0, s, zeros(s))
}
function ones<T>(s: T): Stream<T>
{
Cons(1, s, ones(s))
}
greatest predicate atmost(a: Stream, b: Stream)
{
match a
case Nil => true
case Cons(h,s,t) => b.Cons? && h <= b.head && atmost(t, b.tail)
}
greatest lemma {:induction false} Theorem0<T>(s: T)
ensures atmost(zeros(s), ones(s))
{
// the following shows two equivalent ways to state the
// coinductive hypothesis
if (*) {
Theorem0#<T>[_k-1](s);
} else {
Theorem0(s);
}
}
where the HashCall
is "Theorem0#<T>[_k-1](s);"
.
See Section 5.14.3.5 and Section 5.14.3.6.3.
9.34. Suffix (grammar)
A suffix describes ways of deriving a new value from the entity to which the suffix is appended. The several kinds of suffixes are described below.
9.34.1. Augmented Dot Suffix (grammar)
Examples: (expression with suffix)
a.b
(a).b<int>
a.b#[k]
a.b#<int>[k]
An augmented dot suffix consists of a simple dot suffix optionally followed by either
- a
GenericInstantiation
(for the case where the item selected by theDotSuffix
is generic), or - a
HashCall
for the case where we want to call a prefix predicate or prefix lemma. The result is the result of calling the prefix predicate or prefix lemma.
9.34.2. Datatype Update Suffix (grammar)
Examples: (expression with suffix)
a.(f := e1, g:= e2)
a.(0 := e1)
(e).(f := e1, g:= e2)
A datatype update suffix is used to produce a new datatype value
that is the same as an old datatype value except that the
value corresponding to a given destructor has the specified value.
In a member binding update, the given identifier (or digit sequence) is the
name of a destructor (i.e. the formal parameter name) for one of the
constructors of the datatype. The expression to the right of the
:=
is the new value for that formal.
All of the destructors in a datatype update suffix must be for the same constructor, and if they do not cover all of the destructors for that constructor then the datatype value being updated must have a value derived from that same constructor.
Here is an example:
module NewSyntax {
datatype MyDataType = MyConstructor(myint:int, mybool:bool)
| MyOtherConstructor(otherbool:bool)
| MyNumericConstructor(42:int)
method test(datum:MyDataType, x:int)
returns (abc:MyDataType, def:MyDataType,
ghi:MyDataType, jkl:MyDataType)
requires datum.MyConstructor?
ensures abc == datum.(myint := x + 2)
ensures def == datum.(otherbool := !datum.mybool) // error
ensures ghi == datum.(myint := 2).(mybool := false)
// Resolution error: no non_destructor in MyDataType
//ensures jkl == datum.(non_destructor := 5) // error
ensures jkl == datum.(42 := 7)
{
abc := MyConstructor(x + 2, datum.mybool);
abc := datum.(myint := x + 2);
def := MyOtherConstructor(!datum.mybool);
ghi := MyConstructor(2, false);
jkl := datum.(42 := 7); // error
assert abc.(myint := abc.myint - 2) == datum.(myint := x);
}
}
9.34.3. Subsequence Suffix (grammar)
Examples: (with leading expression)
a[lo .. hi ]
(e)[ lo .. ]
e[ .. hi ]
e[ .. ]
A subsequence suffix applied to a sequence produces a new sequence whose
elements are taken from a contiguous part of the original sequence. For
example, expression s[lo..hi]
for sequence s
, and integer-based
numerics lo
and hi
satisfying 0 <= lo <= hi <= |s|
. See
the section about other sequence expressions for details.
A subsequence suffix applied to an array produces a sequence consisting of
the values of the designated elements. A concise way of converting a whole
array to a sequence is to write a[..]
.
9.34.4. Subsequence Slices Suffix (grammar)
Examples: (with leading expression)
a[ 0 : 2 : 3 ]
a[ e1 : e2 : e3 ]
a[ 0 : 2 : ]
Applying a subsequence slices suffix to a sequence produces a sequence of subsequences of the original sequence. See the section about other sequence expressions for details.
9.34.5. Sequence Update Suffix (grammar)
Examples:
s[1 := 2, 3 := 4]
For a sequence s
and expressions i
and v
, the expression
s[i := v]
is the same as the sequence s
except that at
index i
it has value v
.
If the type of s
is seq<T>
, then v
must have type T
.
The index i
can have any integer- or bit-vector-based type
(this is one situation in which Dafny implements implicit
conversion, as if an as int
were appended to the index expression).
The expression s[i := v]
has the same type as s
.
9.34.6. Selection Suffix (grammar)
Examples:
a[9]
a[i.j.k]
If a selection suffix has only one expression in it, it is a zero-based index that may be used to select a single element of a sequence or from a single-dimensional array.
If a selection suffix has more than one expression in it, then it is a list of indices to index into a multi-dimensional array. The rank of the array must be the same as the number of indices.
If the selection suffix is used with an array or a sequence,
then each index expression can have any integer- or bit-vector-based
type
(this is one situation in which Dafny implements implicit
conversion, as if an as int
were appended to the index expression).
9.34.7. Argument List Suffix (grammar)
Examples:
()
(a)
(a, b)
An argument list suffix is a parenthesized list of expressions that are the arguments to pass to a method or function that is being called. Applying such a suffix causes the method or function to be called and the result is the result of the call.
Note that method calls may only appear in right-hand-side locations, whereas function calls may appear in expressions and specifications; this distinction can be made only during name and type resolution, not by the parser.
9.35. Expression Lists (grammar)
Examples:
// empty list
a
a, b
An expression list is a comma-separated sequence of expressions, used, for example, as actual araguments in a method or function call or in parallel assignment.
9.36. Parameter Bindings (grammar)
Examples:
a
a, b
a, optimize := b
Method calls, object-allocation calls (new
), function calls, and
datatype constructors can be called with both positional arguments
and named arguments.
Formal parameters have three ways to indicate how they are to be passed in:
- nameonly: the only way to give a specific argument value is to name the parameter
- positional only: these are nameless parameters (which are allowed only for datatype constructor parameters)
- either positional or by name: this is the most common parameter
A parameter is either required or optional:
- required: a caller has to supply an argument
- optional: the parameter has a default value that is used if a caller omits passing a specific argument
The syntax for giving a positional-only (i.e., nameless) parameter does not allow a default-value expression, so a positional-only parameter is always required.
At a call site, positional arguments are not allowed to follow named arguments. Therefore, if x
is a nameonly parameter, then there is no way to supply the parameters after x
by position.
Thus, any parameter that follows x
must either be passed by name or have a default value.
That is, if a later (in the formal parameter declaration) parameter does not have a default value, it is effectively nameonly.
Positional arguments must be given before any named arguments.
Positional arguments are passed to the formals in the corresponding
position. Named arguments are passed to the formal of the given
name. Named arguments can be given out of order from how the corresponding
formal parameters are declared. A formal declared with the modifier
nameonly
is not allowed to be passed positionally.
The list of bindings for a call must
provide exactly one value for every required parameter and at most one
value for each optional parameter, and must never name
non-existent formals. Any optional parameter that is not given a value
takes on the default value declared in the callee for that optional parameter.
9.37. Compile-Time Constants
In certain situations in Dafny it is helpful to know what the value of a
constant is during program analysis, before verification or execution takes
place. For example, a compiler can choose an optimized representation of a
newtype
that is a subset of int
if it knows the range of possible values
of the subset type: if the range is within 0 to less than 256, then an
unsigned 8-bit representation can be used.
To continue this example, suppose a new type is defined as
const MAX := 47
newtype mytype = x | 0 <= x < MAX*4
In this case, we would prefer that Dafny recognize that MAX*4
is
known to be constant with a value of 188
. The kinds of expressions
for which such an optimization is possible are called
compile-time constants. Note that the representation of mytype
makes
no difference semantically, but can affect how compiled code is represented at run time.
In addition, though, using a symbolic constant (which may
well be used elsewhere as well) improves the self-documentation of the code.
In Dafny, the following expressions are compile-time constants12, recursively (that is, the arguments of any operation must themselves be compile-time constants):
- int, bit-vector, real, boolean, char and string literals
- int operations:
+ - * / %
and unary-
and comparisons< <= > >= == !=
- real operations:
+ - *
and unary-
and comparisons< <= > >= == !=
- bool operations:
&& || ==> <== <==> == !=
and unary!
- bit-vector operations:
+ - * / % << >> & | ^
and unary! -
and comparisons< <= > >= == !=
- char operations:
< <= > >= == !=
- string operations: length:
|...|
, concatenation:+
, comparisons< <= == !=
, indexing[]
- conversions between:
int
real
char
bit-vector - newtype operations: newtype arguments, but not newtype results
- symbolic values that are declared
const
and have an explicit initialization value that is a compile-time constant - conditional (if-then-else) expressions
- parenthesized expressions
9.38. List of specification expressions
The following is a list of expressions that can only appear in specification contexts or in ghost blocks.
- Fresh expressions
- Allocated expressions
- Unchanged expressions
- Old expressions
- Assert and calc expressions
- Hash Calls
10. Refinement
Refinement is the process of replacing something somewhat abstract with something somewhat more concrete.
For example, in one module one might declare a type name, with no definition,
such as type T
, and then in a refining module, provide a definition.
One could prove general properties about the contents of an (abstract) module,
and use that abstract module, and then later provide a more concrete implementation without having to redo all of the proofs.
Dafny supports module refinement, where one module is created from another, and in that process the new module may be made more concrete than the previous. More precisely, refinement takes the following form in Dafny. One module declares some program entities. A second module refines the first by declaring how to augment or replace (some of) those program entities. The first module is called the refinement parent; the second is the refining module; the result of combining the two (the original declarations and the augmentation directives) is the assembled module or refinement result.
Syntactically, the refinement parent is a normal module declaration.
The refining module declares which module is its refinement parent with the
refines
clause:
module P { // refinement parent
}
module M refines P { // refining module
}
The refinement result is created as follows.
0) The refinement result is a module within the same enclosing module as the refining module, has the same name, and in fact replaces the refining module in their shared scope.
1) All the declarations (including import and export declarations) of the parent are copied into the refinement result. These declarations are not re-resolved. That is, the assignment of declarations and types to syntactic names is not changed. The refinement result may exist in a different enclosing module and with a different set of imports than the refinement parent, so that if names were reresolved, the result might be different (and possibly not semantically valid). This is why Dafny does not re-resolve the names in their new context.
2) All the declarations of the refining module that have different names than the declarations in the refinement parent are also copied into the refinement result. However, because the refining module is just a set of augmentation directives and may refer to names copied from the refinement parent, resolution of names and types of the declarations copied in this step is performed in the context of the full refinement result.
3) Where declarations in the parent and refinement module have the same name, the second refines the first and the combination, a refined declaration, is the result placed in the refinement result module, to the exclusion of the declarations with the same name from the parent and refinement modules.
The way the refinement result declarations are assembled depends on the kind of declaration; the rules are described in subsections below.
So that it is clear that refinement is taking place, refining declarations
have some syntactic indicator that they are refining some parent declaration.
Typically this is the presence of a ...
token.
10.1. Export set declarations
A refining export set declaration begins with the syntax
"export" Ident ellipsis
but otherwise contains the same provides
, reveals
and extends
sections,
with the ellipsis indicating that it is a refining declaration.
The result declaration has the same name as the two input declarations and the unions of names from each of the provides
, reveals
, and extends
sections, respectively.
An unnamed export set declaration from the parent is copied into the result module with the name of the parent module. The result module has a default export set according to the general rules for export sets, after all of the result module’s export set declarations have been assembled.
10.2. Import declarations
Aliasing import declarations are not refined. The result module contains the union of the import declarations from the two input modules. There must be no names in common among them.
Abstract import declarations (declared with :
instead of =
, Section 4.6) are refined. The refinement parent contains the
abstract import and the refining module contains a regular aliasing
import for the same name. Dafny checks that the refining import adheres to
the abstract import.
10.3. Sub-module declarations
With respect to refinement, a nested module behaves just like a top-level module. It may be declared abstract and it may be declared to refine
some refinement parent. If the nested module is not refining anything and not being refined, then it is copied into the refinement result like any other declaration.
Here is some example code:
abstract module P {
module A { const i := 5 }
abstract module B { type T }
}
module X refines P {
module B' refines P.B { type T = int }
module C { const k := 6}
}
module M {
import X
method m() {
var z: X.B'.T := X.A.i + X.C.k;
}
}
The refinement result of P
and X
contains nested modules A
, B'
, and C
. It is this refinement result that is imported into M
.
Hence the names X.B'.T
, X.A.i
and X.C.k
are all valid.
10.4. Const declarations
Const declarations can be refined as in the following example.
module A {
const ToDefine: int
const ToDefineWithoutType: int
const ToGhost: int := 1
}
module B refines A {
const ToDefine: int := 2
const ToDefineWithoutType ... := 3
ghost const ToGhost: int
const NewConst: int
}
Formally, a child const
declaration may refine a const
declaration
from a parent module if
- the parent has no initialization,
- the child has the same type as the parent, and
- one or both of the following holds:
- the child has an initializing expression
- the child is declared
ghost
and the parent is notghost
.
A refining module can also introduce new const
declarations that do
not exist in the refinement parent.
10.5. Method declarations
Method declarations can be refined as in the following example.
module A {
method ToImplement(x: int) returns (r: int)
ensures r > x
method ToStrengthen(x: int) returns (r: int)
method ToDeterminize(x: int) returns (r: int)
ensures r >= x
{
var y :| y >= x;
return y;
}
}
module B refines A {
method ToImplement(x: int) returns (r: int)
{
return x + 2;
}
method ToStrengthen ...
ensures r == x*2
{
return x*2;
}
method ToDeterminize(x: int) returns (r: int)
{
return x;
}
}
Formally, a child method
definition may refine a parent method
declaration or definition by performing one or more of the following
operations:
- provide a body missing in the parent (as in
ToImplement
), - strengthen the postcondition of the parent method by adding one or more
ensures
clauses (as inToStrengthen
), - provide a more deterministic version of a non-deterministic parent
body (as in
ToDeterminize
), or
The type signature of a child method must be the same as that of the
parent method it refines. This can be ensured by providing an explicit
type signature equivalent to that of the parent (with renaming of
parameters allowed) or by using an ellipsis (...
) to indicate copying
of the parent type signature. The body of a child method must satisfy
any ensures clauses from its parent in addition to any it adds.
A refined method is allowed only if it does not invalidate any parent lemmas that mention it.
A refining module can also introduce new method
declarations or
definitions that do not exist in the refinement parent.
10.6. Lemma declarations
As lemmas are (ghost) methods, the description of method refinement from the previous section also applies to lemma refinement.
A valid refinement is one that does not invalidate any proofs. A lemma from a refinement parent must still be valid for the refinement result of any method or lemma it mentions.
10.7. Function and predicate declarations
Function (and equivalently predicate) declarations can be refined as in the following example.
module A {
function F(x: int): (r: int)
ensures r > x
function G(x: int): (r: int)
ensures r > x
{ x + 1 }
}
module B refines A {
function F ...
{ x + 1 }
function G ...
ensures r == x + 1
}
Formally, a child function
(or predicate
) definition can refine a
parent function
(or predicate
) declaration or definition to
- provide a body missing in the parent,
- strengthen the postcondition of the parent function by adding one or more
ensures
clauses.
The relation between the type signature of the parent and child function is the same as for methods and lemmas, as described in the previous section.
A refining module can also introduce new function
declarations or
definitions that do not exist in the refinement parent.
10.8. Class, trait and iterator declarations
Class, trait, and iterator declarations are refined as follows:
- If a class (or trait or iterator, respectively)
C
in a refining parent contains a member that is not matched by a same-named member in the classC
in the refining module, or vice-versa, then that class is copied as is to the refinement result. - When there are members with the same name in the class in the refinement parent and in the refining module, then the combination occurs according to the rules for that category of member.
Here is an example code snippet:
abstract module P {
class C {
function F(): int
ensures F() > 0
}
}
module X refines P {
class C ... {
function F...
ensures F() > 0
{ 1 }
}
}
10.9. Type declarations
Types can be refined in two ways:
- Turning an abstract type into a concrete type;
- Adding members to a datatype or a newtype.
For example, consider the following abstract module:
abstract module Parent {
type T
type B = bool
type S = s: string | |s| > 0 witness "!"
newtype Pos = n: nat | n > 0 witness 1
datatype Bool = True | False
}
In this module, type T
is opaque and hence can be refined with any type,
including class types. Types B
, S
, Pos
, and Bool
are concrete and
cannot be refined further, except (for Pos
and Bool
) by giving them
additional members or attributes (or refining their existing members, if any).
Hence, the following are valid refinements:
module ChildWithTrait refines Parent {
trait T {}
}
module ChildWithClass refines Parent {
class T {}
}
module ChildWithSynonymType refines Parent {
type T = bool
}
module ChildWithSubsetType refines Parent {
type T = s: seq<int> | s != [] witness [0]
}
module ChildWithDataType refines Parent {
datatype T = True | False
}
abstract module ChildWithExtraMembers refines Parent {
newtype Pos ... {
method Print() { print this; }
}
datatype Bool ... {
function AsDafnyBool() : bool { this.True? }
}
}
(The last example is marked abstract
because it leaves T
opaque.)
Note that datatype constructors, codatatype destructors, and newtype definitions
cannot be refined: it is not possible to add or remove datatype
constructors,
nor to change destructors of a codatatype
, nor to change the base
type, constraint, or witness of a newtype
.
When a type takes arguments, its refinement must use the same type arguments with the same type constraints and the same variance.
When a type has type constraints, these type constraints must be preserved by
refinement. This means that a type declaration type T(!new)
cannot be refined
by a class T
, for example. Similarly, a type T(00)
cannot be refined by a
subset type with a witness *
clause.
The refinement of an abstract type with body-less members can include both a definition for the type along with a body for the member, as in this example:
abstract module P {
type T3 {
function ToString(): string
}
}
module X refines P {
newtype T3 = i | 0 <= i < 10 {
function ToString... { "" }
}
}
Note that type refinements are not required to include the ...
indicator that they are refining a parent type.
10.10. Statements
The refinement syntax (...
) in statements is deprecated.
11. Attributes
Dafny allows many of its entities to be annotated with Attributes.
Attributes are declared between {:
and }
like this:
{:attributeName "argument", "second" + "argument", 57}
(White-space may follow but not precede the :
in {:
.)
In general an attribute may have any name the user chooses. It may be followed by a comma-separated list of expressions. These expressions will be resolved and type-checked in the context where the attribute appears.
Any Dafny entity may have a list of attributes. Dafny does not check that the attributes listed for an entity are appropriate for it (which means that misspellings may go silently unnoticed).
The grammar shows where the attribute annotations may appear:
Attribute = "{:" AttributeName [ Expressions ] "}"
Dafny has special processing for some attributes13. Of those, some apply only to the entity bearing the attribute, while others (inherited attributes) apply to the entity and its descendants (such as nested modules, types, or declarations). The attribute declaration closest to the entity overrides those further away.
For attributes with a single boolean expression argument, the attribute with no argument is interpreted as if it were true.
11.1. Attributes on top-level declarations
11.1.1. {:autocontracts}
Dynamic frames [@Kassios:FM2006;@SmansEtAl:VeriCool;@SmansEtAl:ImplicitDynamicFrames; @LEINO:Dafny:DynamicFrames] are frame expressions that can vary dynamically during program execution. AutoContracts is an experimental feature that will fill much of the dynamic-frames boilerplate into a class.
From the user’s perspective, what needs to be done is simply:
- mark the class with
{:autocontracts}
- declare a function (or predicate) called
Valid()
AutoContracts will then:
- Declare:
ghost var Repr: set<object>
- For function/predicate
Valid()
, insert:reads this, Repr
- Into body of
Valid()
, insert (at the beginning of the body):this in Repr && null !in Repr
- and also insert, for every array-valued field
A
declared in the class:&& (A != null ==> A in Repr)
- and for every field
F
of a class typeT
whereT
has a field calledRepr
, also insert:(F != null ==> F in Repr && F.Repr <= Repr && this !in F.Repr)
Except, if A or F is declared with
{:autocontracts false}
, then the implication will not be added. - For every constructor, add:
modifies this ensures Valid() && fresh(Repr - {this})
- At the end of the body of the constructor, add:
Repr := {this}; if (A != null) { Repr := Repr + {A}; } if (F != null) { Repr := Repr + {F} + F.Repr; }
- For every method, add:
requires Valid() modifies Repr ensures Valid() && fresh(Repr - old(Repr))
- At the end of the body of the method, add:
if (A != null) { Repr := Repr + {A}; } if (F != null) { Repr := Repr + {F} + F.Repr; }
11.1.2. {:nativeType}
The {:nativeType}
attribute is only recognized by a newtype
declaration
where the base type is an integral type or a real type. For example:
newtype {:nativeType "byte"} ubyte = x : int | 0 <= x < 256
newtype {:nativeType "byte"} bad_ubyte = x : int | 0 <= x < 257 // Fails
It can take one of the following forms:
{:nativeType}
- With no parameters it has no effect and the declaration will have its default behavior, which is to choose a native type that can hold any value satisfying the constraints, if possible, and otherwise to use BigInteger.{:nativeType true}
- Also gives default behavior, but gives an error if the base type is not integral.{:nativeType false}
- Inhibits using a native type. BigInteger is used.{:nativeType "typename"}
- This form has an native integral type name as a string literal. Acceptable values are:"byte"
8 bits, unsigned"sbyte"
8 bits, signed"ushort"
16 bits, unsigned"short"
16 bits, signed"uint"
32 bits, unsigned"int"
32 bits, signed"number"
53 bits, signed"ulong"
64 bits, unsigned"long"
64 bits, signed
If the target compiler does not support a named native type X, then an error is generated. Also, if, after scrutinizing the constraint predicate, the compiler cannot confirm that the type’s values will fit in X, an error is generated. The names given above do not have to match the names in the target compilation language, just the characteristics of that type.
11.1.3. {:ignore}
(deprecated)
Ignore the declaration (after checking for duplicate names).
11.1.4. {:extern}
{:extern}
is a target-language dependent modifier used
- to alter the
CompileName
of entities such as modules, classes, methods, etc., - to alter the
ReferenceName
of the entities, - to decide how to define external abstract types,
- to decide whether to emit target code or not, and
- to decide whether a declaration is allowed not to have a body.
The CompileName
is the name for the entity when translating to one of the target languages.
The ReferenceName
is the name used to refer to the entity in the target language.
A common use case of {:extern}
is to avoid name clashes with existing library functions.
{:extern}
takes 0, 1, or 2 (possibly empty) string arguments:
{:extern}
: Dafny will use the Dafny-determined name as theCompileName
and not affect theReferenceName
{:extern s1}
: Dafny will uses1
as theCompileName
, and replaces the last portion of theReferenceName
bys1
. When used on an abstract type, s1 is used as a hint as to how to declare that type when compiling.{:extern s1, s2}
Dafny will uses2
as theCompileName
. Dafny will use a combination ofs1
ands2
such as for examples1.s2
as theReferenceName
It may also be the case that one of the arguments is simply ignored.
Dafny does not perform sanity checks on the arguments—it is the user’s responsibility not to generate malformed target code.
For more detail on the use of {:extern}
, see the corresponding section in the user’s guide.
11.2. Attributes on functions and methods
11.2.1. {:abstemious}
The {:abstemious}
attribute is appropriate for functions on codatatypes.
If appropriate to a function, the attribute can aid in proofs that the function is productive.
See the section on abstemious functions for more description.
11.2.2. {:autoReq}
For a function declaration, if this attribute is set true at the nearest
level, then its requires
clause is strengthened sufficiently so that
it may call the functions that it calls.
For following example
function f(x:int) : bool
requires x > 3
{
x > 7
}
// Should succeed thanks to auto_reqs
function {:autoReq} g(y:int, b:bool) : bool
{
if b then f(y + 2) else f(2*y)
}
the {:autoReq}
attribute causes Dafny to
deduce a requires
clause for g as if it had been
declared
function f(x:int) : bool
requires x > 3
{
x > 7
}
function g(y:int, b:bool) : bool
requires if b then y + 2 > 3 else 2 * y > 3
{
if b then f(y + 2) else f(2*y)
}
11.2.3. {:axiom}
The {:axiom}
attribute may be placed on a function or method.
It means that the post-condition may be assumed to be true
without proof. In that case also the body of the function or
method may be omitted.
The {:axiom}
attribute only prevents Dafny from verifying that the body matches the post-condition.
Dafny still verifies the well-formedness of pre-conditions, of post-conditions, and of the body if provided.
To prevent Dafny from running all these checks, one would use {:verify false}
, which is not recommended.
The compiler will still emit code for an {:axiom}
, if it is a function
, a method
or a function by method
with a body.
11.2.4. {:compile}
The {:compile}
attribute takes a boolean argument. It may be applied to
any top-level declaration. If that argument is false, then that declaration
will not be compiled at all.
The difference with {:extern}
is that {:extern}
will still emit declaration code if necessary,
whereas {:compile false}
will just ignore the declaration for compilation purposes.
11.2.5. {:extern <name>}
See {:extern <name>}
.
11.2.6. {:fuel X}
The fuel attribute is used to specify how much “fuel” a function should have,
i.e., how many times the verifier is permitted to unfold its definition. The
{:fuel}
annotation can be added to the function itself, in which
case it will apply to all uses of that function, or it can be overridden
within the scope of a module, function, method, iterator, calc, forall,
while, assert, or assume. The general format is:
{:fuel functionName,lowFuel,highFuel}
When applied as an annotation to the function itself, omit functionName. If highFuel is omitted, it defaults to lowFuel + 1.
The default fuel setting for recursive functions is 1,2. Setting the fuel higher, say, to 3,4, will give more unfoldings, which may make some proofs go through with less programmer assistance (e.g., with fewer assert statements), but it may also increase verification time, so use it with care. Setting the fuel to 0,0 is similar to making the definition opaque, except when used with all literal arguments.
11.2.7. {:id <string>}
Assign a custom unique ID to a function or a method to be used for verification result caching.
11.2.8. {:induction}
The {:induction}
attribute controls the application of
proof by induction to two contexts. Given a list of
variables on which induction might be applied, the
{:induction}
attribute selects a sub-list of those
variables (in the same order) to which to apply induction.
Dafny issue 34
proposes to remove the restriction that the sub-list
be in the same order, and would apply induction in the
order given in the {:induction}
attribute.
The two contexts are:
- A method, in which case the bound variables are all the in-parameters of the method.
- A quantifier expression, in which case the bound variables are the bound variables of the quantifier expression.
The form of the {:induction}
attribute is one of the following:
{:induction}
– apply induction to all bound variables{:induction false}
– suppress induction, that is, don’t apply it to any bound variable{:induction L}
whereL
is a list consisting entirely of bound variables – apply induction to the specified bound variables{:induction X}
whereX
is anything else – treat the same as{:induction}
, that is, apply induction to all bound variables. For this usage conventionallyX
istrue
.
Here is an example of using it on a quantifier expression:
datatype Unary = Zero | Succ(Unary)
function UnaryToNat(n: Unary): nat {
match n
case Zero => 0
case Succ(p) => 1 + UnaryToNat(p)
}
function NatToUnary(n: nat): Unary {
if n == 0 then Zero else Succ(NatToUnary(n - 1))
}
lemma Correspondence()
ensures forall n: nat {:induction n} :: UnaryToNat(NatToUnary(n)) == n
{
}
11.2.9. {:only}
method {:only} X() {}
or function {:only} X() {}
temporarily disables the verification of all other non-{:only}
members, e.g. other functions and methods, in the same file, even if they contain assertions with {:only}
.
method {:only} TestVerified() {
assert true; // Unchecked
assert {:only} true by { // Checked
assert true; // Checked
}
assert true; // Unchecked
}
method TestUnverified() {
assert true; // Unchecked
assert {:only} true by { // Unchecked because of {:only} Test()
assert true; // Unchecked
}
assert true; // Unchecked
}
{:only}
can help focusing on a particular member, for example a lemma or a function, as it simply disables the verification of all other lemmas, methods and functions in the same file. It’s equivalent to adding {:verify false}
to all other declarations simulatenously on the same file. Since it’s meant to be a temporary construct, it always emits a warning.
11.2.10. {:opaque}
This attribute is replaced by the keyword opaque
; the attribute is deprecated.
Ordinarily, the body of a function is transparent to its users, but
sometimes it is useful to hide it. If a function foo
or bar
is given the
{:opaque}
attribute, then Dafny hides the body of the function,
so that it can only be seen within its recursive clique (if any),
or if the programmer specifically asks to see it via the statement reveal foo(), bar();
.
More information about the Boogie implementation of {:opaque}
is here.
11.2.11. {:print}
This attribute declares that a method may have print effects,
that is, it may use print
statements and may call other methods
that have print effects. The attribute can be applied to compiled
methods, constructors, and iterators, and it gives an error if
applied to functions or ghost methods. An overriding method is
allowed to use a {:print}
attribute only if the overridden method
does.
Print effects are enforced only with --track-print-effects
.
11.2.12. {:priority}
{:priority N}
assigns a positive priority ‘N’ to a method or function to control the order
in which methods or functions are verified (default: N = 1).
11.2.13. {:rlimit}
{:rlimit N}
limits the verifier resource usage to verify the method or function at N * 1000
.
This is the per-method equivalent of the command-line flag /rlimit:N
.
If using {:vcs_split_on_every_assert}
as well, the limit will be set for each assertion.
To give orders of magnitude about resource usage, here is a list of examples indicating how many resources are used to verify each method:
- 8K resource usage
method f() { assert true; }
- 10K resource usage using assertions that do not add assumptions:
method f(a: bool, b: bool) { assert a: (a ==> b) <==> (!b ==> !a); assert b: (a ==> b) <==> (!b ==> !a); assert c: (a ==> b) <==> (!b ==> !a); assert d: (a ==> b) <==> (!b ==> !a); }
- 40K total resource usage using
{:vcs_split_on_every_assert}
method {:vcs_split_on_every_assert} f(a: bool, b: bool) { assert a: (a ==> b) <==> (!b ==> !a); assert b: (a ==> b) <==> (!b ==> !a); assert c: (a ==> b) <==> (!b ==> !a); assert d: (a ==> b) <==> (!b ==> !a); }
- 37K total resource usage and thus fails with
out of resource
.method {:rlimit 30} f(a: int, b: int, c: int) { assert ((1 + a*a)*c) / (1 + a*a) == c; }
Note that, the default solver Z3 tends to overshoot by 7K
to 8K
, so if you put {:rlimit 20}
in the last example, the total resource usage would be 27K
.
11.2.14. {:selective_checking}
Turn all assertions into assumptions except for the ones reachable from after the
assertions marked with the attribute {:start_checking_here}
.
Thus, assume {:start_checking_here} something;
becomes an inverse
of assume false;
: the first one disables all verification before
it, and the second one disables all verification after.
11.2.15. {:tailrecursion}
This attribute is used on method declarations. It has a boolean argument.
If specified with a false
value, it means the user specifically
requested no tail recursion, so none is done.
If specified with a true
value, or if no argument is specified,
then tail recursive optimization will be attempted subject to
the following conditions:
- It is an error if the method is a ghost method and tail recursion was explicitly requested.
- Only direct recursion is supported, not mutually recursive methods.
- If
{:tailrecursion true}
was specified but the code does not allow it, an error message is given.
11.2.16. {:test}
This attribute indicates the target function or method is meant to be executed at runtime in order to test that the program is working as intended.
There are two different ways to dynamically test functionality in a test:
- A test can optionally return a single value to indicate success or failure.
If it does, this must be a failure-compatible type
just as the update-with-failure statement requires. That is,
the returned type must define a
IsFailure()
function method. IfIsFailure()
evaluates totrue
on the return value, the test will be marked a failure, and this return value used as the failure message. - Code in the control flow of the test can use
expect
statements to dynamically test if a boolean expression is true, and cause the test to halt if not (but not the overall testing process). The optional second argument to a failedexpect
statement will be used as the test failure message.
Note that the expect
keyword can also be used to form “assign or halt” statements
such as var x :- expect CalculateX();
, which is a convenient way to invoke a method
that may produce a failure within a test without having to return a value from the test.
There are also two different approaches to executing all tests in a program:
- By default, the compiler will mark each compiled method as necessary so that
a designated target language testing framework will discover and run it.
This is currently only implemented for C#, using the xUnit
[Fact]
annotation. - If
dafny test
is used, Dafny will instead produce a main method that invokes each test and prints the results. This runner is currently very basic, but avoids introducing any additional target language dependencies in the compiled code.
A method marked {:test}
may not have any input arguments. If there is an
output value that does not have a failure-compatible type, that value is
ignored. A method that does have input arguments can be wrapped in a test
harness that supplies input arguments but has no inputs of its own and that
checks any output values, perhaps with expect
statements. The test harness
is then the method marked with {:test}
.
11.2.17. {:timeLimit N}
Set the time limit for verifying a given function or method.
11.2.18. {:timeLimitMultiplier X}
This attribute may be placed on a method or function declaration
and has an integer argument. If {:timeLimitMultiplier X}
was
specified a {:timeLimit Y}
attribute is passed on to Boogie
where Y
is X
times either the default verification time limit
for a function or method, or times the value specified by the
Boogie -timeLimit
command-line option.
11.2.19. {:verify false}
Skip verification of a function or a method altogether,
not even trying to verify the well-formedness of postconditions and preconditions.
We discourage using this attribute and prefer {:axiom}
,
which performs these minimal checks while not checking that the body satisfies the postconditions.
If you simply want to temporarily disable all verification except on a single function or method, use the {:only}
attribute on that function or method.
11.2.20. {:vcs_max_cost N}
Per-method version of the command-line option /vcsMaxCost
.
The assertion batch of a method
will not be split unless the cost of an assertion batch exceeds this
number, defaults to 2000.0. In
keep-going mode, only applies to the first round.
If {:vcs_split_on_every_assert}
is set, then this parameter is useless.
11.2.21. {:vcs_max_keep_going_splits N}
Per-method version of the command-line option /vcsMaxKeepGoingSplits
.
If set to more than 1, activates the keep going mode where, after the first round of splitting,
assertion batches that timed out are split into N assertion batches and retried
until we succeed proving them, or there is only one
single assertion that it timeouts (in which
case an error is reported for that assertion).
Defaults to 1.
If {:vcs_split_on_every_assert}
is set, then this parameter is useless.
11.2.22. {:vcs_max_splits N}
Per-method version of the command-line option /vcsMaxSplits
.
Maximal number of assertion batches generated for this method.
In keep-going mode, only applies to the first round.
Defaults to 1.
If {:vcs_split_on_every_assert}
is set, then this parameter is useless.
11.2.23. {:vcs_split_on_every_assert}
Per-method version of the command-line option /vcsSplitOnEveryAssert
.
In the first and only verification round, this option will split the original assertion batch into one assertion batch per assertion. This is mostly helpful for debugging which assertion is taking the most time to prove, e.g. to profile them.
11.2.24. {:synthesize}
The {:synthesize}
attribute must be used on methods that have no body and
return one or more fresh objects. During compilation,
the postconditions associated with such a
method are translated to a series of API calls to the target languages’s
mocking framework. The object returned, therefore, behaves exactly as the
postconditions specify. If there is a possibility that this behavior violates
the specifications on the object’s instance methods or hardcodes the values of
its fields, the compiler will throw an error but the compilation will go
through. Currently, this compilation pass is only supported in C# and requires
adding the latest version of the Moq library to the .csproj file before
generating the binary.
Not all Dafny postconditions can be successfully compiled - below is the
grammar for postconditions that are supported (S
is the start symbol, EXPR
stands for an arbitrary Dafny expression, and ID
stands for
variable/method/type identifiers):
S = FORALL
| EQUALS
| S && S
EQUALS = ID.ID (ARGLIST) == EXPR // stubs a function call
| ID.ID == EXPR // stubs field access
| EQUALS && EQUALS
FORALL = forall BOUNDVARS :: EXPR ==> EQUALS
ARGLIST = ID // this can be one of the bound variables
| EXPR // this expr may not reference any of the bound variables
| ARGLIST, ARGLIST
BOUNDVARS = ID : ID
| BOUNDVARS, BOUNDVARS
11.2.25. {:options OPT0, OPT1, ... }
This attribute applies only to modules. It configures Dafny as if
OPT0
, OPT1
, … had been passed on the command line. Outside of the module,
options revert to their previous values.
Only a small subset of Dafny’s command line options is supported. Use the
/attrHelp
flag to see which ones.
11.3. Attributes on assertions, preconditions and postconditions
11.3.1. {:only}
assert {:only} X;
temporarily transforms all other non-{:only}
assertions in the surrounding declaration into assumptions.
method Test() {
assert true; // Unchecked
assert {:only} true by { // Checked
assert true; // Checked
}
assert true; // Unchecked
assert {:only "after"} true; // Checked
assert true; // Checked
assert {:only "before"} true; // Checked
assert true; // Unchecked
}
{:only}
can help focusing on a particular proof or a particular branch, as it transforms not only other explicit assertions, but also other implicit assertions, and call requirements, into assumptions.
Since it’s meant to be a temporary construct, it always emits a warning.
It also has two variants assert {:only "before"}
and assert {:only "after"}
.
Here is precisely how Dafny determines what to verify or not.
Each {:only}
annotation defines a “verification interval” which is visual:
assert {:only} X [by {...} | ;]
sets a verification interval that starts at the keywordassert
and ends either at the end of the proof}
or the semicolon;
, depending on which variant ofassert
is being used.assert {:only} ...
inside another verification interval removes that verification interval and sets a new one.assert {:only "before"} ...
inside another verification interval finishes that verification interval earlier at the end of this assertion. Outside a verification interval, it sets a verification interval from the beginning of the declaration to the end of this assertion, but only if there were no other verification intervals before.assert {:only "after"} ...
inside another verification interval moves the start of that verification interval to the start of this new assert. Outside a verification interval, it sets a verification interval from the beginning of thisassert
to the end of the declaration.
The start of an asserted expression is used to determines if it’s inside a verification interval or not.
For example, in assert B ==> (assert {:only "after"} true; C)
, C
is actually the start of the asserted expression, so it is verified because it’s after assert {:only "after"} true
.
As soon as a declaration contains one assert {:only}
, none of the postconditions are verified; you’d need to make them explicit with assertions if you wanted to verify them at the same time.
You can also isolate the verification of a single member using a similar {:only}
attribute.
11.3.2. {:focus}
assert {:focus} X;
splits verification into two assertion batches.
The first batch considers all assertions that are not on the block containing the assert {:focus} X;
The second batch considers all assertions that are on the block containing the assert {:focus} X;
and those that will always follow afterwards.
Hence, it might also occasionally double-report errors.
If you truly want a split on the batches, prefer {:split_here}
.
Here are two examples illustrating how {:focus}
works, where --
in the comments stands for Assumption
:
method doFocus1(x: bool) returns (y: int) {
y := 1; // Batch 1 Batch 2
assert y == 1; // Assertion --
if x {
if false {
assert y >= 0; // -- Assertion
assert {:focus} y <= 2; // -- Assertion
y := 2;
assert y == 2; // -- Assertion
}
} else {
assert y == 1; // Assertion --
}
assert y == 1; // Assertion Assertion
if !x {
assert y >= 1; // Assertion Assertion
} else {
assert y <= 1; // Assertion Assertion
}
}
And another one where the focused block is guarded with a while
, resulting in remaining assertions not being part of the first assertion batch:
method doFocus2(x: bool) returns (y: int) {
y := 1; // Batch 1 Batch 2
assert y == 1; // Assertion --
if x {
while false {
assert y >= 0; // -- Assertion
assert {:focus} y <= 2; // -- Assertion
y := 2;
assert y == 2; // -- Assertion
}
} else {
assert y == 1; // Assertion --
}
assert y == 1; // Assertion --
if !x {
assert y >= 1; // Assertion --
} else {
assert y <= 1; // Assertion --
}
}
11.3.3. {:split_here}
assert {:split_here} X;
splits verification into two assertion batches.
It verifies the code leading to this point (excluded) in a first assertion batch,
and the code leading from this point (included) to the next {:split_here}
or until the end in a second assertion batch.
It might help with timeouts.
Here is one example, where --
in the comments stands for Assumption
:
method doSplitHere(x: bool) returns (y: int) {
y := 1; // Batch 1 Batch 2 Batch 3
assert y >= 0; // Assertion -- --
if x {
assert y <= 1; // Assertion -- --
assert {:split_here} true; // -- Assertion --
assert y <= 2; // -- Assertion --
assert {:split_here} true; // -- -- Assertion
if x {
assert y == 1; // -- -- Assertion
} else {
assert y >= 1; // -- -- Assertion
}
} else {
assert y <= 3; // Assertion -- --
}
assert y >= -1; // Assertion -- --
}
11.3.4. {:subsumption n}
Overrides the /subsumption
command-line setting for this assertion.
{:subsumption 0}
checks an assertion but does not assume it after proving it.
You can achieve the same effect using labelled assertions.
11.3.5. {:error "errorMessage", "successMessage"}
Provides a custom error message in case the assertion fails. As a hint, messages indicating what the user needs to do to fix the error are usually better than messages that indicate the error only. For example:
method Process(instances: int, price: int)
requires {:error "There should be an even number of instances", "The number of instances is always even"} instances % 2 == 0
requires {:error "Could not prove that the price is positive", "The price is always positive"} price >= 0
{
}
method Test()
{
if * {
Process(1, 0); // Error: There should be an even number of instances
}
if * {
Process(2, -1); // Error: Could not prove that the price is positive
}
if * {
Process(2, 5); // Success: The number of instances is always even
// Success: The price is always positive
}
}
The success message is optional but is recommended if errorMessage is set.
11.4. Attributes on variable declarations
11.4.1. {:assumption}
This attribute can only be placed on a local ghost bool
variable of a method. Its declaration cannot have a rhs, but it is
allowed to participate as the lhs of exactly one assignment of the
form: b := b && expr;
. Such a variable declaration translates in the
Boogie output to a declaration followed by an assume b
command.
See [@LeinoWuestholz2015], Section 3, for example uses of the {:assumption}
attribute in Boogie.
11.5. Attributes on quantifier expressions (forall, exists)
11.5.1. {:heapQuantifier}
This attribute has been removed.
11.5.2. {:induction}
See {:induction}
for functions and methods.
11.5.3. {:trigger}
Trigger attributes are used on quantifiers and comprehensions.
The verifier instantiates the body of a quantified expression only when it can find an expression that matches the provided trigger.
Here is an example:
predicate P(i: int)
predicate Q(i: int)
lemma PHoldEvenly()
ensures forall i {:trigger Q(i)} :: P(i) ==> P(i + 2) && Q(i)
lemma PHoldsForTwo()
ensures forall i :: P(i) ==> P(i + 4)
{
forall j: int
ensures P(j) ==> P(j + 4)
{
if P(j) {
assert P(j); // Trivial assertion
PHoldEvenly();
// Invoking the lemma assumes `forall i :: P(i) ==> P(i + 4)`,
// but it's not instantiated yet
// The verifier sees `Q(j)`, so it instantiates
// `forall i :: P(i) ==> P(i + 4)` with `j`
// and we get the axiom `P(j) ==> P(j + 2) && Q(j)`
assert Q(j); // hence it can prove `Q(j)`
assert P(j + 2); // and it can prove `P(j + 2)`
assert P(j + 4); // But it cannot prove this
// because it did not instantiate `forall i :: P(i) ==> P(i + 4)` with `j+2`
}
}
}
Here are ways one can prove assert P(j + 4);
:
- Add
assert Q(j + 2);
just beforeassert P(j + 4);
, so that the verifier sees the trigger. - Change the trigger
{:trigger Q(i)}
to{:trigger P(i)}
(replace the trigger) - Change the trigger
{:trigger Q(i)}
to{:trigger Q(i)} {:trigger P(i)}
(add a trigger) - Remove
{:trigger Q(i)}
so that it will automatically determine all possible triggers thanks to the option/autoTriggers:1
which is the default.
11.6. Deprecated attributes
These attributes have been deprecated or removed. They are no longer useful (or perhaps never were) or were experimental. They will likely be removed entirely sometime soon after the release of Dafny 4.
Removed:
- :heapQuantifier
- :dllimport
- :handle
11.7. Other undocumented verification attributes
A scan of Dafny’s sources shows it checks for the following attributes.
{:$}
{:$renamed$}
{:InlineAssume}
{:PossiblyUnreachable}
{:__dominator_enabled}
{:__enabled}
{:a##post##}
{:absdomain}
{:ah}
{:assumption}
{:assumption_variable_initialization}
{:atomic}
{:aux}
{:both}
{:bvbuiltin}
{:candidate}
{:captureState}
{:checksum}
{:constructor}
{:datatype}
{:do_not_predicate}
{:entrypoint}
{:existential}
{:exitAssert}
{:expand}
{:extern}
{:focus}
{:hidden}
{:ignore}
{:inline}
{:left}
{:linear}
{:linear_in}
{:linear_out}
{:msg}
{:name}
{:originated_from_invariant}
{:partition}
{:positive}
{:post}
{:pre}
{:precondition_previous_snapshot}
{:qid}
{:right}
{:selective_checking}
{:si_fcall}
{:si_unique_call}
{:sourcefile}
{:sourceline}
{:split_here}
{:stage_active}
{:stage_complete}
{:staged_houdini_tag}
{:start_checking_here}
{:subsumption}
{:template}
{:terminates}
{:upper}
{:verified_under}
{:weight}
{:yields}
12. Advanced Topics
12.1. Type Parameter Completion
Generic types, like A<T,U>
, consist of a type constructor, here A
, and type parameters, here T
and U
.
Type constructors are not first-class entities in Dafny, they are always used syntactically to construct
type names; to do so, they must have the requisite number of type parameters, which must be either concrete types, type parameters, or
a generic type instance.
However, those type parameters do not always have to be explicit; Dafny can often infer what they ought to be. For example, here is a fully parameterized function signature:
type List<T>
function Elements<T>(list: List<T>): set<T>
However, Dafny also accepts
type List<T>
function Elements(list: List): set
In the latter case, Dafny knows that the already defined types set
and List
each take one type parameter
so it fills in <T>
(using some unique type parameter name) and then determines that the function itself needs
a type parameter <T>
as well.
Dafny also accepts
type List<T>
function Elements<T>(list: List): set
In this case, the function already has a type parameter list. List
and set
are each known to need type parameters,
so Dafny takes the first n
parameters from the function signature and applies them to List
and set
, where n
(here 1
) is the
number needed by those type constructors.
It never hurts to simply write in all the type parameters, but that can reduce readability. Omitting them in cases where Dafny can intuit them makes a more compact definition.
This process is described in more detail with more examples in this paper: http://leino.science/papers/krml270.html.
12.2. Type Inference
Signatures of methods, functions, fields (except const
fields with a
RHS), and datatype constructors have to declare the types of their
parameters. In other places, types can be omitted, in which case
Dafny attempts to infer them. Type inference is “best effort” and may
fail. If it fails to infer a type, the remedy is simply for the
program to give the type explicitly.
Despite being just “best effort”, the types of most local variables, bound variables, and the type parameters of calls are usually inferred without the need for a program to give the types explicitly. Here are some notes about type inference:
- With some exceptions, type inference is performed across a whole method body. In some cases, the information needed to infer a local variable’s type may be found after the variable has been declared and used. For example, the nonsensical program
```dafny
method M(n: nat) returns (y: int)
{
var a, b;
for i := 0 to n {
if i % 2 == 0 {
a := a + b;
}
}
y := a;
}
```
uses a
and b
after their declarations. Still, their types are
inferred to be int
, because of the presence of the assignment y := a;
.
A more useful example is this:
```dafny
class Cell {
var data: int
}
method LastFive(a: array<int>) returns (r: int)
{
var u := null;
for i := 0 to a.Length {
if a[i] == 5 {
u := new Cell;
u.data := i;
}
}
r := if u == null then a.Length else u.data;
}
```
Here, using only the assignment u := null;
to infer the type of
u
would not be helpful. But Dafny looks past the initial
assignment and infers the type of u
to be Cell?
.
- The primary example where type inference does not inspect the entire context before giving up on inference is when there is a member lookup. For example,
```dafny
datatype List<T> = Nil | Cons(T, List<T>)
method Tutone() {
assert forall pair :: pair.0 == 867 && pair.1 == 5309 ==> pair == (867, 5309); // error: members .0 and .1 not found
assert forall pair: (int, int) :: pair.0 == 867 && pair.1 == 5309 ==> pair == (867, 5309);
}
```
In the first quantifier, type inference fails to infer the type of
pair
before it tries to look up the members .0
and .1
, which
results in a “type of the receiver not fully determined” error. The
remedy is to provide the type of pair
explicitly, as is done in the
second quantifier.
(In the future, Dafny may do more type inference before giving up on the member lookup.)
- If type parameters cannot be inferred, then they can be given explicitly in angle brackets. For example, in
```dafny
datatype Option<T> = None | Some(T)
method M() {
var a: Option<int> := None;
var b := None; // error: type is underspecified
var c := Option<int>.None;
var d := None;
d := Some(400);
}
```
the type of b
cannot be inferred, because it is underspecified.
However, the types of c
and d
are inferred to be Option<int>
.
Here is another example:
```dafny
function EmptySet<T>(): set<T> {
{}
}
method M() {
var a := EmptySet(); // error: type is underspecified
var b := EmptySet();
b := b + {2, 3, 5};
var c := EmptySet<int>();
}
```
The type instantiation in the initial assignment to a
cannot
be inferred, because it is underspecified. However, the type
instantiation in the initial assignment to b
is inferred to
be int
, and the types of b
and c
are inferred to be
set<int>
.
- Even the element type of
new
is optional, if it can be inferred. For example, in
```dafny
method NewArrays()
{
var a := new int[3];
var b: array<int> := new [3];
var c := new [3];
c[0] := 200;
var d := new [3] [200, 800, 77];
var e := new [] [200, 800, 77];
var f := new [3](_ => 990);
}
```
the omitted types of local variables are all inferred as
array<int>
and the omitted element type of each new
is inferred
to be int
.
-
In the absence of any other information, integer-looking literals (like
5
and7
) are inferred to have typeint
(and not, say,bv128
orORDINAL
). -
Many of the types inferred can be inspected in the IDE.
12.3. Ghost Inference
After14 type inference, Dafny revisits the program and makes a final decision about which statements are to be compiled, and which statements are ghost. The ghost statements form what is called the ghost context of expressions.
These statements are determined to be ghost:
assert
,assume
,reveal
, andcalc
statements.- The body of the
by
of anassert
statement. - Calls to ghost methods, including lemmas.
if
,match
, andwhile
statements with condition expressions or alternatives containing ghost expressions. Their bodies are also ghost.for
loops whose start expression contains ghost expressions.- Variable declarations if they are explicitly ghost or if their respective right-hand side is a ghost expression.
- Assignments or update statement if all updated variables are ghost.
forall
statements, unless there is exactly one assignment to a non-ghost array in its body.
These statements always non-ghost:
The following expressions are ghost, which is used in some of the tests above:
- All specification expressions
- All calls to functions and predicates marked as
ghost
- All variables, constants and fields declared using the
ghost
keyword
Note that inferring ghostness can uncover other errors, such as updating non-ghost variables in ghost contexts.
For example, if f
is a ghost function, in the presence of the following code:
var x := 1;
if(f(x)) {
x := 2;
}
Dafny will infer that the entire if
is ghost because the condition uses a ghost function,
and will then raise the error that it’s not possible to update the non-ghost variable x
in a ghost context.
12.4. Well-founded Functions and Extreme Predicates
Recursive functions are a core part of computer science and mathematics. Roughly speaking, when the definition of such a function spells out a terminating computation from given arguments, we may refer to it as a well-founded function. For example, the common factorial and Fibonacci functions are well-founded functions.
There are also other ways to define functions. An important case regards the definition of a boolean function as an extreme solution (that is, a least or greatest solution) to some equation. For computer scientists with interests in logic or programming languages, these extreme predicates are important because they describe the judgments that can be justified by a given set of inference rules (see, e.g., [@CamilleriMelham:InductiveRelations; @Winskel:FormalSemantics; @LeroyGrall:CoinductiveBigStep; @Pierce:SoftwareFoundations; @NipkowKlein:ConcreteSemantics]).
To benefit from machine-assisted reasoning, it is necessary not just to understand extreme predicates but also to have techniques for proving theorems about them. A foundation for this reasoning was developed by Paulin-Mohring [@PaulinMohring:InductiveCoq] and is the basis of the constructive logic supported by Coq [@Coq:book] as well as other proof assistants [@BoveDybjerNorell:BriefAgda; @SwamyEtAl:Fstar2011]. Essentially, the idea is to represent the knowledge that an extreme predicate holds by the proof term by which this knowledge was derived. For a predicate defined as the least solution, such proof terms are values of an inductive datatype (that is, finite proof trees), and for the greatest solution, a coinductive datatype (that is, possibly infinite proof trees). This means that one can use induction and coinduction when reasoning about these proof trees. These extreme predicates are known as, respectively, least predicates and greatest predicates. Support for extreme predicates is also available in the proof assistants Isabelle [@Paulson:CADE1994] and HOL [@Harrison:InductiveDefs].
Dafny supports both well-founded functions and extreme predicates. This section describes the difference in general terms, and then describes novel syntactic support in Dafny for defining and proving lemmas with extreme predicates. Although Dafny’s verifier has at its core a first-order SMT solver, Dafny’s logical encoding makes it possible to reason about fixpoints in an automated way.
The encoding for greatest predicates in Dafny was described previously [@LeinoMoskal:Coinduction] and is here described in the section about datatypes.
12.4.1. Function Definitions
To define a function $f \colon X \to Y$ in terms of itself, one can write a general equation like
$$f = \mathcal{F}(f)$$
where $\mathcal{F}$ is a non-recursive function of type
$(X \to Y) \to X \to Y$.
Because it takes a function as an argument,
$\mathcal{F}$
is referred to as a functor (or functional, but not to be
confused by the category-theory notion of a functor).
Throughout, assume that
$\mathcal{F}(f)$
by itself is well defined,
for example that it does not divide by zero. Also assume that
$f$
occurs
only in fully applied calls in
$\mathcal{F}(f)$;
eta expansion can be applied to
ensure this. If
$f$
is a boolean
function, that is, if
$Y$
is
the type of booleans, then
$f$ is called
a predicate.
For example, the common Fibonacci function over the natural numbers can be defined by the equation
$$ \mathit{fib} = \lambda n \bullet\: \mathbf{if}\:n < 2 \:\mathbf{then}\: n \:\mathbf{else}\: \mathit{fib}(n-2) + \mathit{fib}(n-1) $$
With the understanding that the argument $n$ is universally quantified, we can write this equation equivalently as
$$ \mathit{fib}(n) = \mathbf{if}\:n < 2\:\mathbf{then}\:n\:\mathbf{else}\:\mathit{fib}(n-2)%2B\mathit{fib}(n-1) $$
The fact that the function being defined occurs on both sides of the equation causes concern that we might not be defining the function properly, leading to a logical inconsistency. In general, there could be many solutions to an equation like the general equation or there could be none. Let’s consider two ways to make sure we’re defining the function uniquely.
12.4.1.1. Well-founded Functions
A standard way to ensure that the general equation has a unique solution in $f$ is to make sure the recursion is well-founded, which roughly means that the recursion terminates. This is done by introducing any well-founded relation $\ll$ on the domain of $f$ and making sure that the argument to each recursive call goes down in this ordering. More precisely, if we formulate the general equation as
$$ f(x) = \mathcal{F}{'}(f) $$
then we want to check $E \ll x$ for each call $f(E)$ in $f(x) = \mathcal{F}’(f)$. When a function definition satisfies this decrement condition, then the function is said to be well-founded.
For example, to check the decrement condition for $\mathit{fib}$ in the fib equation, we can pick $\ll$ to be the arithmetic less-than relation on natural numbers and check the following, for any $n$:
$$ 2 \leq n \;\Longrightarrow\; n-2 \ll n \;\wedge\; n-1 \ll n $$
Note that we are entitled to use the antecedent $2 \leq n$ because that is the condition under which the else branch in the fib equation is evaluated.
A well-founded function is often thought of as “terminating” in the sense that the recursive depth in evaluating $f$ on any given argument is finite. That is, there are no infinite descending chains of recursive calls. However, the evaluation of $f$ on a given argument may fail to terminate, because its width may be infinite. For example, let $P$ be some predicate defined on the ordinals and let $\mathit{P}_\downarrow$ be a predicate on the ordinals defined by the following equation:
$\mathit{P}_\downarrow = P(o) \;\wedge\; \forall p \bullet\; p \ll o \;\Longrightarrow\; \mathit{P}_\downarrow(p)$
With $\ll$ as the usual ordering on ordinals, this equation satisfies the decrement condition, but evaluating $\mathit{P}_\downarrow(\omega)$ would require evaluating $\mathit{P}_\downarrow(n)$ for every natural number $n$. However, what we are concerned about here is to avoid mathematical inconsistencies, and that is indeed a consequence of the decrement condition.
12.4.1.2. Example with Well-founded Functions
So that we can later see how inductive proofs are done in Dafny, let’s prove that for any $n$, $\mathit{fib}(n)$ is even iff $n$ is a multiple of $3$. We split our task into two cases. If $n < 2$, then the property follows directly from the definition of $\mathit{fib}$. Otherwise, note that exactly one of the three numbers $n-2$, $n-1$, and $n$ is a multiple of 3. If $n$ is the multiple of 3, then by invoking the induction hypothesis on $n-2$ and $n-1$, we obtain that $\mathit{fib}(n-2) + \mathit{fib}(n-1)$ is the sum of two odd numbers, which is even. If $n-2$ or $n-1$ is a multiple of 3, then by invoking the induction hypothesis on $n-2$ and $n-1$, we obtain that $\mathit{fib}(n-2) + \mathit{fib}(n-1)$ is the sum of an even number and an odd number, which is odd. In this proof, we invoked the induction hypothesis on $n-2$ and on $n-1$. This is allowed, because both are smaller than $n$, and hence the invocations go down in the well-founded ordering on natural numbers.
12.4.1.3. Extreme Solutions
We don’t need to exclude the possibility of the general equation having multiple solutions—instead, we can just be clear about which one of them we want. Let’s explore this, after a smidgen of lattice theory.
For any complete lattice $(Y,\leq)$ and any set $X$, we can by pointwise extension define a complete lattice $(X \to Y, \dot{\Rightarrow})$, where for any $f,g \colon X \to Y$,
$$ f \dot{\Rightarrow} g \;\;\equiv\;\; \forall x \bullet\; f(x) \leq g(x) $$
In particular, if $Y$ is the set of booleans ordered by implication (false
$\leq$ true
),
then the set of predicates over any domain $X$ forms a complete lattice.
Tarski’s Theorem [@Tarski:theorem] tells us that any monotonic function over a
complete lattice has a least and a greatest fixpoint. In particular, this means that
$\mathcal{F}$ has a least fixpoint and a greatest fixpoint, provided $\mathcal{F}$ is monotonic.
Speaking about the set of solutions in $f$ to the general equation is the same as speaking about the set of fixpoints of functor $\mathcal{F}$. In particular, the least and greatest solutions to the general equation are the same as the least and greatest fixpoints of $\mathcal{F}$. In casual speak, it happens that we say “fixpoint of the general equation”, or more grotesquely, “fixpoint of $f$” when we really mean “fixpoint of $\mathcal{F}$”.
To conclude our little excursion into lattice theory, we have that, under the
proviso of $\mathcal{F}$ being monotonic, the set of solutions in $f$ to the general equation is nonempty,
and among these solutions, there is in the $\dot{\Rightarrow}$ ordering a least solution (that is,
a function that returns false
more often than any other) and a greatest solution (that
is, a function that returns true
more often than any other).
When discussing extreme solutions, let’s now restrict our attention to boolean functions (that is, with $Y$ being the type of booleans). Functor $\mathcal{F}$ is monotonic if the calls to $f$ in $\mathcal{F}’(f)$ are in positive positions (that is, under an even number of negations). Indeed, from now on, we will restrict our attention to such monotonic functors $\mathcal{F}$.
Here is a running example. Consider the following equation, where $x$ ranges over the integers:
$$ g(x) = (x = 0 \:\vee\: g(x-2)) $$
This equation has four solutions in $g$. With $w$ ranging over the integers, they are:
$$ \begin{array}{r@{}l} g(x) \;\;\equiv\;\;{}& x \in \{w \;|\; 0 \leq w \;\wedge\; w\textrm{ even}\} \\ g(x) \;\;\equiv\;\;{}& x \in \{w \;|\; w\textrm{ even}\} \\ g(x) \;\;\equiv\;\;{}& x \in \{w \;|\; (0 \leq w \;\wedge\; w\textrm{ even}) \:\vee\: w\textrm{ odd}\} \\ g(x) \;\;\equiv\;\;{}& x \in \{w \;|\; \mathit{true}\} \end{array} $$
The first of these is the least solution and the last is the greatest solution.
In the literature, the definition of an extreme predicate is often given as a set of inference rules. To designate the least solution, a single line separating the antecedent (on top) from conclusion (on bottom) is used:
$$\dfrac{}{g(0)} \qquad\qquad \dfrac{g(x-2)}{g(x)}$$
Through repeated applications of such rules, one can show that the predicate holds for a particular value. For example, the derivation, or proof tree, to the left in the proof tree figure shows that $g(6)$ holds. (In this simple example, the derivation is a rather degenerate proof “tree”.) The use of these inference rules gives rise to a least solution, because proof trees are accepted only if they are finite.
When inference rules are to designate the greatest solution, a thick line is used:
$$\genfrac{}{}{1.2pt}0{}{g(0)} \qquad\qquad \genfrac{}{}{1.2pt}0{g(x-2)}{g(x)}$$
In this case, proof trees are allowed to be infinite. For example, the left-hand example below shows a finite proof tree that uses the inductive rules to establish $g(6)$. On the right is a partial depiction of an infinite proof tree that uses the coinductive rules to establish $g(1)$.
$$\dfrac{ \dfrac{ \dfrac{ \dfrac{}{g(0)} }{g(2)} }{g(4)} }{g(6)} \qquad\qquad \genfrac{}{}{1.2pt}0{ \genfrac{}{}{1.2pt}0{ \genfrac{}{}{1.2pt}0{ \genfrac{}{}{1.2pt}0{ {} {\vdots } }{g(-5)} }{g(-3)} }{g(-1)} }{g(1)}$$
Note that derivations may not be unique. For example, in the case of the greatest solution for $g$, there are two proof trees that establish $g(0)$: one is the finite proof tree that uses the left-hand rule of these coinductive rules once, the other is the infinite proof tree that keeps on using the right-hand rule of these coinductive rules.
12.4.2. Working with Extreme Predicates
In general, one cannot evaluate whether or not an extreme predicate holds for some input, because doing so may take an infinite number of steps. For example, following the recursive calls in the definition the EvenNat equation to try to evaluate $g(7)$ would never terminate. However, there are useful ways to establish that an extreme predicate holds and there are ways to make use of one once it has been established.
For any $\mathcal{F}$ as in the general equation, define two infinite series of well-founded functions, ${ {}^{\flat}\kern-1mm f}_k$ and ${ {}^{\sharp}\kern-1mm f}_k$ where $k$ ranges over the natural numbers:
$$ { {}^{\flat}\kern-1mm f}_k(x) = \left\{ \begin{array}{ll} \mathit{false} & \textrm{if } k = 0 \\ \mathcal{F}({ {}^{\flat}\kern-1mm f}_{k-1})(x) & \textrm{if } k > 0 \end{array} \right\} $$
$$ { {}^{\sharp}\kern-1mm f}_k(x) = \left\{ \begin{array}{ll} \mathit{true} & \textrm{if } k = 0 \\ \mathcal{F}({ {}^{\sharp}\kern-1mm f}_{k-1})(x) & \textrm{if } k > 0 \end{array} \right\} $$
These functions are called the iterates of $f$, and we will also refer to them
as the prefix predicates of $f$ (or the prefix predicate of $f$, if we think
of $k$ as being a parameter).
Alternatively, we can define ${ {}^{\flat}\kern-1mm f}_k$ and ${ {}^{\sharp}\kern-1mm f}_k$ without mentioning $x$:
let $\bot$ denote the function that always returns false
, let $\top$
denote the function that always returns true
, and let a superscript on $\mathcal{F}$ denote
exponentiation (for example, $\mathcal{F}^0(f) = f$ and $\mathcal{F}^2(f) = \mathcal{F}(\mathcal{F}(f))$).
Then, the least approx definition and the greatest approx definition can be stated equivalently as
${ {}^{\flat}\kern-1mm f}_k = \mathcal{F}^k(\bot)$ and ${ {}^{\sharp}\kern-1mm f}_k = \mathcal{F}^k(\top)$.
For any solution $f$ to the general equation, we have, for any $k$ and $\ell$ such that $k \leq \ell$:
$$ {\;{}^{\flat}\kern-1mm f}_k \quad\;\dot{\Rightarrow}\;\quad {\;{}^{\flat}\kern-1mm f}_\ell \quad\;\dot{\Rightarrow}\;\quad f \quad\;\dot{\Rightarrow}\;\quad {\;{}^{\sharp}\kern-1mm f}_\ell \quad\;\dot{\Rightarrow}\;\quad { {}^{\sharp}\kern-1mm f}_k $$
In other words, every ${\;{}^{\flat}\kern-1mm f}_{k}$ is a pre-fixpoint of $f$ and every ${\;{}^{\sharp}\kern-1mm f}_{k}$ is a post-fixpoint of $f$. Next, define two functions, $f^{\downarrow}$ and $f^{\uparrow}$, in terms of the prefix predicates:
$$ f^{\downarrow}(x) \;=\; \exists k \bullet\; { {}^{\flat}\kern-1mm f}_k(x) $$
$$ f^{\uparrow}(x) \;=\; \forall k \bullet\; { {}^{\sharp}\kern-1mm f}_k(x) $$
By the prefix postfix result, we also have that $f^{\downarrow}$ is a pre-fixpoint of $\mathcal{F}$ and $f^{\uparrow}$ is a post-fixpoint of $\mathcal{F}$. The marvelous thing is that, if $\mathcal{F}$ is continuous, then $f^{\downarrow}$ and $f^{\uparrow}$ are the least and greatest fixpoints of $\mathcal{F}$. These equations let us do proofs by induction when dealing with extreme predicates. The extreme predicate section explains how to check for continuity.
Let’s consider two examples, both involving function $g$ in the EvenNat equation. As it turns out, $g$’s defining functor is continuous, and therefore I will write $g^{\downarrow}$ and $g^{\uparrow}$ to denote the least and greatest solutions for $g$ in the EvenNat equation.
12.4.2.1. Example with Least Solution
The main technique for establishing that $g^{\downarrow}(x)$ holds for some $x$, that is, proving something of the form $Q \Longrightarrow g^{\downarrow}(x)$, is to construct a proof tree like the one for $g(6)$ in the proof tree figure. For a proof in this direction, since we’re just applying the defining equation, the fact that we’re using a least solution for $g$ never plays a role (as long as we limit ourselves to finite derivations).
The technique for going in the other direction, proving something from an established $g^{\downarrow}$ property, that is, showing something of the form $g^{\downarrow}(x) \Longrightarrow R$, typically uses induction on the structure of the proof tree. When the antecedent of our proof obligation includes a predicate term $g^{\downarrow}(x)$, it is sound to imagine that we have been given a proof tree for $g^{\downarrow}(x)$. Such a proof tree would be a data structure—to be more precise, a term in an inductive datatype. Least solutions like $g^{\downarrow}$ have been given the name least predicate.
Let’s prove $g^{\downarrow}(x) \Longrightarrow 0 \leq x \wedge x \text{ even}$. We split our task into two cases, corresponding to which of the two proof rules in the inductive rules was the last one applied to establish $g^{\downarrow}(x)$. If it was the left-hand rule, then $x=0$, which makes it easy to establish the conclusion of our proof goal. If it was the right-hand rule, then we unfold the proof tree one level and obtain $g^{\downarrow}(x-2)$. Since the proof tree for $g^{\downarrow}(x-2)$ is smaller than where we started, we invoke the induction hypothesis and obtain $0 \leq (x-2) \wedge (x-2) \textrm{ even}$, from which it is easy to establish the conclusion of our proof goal.
Here’s how we do the proof formally using the least exists definition. We massage the general form of our proof goal:
$$ \begin{array}{lll} & f^{\uparrow}(x) \;\Longrightarrow\; R & \\ = & & \textrm{ (the least exists definition) } \\ & (\exists k \bullet\; { {}^{\flat}\kern-1mm f}_k(x)) \;\Longrightarrow\; R & \\ = & & \text{distribute} \;\Longrightarrow\; \text{over} \;\exists\; \text{to the left} \\ & \forall k \bullet\; ({ {}^{\flat}\kern-1mm f}_k(x) \;\Longrightarrow\; R) & \end{array} $$
The last line can be proved by induction over $k$. So, in our case, we prove
${ {}^{\flat}\kern-1mm g}_k(x) \Longrightarrow 0 \leq x \wedge x \textrm{ even}$ for every $k$.
If $k = 0$, then ${ {}^{\flat}\kern-1mm g}_k(x)$ is false
, so our goal holds trivially.
If $k > 0$, then ${ {}^{\flat}\kern-1mm g}_k(x) = (x = 0 :\vee: { {}^{\flat}\kern-1mm g}_{k-1}(x-2))$. Our goal holds easily
for the first disjunct ($x=0$). For the other disjunct,
we apply the induction hypothesis (on the smaller $k-1$ and with $x-2$) and
obtain $0 \leq (x-2)\;\wedge\; (x-2) \textrm{ even}$, from which our proof goal
follows.
12.4.2.2. Example with Greatest Solution
We can think of a predicate $g^{\uparrow}(x)$ as being represented by a proof tree—in this case a term in a coinductive datatype, since the proof may be infinite. Greatest solutions like $g^{\uparrow}$ have been given the name greatest predicate. The main technique for proving something from a given proof tree, that is, to prove something of the form $g^{\uparrow}(x) \;\Longrightarrow\; R$, is to destruct the proof. Since this is just unfolding the defining equation, the fact that we’re using a greatest solution for $g$ never plays a role (as long as we limit ourselves to a finite number of unfoldings).
To go in the other direction, to establish a predicate defined as a greatest solution, like $Q \Longrightarrow g^{\uparrow}(x)$, we may need an infinite number of steps. For this purpose, we can use induction’s dual, coinduction. Were it not for one little detail, coinduction is as simple as continuations in programming: the next part of the proof obligation is delegated to the coinduction hypothesis. The little detail is making sure that it is the “next” part we’re passing on for the continuation, not the same part. This detail is called productivity and corresponds to the requirement in induction of making sure we’re going down a well-founded relation when applying the induction hypothesis. There are many sources with more information, see for example the classic account by Jacobs and Rutten [@JacobsRutten:IntroductionCoalgebra] or a new attempt by Kozen and Silva that aims to emphasize the simplicity, not the mystery, of coinduction [@KozenSilva:Coinduction].
Let’s prove $\mathit{true} \Longrightarrow g^{\uparrow}(x)$. The intuitive coinductive proof goes like this: According to the right-hand rule of these coinductive rules, $g^{\uparrow}(x)$ follows if we establish $g^{\uparrow}(x-2)$, and that’s easy to do by invoking the coinduction hypothesis. The “little detail”, productivity, is satisfied in this proof because we applied a rule in these coinductive rules before invoking the coinduction hypothesis.
For anyone who may have felt that the intuitive proof felt too easy, here is a formal proof using the greatest forall definition, which relies only on induction. We massage the general form of our proof goal:
$$ \begin{array}{lll} & Q \;\Longrightarrow\; f^{\uparrow}(x) & \\ = & & \textrm{ (the greatest forall definition) } \\ & Q \;\Longrightarrow\; \forall k \bullet\; { {}^{\sharp}\kern-1mm f}_k(x) & \\ = & & \text{distribute} \;\Longrightarrow\; \text{over} \;\forall\; \text{to the right } \\ & \forall k \bullet\; Q \;\Longrightarrow\; { {}^{\sharp}\kern-1mm f}_k(x) & \end{array} $$
The last line can be proved by induction over $k$. So, in our case, we prove $\mathit{true} \;\Longrightarrow\; { {}^{\sharp}\kern-1mm g}_k(x)$ for every $k$. If $k=0$, then ${ {}^{\sharp}\kern-1mm g}_k(x)$ is $\mathit{true}$, so our goal holds trivially. If $k > 0$, then ${ {}^{\sharp}\kern-1mm g}_k(x) = (x = 0 :\vee: { {}^{\sharp}\kern-1mm g}_{k-1}(x-2))$. We establish the second disjunct by applying the induction hypothesis (on the smaller $k-1$ and with $x-2$).
12.4.3. Other Techniques
Although this section has considered only well-founded functions and extreme predicates, it is worth mentioning that there are additional ways of making sure that the set of solutions to the general equation is nonempty. For example, if all calls to $f$ in $\mathcal{F}’(f)$ are tail-recursive calls, then (under the assumption that $Y$ is nonempty) the set of solutions is nonempty. To see this, consider an attempted evaluation of $f(x)$ that fails to determine a definite result value because of an infinite chain of calls that applies $f$ to each value of some subset $X’$ of $X$. Then, apparently, the value of $f$ for any one of the values in $X’$ is not determined by the equation, but picking any particular result value for these makes for a consistent definition. This was pointed out by Manolios and Moore [@ManoliosMoore:PartialFunctions]. Functions can be underspecified in this way in the proof assistants ACL2 [@ACL2:book] and HOL [@Krauss:PhD].
12.5. Functions in Dafny
This section explains with examples the support in Dafny for well-founded functions, extreme predicates, and proofs regarding these, building on the concepts explained in the previous section.
12.5.1. Well-founded Functions in Dafny
Declarations of well-founded functions are unsurprising. For example, the Fibonacci function is declared as follows:
function fib(n: nat): nat
{
if n < 2 then n else fib(n-2) + fib(n-1)
}
Dafny verifies that the body (given as an expression in curly braces) is well defined.
This includes decrement checks for recursive (and mutually recursive) calls. Dafny
predefines a well-founded relation on each type and extends it to lexicographic tuples
of any (fixed) length. For example, the well-founded relation $x \ll y$ for integers
is $x < y \;\wedge\; 0 \leq y$, the one for reals is $x \leq y - 1.0 \;\wedge\; 0.0 \leq y$
(this is the same ordering as for integers, if you read the integer
relation as $x \leq y - 1 \;\wedge\; 0 \leq y$), the one for inductive
datatypes is structural inclusion,
and the one for coinductive datatypes is false
.
Using a decreases
clause, the programmer can specify the term in this predefined
order. When a function definition omits a decreases
clause, Dafny makes a simple
guess. This guess (which can be inspected by hovering over the function name in the
Dafny IDE) is very often correct, so users are rarely bothered to provide explicit
decreases
clauses.
If a function returns bool
, one can drop the result type : bool
and change the
keyword function
to predicate
.
12.5.2. Proofs in Dafny
Dafny has lemma
declarations, as described in Section 6.3.3:
lemmas can have pre- and postcondition specifications and their body is a code block.
Here is the lemma we stated and proved in the fib example in the previous section:
lemma FibProperty(n: nat)
ensures fib(n) % 2 == 0 <==> n % 3 == 0
{
if n < 2 {
} else {
FibProperty(n-2); FibProperty(n-1);
}
}
function fib(n: nat): nat
{
if n < 2 then n else fib(n-2) + fib(n-1)
}
The postcondition of this lemma (keyword ensures
) gives the proof
goal. As in any program-correctness logic (e.g.,
[@Hoare:AxiomaticBasis]), the postcondition must
be established on every control path through the lemma’s body. For
FibProperty
, I give the proof by
an if
statement, hence introducing a case split. The then branch is empty, because
Dafny can prove the postcondition automatically in this case. The else branch
performs two recursive calls to the lemma. These are the invocations of the induction
hypothesis and they follow the usual program-correctness rules,
namely: the precondition must hold at the call site, the call must terminate, and then
the caller gets to assume the postcondition upon return. The “proof glue” needed
to complete the proof is done automatically by Dafny.
Dafny features an aggregate statement using which it is possible to make (possibly
infinitely) many calls at once. For example, the induction hypothesis can be called
at once on all values n'
smaller than n
:
forall n' | 0 <= n' < n {
FibProperty(n');
}
For our purposes, this corresponds to strong induction. More
generally, the forall
statement has the form
forall k | P(k)
ensures Q(k)
{ Statements; }
Logically, this statement corresponds to universal introduction: the body proves that
Q(k)
holds for an arbitrary k
such that P(k)
, and the conclusion of the forall
statement
is then $\forall k \bullet\; P(k) \;\Longrightarrow\; Q(k)$. When the body of the forall
statement is
a single call (or calc
statement), the ensures
clause is inferred and can be omitted,
like in our FibProperty
example.
Lemma FibProperty
is simple enough that its whole body can be replaced by the one
forall
statement above. In fact, Dafny goes one step further: it automatically
inserts such a forall
statement at the beginning of every lemma [@Leino:induction].
Thus, FibProperty
can be declared and proved simply by:
lemma FibProperty(n: nat)
ensures fib(n) % 2 == 0 <==> n % 3 == 0
{ }
function fib(n: nat): nat
{
if n < 2 then n else fib(n-2) + fib(n-1)
}
Going in the other direction from universal introduction is existential elimination,
also known as Skolemization. Dafny has a statement for this, too:
for any variable x
and boolean expression Q
, the
assign such that statement x :| Q;
says to assign to x
a value such that Q
will hold. A proof obligation when using this statement is to show that there
exists an x
such that Q
holds. For example, if the fact
$\exists k \bullet\; 100 \leq \mathit{fib}(k) < 200$ is known, then the statement
k :| 100 <= fib(k) < 200;
will assign to k
some value (chosen arbitrarily)
for which fib(k)
falls in the given range.
12.5.3. Extreme Predicates in Dafny
The previous subsection explained that a predicate
declaration introduces a
well-founded predicate. The declarations for introducing extreme predicates are
least predicate
and greatest predicate
. Here is the definition of the least and
greatest solutions of $g$ from above; let’s call them g
and G
:
least predicate g[nat](x: int) { x == 0 || g(x-2) }
greatest predicate G[nat](x: int) { x == 0 || G(x-2) }
When Dafny receives either of these definitions, it automatically declares the corresponding
prefix predicates. Instead of the names ${ {}^{\flat}\kern-1mm g}_k$ and ${ {}^{\sharp}\kern-1mm g}_k$ that I used above, Dafny
names the prefix predicates g#[k]
and G#[k]
, respectively, that is, the name of
the extreme predicate appended with #
, and the subscript is given as an argument in
square brackets. The definition of the prefix predicate derives from the body of
the extreme predicate and follows the form in the least approx definition and the greatest approx definition.
Using a faux-syntax for illustrative purposes, here are the prefix
predicates that Dafny defines automatically from the extreme
predicates g
and G
:
predicate g#[_k: nat](x: int) { _k != 0 && (x == 0 || g#[_k-1](x-2)) }
predicate G#[_k: nat](x: int) { _k != 0 ==> (x == 0 || G#[_k-1](x-2)) }
The Dafny verifier is aware of the connection between extreme predicates and their prefix predicates, the least exists definition and the greatest forall definition.
Remember that to be well defined, the defining functor of an extreme predicate must be monotonic, and for the least exists definition and the greatest forall definition to hold, the functor must be continuous. Dafny enforces the former of these by checking that recursive calls of extreme predicates are in positive positions. The continuity requirement comes down to checking that they are also in continuous positions: that recursive calls to least predicates are not inside unbounded universal quantifiers and that recursive calls to greatest predicates are not inside unbounded existential quantifiers [@Milner:CCS; @LeinoMoskal:Coinduction].
12.5.4. Proofs about Extreme Predicates
From what has been presented so far, we can do the formal proofs for the example about the least solution and the example about the greatest solution. Here is the former:
least predicate g[nat](x: int) { x == 0 || g(x-2) }
greatest predicate G[nat](x: int) { x == 0 || G(x-2) }
lemma EvenNat(x: int)
requires g(x)
ensures 0 <= x && x % 2 == 0
{
var k: nat :| g#[k](x);
EvenNatAux(k, x);
}
lemma EvenNatAux(k: nat, x: int)
requires g#[k](x)
ensures 0 <= x && x % 2 == 0
{
if x == 0 { } else { EvenNatAux(k-1, x-2); }
}
Lemma EvenNat
states the property we wish to prove. From its
precondition (keyword requires
) and
the least exists definition, we know there is some k
that will make the condition in the
assign-such-that statement true. Such a value is then assigned to k
and passed to
the auxiliary lemma, which promises to establish the proof goal. Given the condition
g#[k](x)
, the definition of g#
lets us conclude k != 0
as well as the disjunction
x == 0 || g#[k-1](x-2)
. The then branch considers the case of the first disjunct,
from which the proof goal follows automatically. The else branch can then assume
g#[k-1](x-2)
and calls the induction hypothesis with those parameters. The proof
glue that shows the proof goal for x
to follow from the proof goal with x-2
is
done automatically.
Because Dafny automatically inserts the statement
forall k', x' | 0 <= k' < k && g#[k'](x') {
EvenNatAux(k', x');
}
at the beginning of the body of EvenNatAux
, the body can be left empty and Dafny
completes the proof automatically.
Here is the Dafny program that gives the proof from the example of the greatest solution:
least predicate g[nat](x: int) { x == 0 || g(x-2) }
greatest predicate G[nat](x: int) { x == 0 || G(x-2) }
lemma Always(x: int)
ensures G(x)
{ forall k: nat { AlwaysAux(k, x); } }
lemma AlwaysAux(k: nat, x: int)
ensures G#[k](x)
{ }
While each of these proofs involves only basic proof rules, the setup feels a bit clumsy, even with the empty body of the auxiliary lemmas. Moreover, the proofs do not reflect the intuitive proofs described in the example of the least solution and the example of the greatest solution. These shortcomings are addressed in the next subsection.
12.5.5. Nicer Proofs of Extreme Predicates
The proofs we just saw follow standard forms:
use Skolemization to convert the least predicate into a prefix predicate for some k
and then do the proof inductively over k
; respectively,
by induction over k
, prove the prefix predicate for every k
, then use
universal introduction to convert to the greatest predicate.
With the declarations least lemma
and greatest lemma
, Dafny offers to
set up the proofs
in these standard forms. What is gained is not just fewer characters in the program
text, but also a possible intuitive reading of the proofs. (Okay, to be fair, the
reading is intuitive for simpler proofs; complicated proofs may or may not be intuitive.)
Somewhat analogous to the creation of prefix predicates from extreme predicates, Dafny
automatically creates a prefix lemma L#
from each “extreme lemma” L
. The pre-
and postconditions of a prefix lemma are copied from those of the extreme lemma,
except for the following replacements:
- for a least lemma, Dafny looks in the precondition to find calls (in positive, continuous
positions) to least predicates
P(x)
and replaces these withP#[_k](x)
; - for a greatest lemma,
Dafny looks in the postcondition to find calls (in positive, continuous positions)
to greatest predicates
P
(including equality among coinductive datatypes, which is a built-in greatest predicate) and replaces these withP#[_k](x)
. In each case, these predicatesP
are the lemma’s focal predicates.
The body of the extreme lemma is moved to the prefix lemma, but with
replacing each recursive
call L(x)
with L#[_k-1](x)
and replacing each occurrence of a call
to a focal predicate
P(x)
with P#[_k-1](x)
. The bodies of the extreme lemmas are then replaced as shown
in the previous subsection. By construction, this new body correctly leads to the
extreme lemma’s postcondition.
Let us see what effect these rewrites have on how one can write proofs. Here are the proofs of our running example:
least predicate g(x: int) { x == 0 || g(x-2) }
greatest predicate G(x: int) { x == 0 || G(x-2) }
least lemma EvenNat(x: int)
requires g(x)
ensures 0 <= x && x % 2 == 0
{ if x == 0 { } else { EvenNat(x-2); } }
greatest lemma Always(x: int)
ensures G(x)
{ Always(x-2); }
Both of these proofs follow the intuitive proofs given in the example of the least solution and the example of the greatest solution. Note that in these simple examples, the user is never bothered with either prefix predicates nor prefix lemmas—the proofs just look like “what you’d expect”.
Since Dafny automatically inserts calls to the induction hypothesis at the beginning of
each lemma, the bodies of the given extreme lemmas EvenNat
and
Always
can be empty and Dafny still completes the proofs.
Folks, it doesn’t get any simpler than that!
12.6. Variable Initialization and Definite Assignment
The Dafny language semantics ensures that any use (read) of a variable (or constant, parameter, object field, or array element) gives a value of the variable’s type. It is easy to see that this property holds for any variable that is declared with an initializing assignment. However, for many useful programs, it would be too strict to require an initializing assignment at the time a variable is declared. Instead, Dafny ensures the property through auto-initialization and rules for definite assignment.
As explained in section 5.3.1, each type in Dafny is one of the following:
- auto-init type: the type is nonempty and the compiler has some way to emit code that constructs a value
- nonempty type: the type is nonempty, but the compiler does not know how perform automatic initialization
- possibly empty type: the type is not known for sure to have a value
For a variable of an auto-init type, the compiler can initialize the variable automatically. This means that the variable can be used immediately after declaration, even if the program does not explicitly provide an initializing assignment.
In a ghost context, one can an imagine a “ghost” that initializes variables. Unlike the compiler, such a “ghost” does not need to emit code that constructs an initializing value; it suffices for the ghost to know that a value exists. Therefore, in a ghost context, a variable of a nonempty type can be used immediately after declaration.
Before a variable of a possibly empty type can be used, the program must initialize it. The variable need not be given a value when it is declared, but it must have a value by the time it is first used. Dafny uses the precision of the verifier to reason about the control flow between assignments and uses of variables, and it reports an error if it cannot assure itself that the variable has been given a value.
The elements of an array must be assured to have values already in the statement that allocates the array. This is achieved in any of the following four ways:
- If the array is allocated to be empty (that is, one of its dimensions is requested to be 0), then the array allocation trivially satisfies the requirement.
- If the element type of the array is an auto-init type, then nothing further is required by the program.
- If the array allocation occurs in a ghost context and the element type is a nonempty type, then nothing further is required by the program.
- Otherwise, the array allocation must provide an initialization display or an initialization function. See section 5.10 for information about array initialization.
The fields of a class must have values by the end of the first phase of each constructor (that is, at
the explicit or implicit new;
statement in the constructor). If a class has a compiled field that is
not of an auto-init type, or if it has a ghost field of a possibly empty type, then the class is required
to declare a(t least one) constructor.
The yield-parameters of an iterator
turn into fields of the corresponding iterator class, but there
is no syntactic place to give these initial values. Therefore, every compiled yield-parameter must be of
auto-init types and every ghost yield-parameter must be of an auto-init or nonempty type.
For local variables and out-parameters, Dafny supports two definite-assignment modes:
- A strict mode (the default, which is
--relax-definite-assignment=false
; or/definiteAssignment:4
in the legacy CLI), in which local variables and out-parameters are always subject to definite-assignment rules, even for auto-initializable types. - A relaxed mode (enabled by the option
--relax-definite-assignment
; or/definiteAssignment:1
in the legacy CLI), in which the auto-initialization (or, for ghost variables and parametes, nonemptiness) is sufficient to satisfy the definite assignment rules.
A program using the strict mode can still indicate that it is okay with an arbitrary value of a variable x
by using an assignment statement x := *;
, provided the type of x
is an auto-init type (or, if x
is
ghost, a nonempty type). (If x
is of a possibly nonempty type, then x := *;
is still allowed, but it
sets x
to a value of its type only if the type actually contains a value. Therefore, when x
is of
a possibly empty type, x := *;
does not count as a definite assignment to x
.)
Note that auto-initialization is nondeterministic. Dafny only guarantees that each value it assigns to
a variable of an auto-init type is some value of the type. Indeed, a variable may be auto-initialized
to different values in different runs of the program or even at different times during the same run of
the program. In other words, Dafny does not guarantee the “zero-equivalent value” initialization that
some languages do. Along these lines, also note that the witness
value provided in some subset-type
declarations is not necessarily the value chosen by auto-initialization, though it does esstablish that
the type is an auto-init type.
In some programs (for example, in some test programs), it is desirable to avoid nondeterminism.
For that purpose, Dafny provides an --enforce-determinism
option. It forbids use of any program
statement that may have nondeterministic behavior and it disables auto-initialization.
This mode enforces definite assignments everywhere, going beyond what the strict mode does by enforcing
definite assignment also for fields and array elements. It also forbids the use of iterator
declarations
and constructor
-less class
declarations. It is up to a user’s build process to ensure that
--enforce-determinism
is used consistently throughout the program. (In the legacy CLI, this
mode is enabled by /definiteAssignment:3
.)
This document, which is intended for developers of the Dafny tool itself, has more detail on auto-initialization and how it is implemented.
Finally, note that --relax-definite-assignment=false
is the default in the command-based CLI,
but, for backwards compatibility, the relaxed rules (`/definiteAssignment:1) are still the default
in the legacy CLI.
12.7. Well-founded Orders
The well-founded order relations for a variety of built-in types in Dafny are given in the following table:
type of X and x |
x strictly below X |
---|---|
bool |
X && !x |
int |
x < X && 0 <= X |
real |
x <= X - 1.0 && 0.0 <= X |
set<T> |
x is a proper subset of X |
multiset<T> |
x is a proper multiset-subset of X |
seq<T> |
x is a consecutive proper sub-sequence of X |
map<K, V> |
x.Keys is a proper subset of X.Keys |
inductive datatypes | x is structurally included in X |
reference types | x == null && X != null |
coinductive datatypes | false |
type parameter | false |
arrow types | false |
Also, there are a few relations between the rows in the table above. For example, a datatype value x
sitting inside a set that sits inside another datatype value X
is considered to be strictly below X
. Here’s an illustration of that order, in a program that verifies:
datatype D = D(s: set<D>)
method TestD(dd: D) {
var d := dd;
while d != D({})
decreases d
{
var x :| x in d.s;
d := x;
}
}
13. Dafny User’s Guide
Most of this document describes the Dafny programming language.
This section describes the dafny
tool, a combined verifier and compiler
that implements the Dafny language.
The development of the Dafny language and tool is a GitHub project at https://github.com/dafny-lang/dafny. The project is open source, with collaborators from various organizations; additional contributors are welcome. The software itself is licensed under the MIT license.
13.1. Introduction
The dafny
tool implements the following primary capabilities, implemented as various commands within the dafny
tool:
- checking that the input files represent a valid Dafny program (i.e., syntax, grammar and name and type resolution);
- verifying that the program meets its specifications, by translating the program to verification conditions and checking those with Boogie and an SMT solver, typically Z3;
- compiling the program to a target language, such as C#, Java, Javascript, Go (and others in development);
- running the executable produced by the compiler.
In addition there are a variety of other capabilities, such as formatting files, also implemented as commands; more such commands are expected in the future.
13.2. Installing Dafny
13.2.1. Command-line tools
The instructions for installing dafny
and the required dependencies and environment
are described on the Dafny wiki:
https://github.com/dafny-lang/dafny/wiki/INSTALL.
They are not repeated here to avoid replicating information that
easily becomes inconsistent and out of date.
The dafny tool can also be installed using dotnet tool install --global dafny
(presuming that dotnet
is already installed on your system).
Most users will find it most convenient to install the pre-built Dafny binaries available on the project release site or using the dotnet
CLI.
As is typical for Open Source projects, dafny can also be built directly from the source files maintained in the github project.
Current and past Dafny binary releases can be found at https://github.com/dafny-lang/dafny/releases for each supported platform. Each release is a .zip file with a name combining the release name and the platform. Current platforms are Windows 11, Ubuntu 20 and later, and MacOS 10.14 and later.
The dafny tool is distributed as a standalone executable. A compatible version of the required Z3 solver is included in the release. There are additional dependencies that are needed to compile dafny to particular target languages, as described in the release instructions. A development environment to build dafny from source requires additional dependencies, described here.
13.2.2. IDEs for Dafny
Dafny source files are text files and can of course be edited with any text editor. However, some tools provide syntax-aware features:
- VSCode, a cross-platform editor for many programming languages has an extension for Dafny. VSCode is available here and the Dafny extension can be installed from within VSCode. The extension provides syntax highlighting, in-line parser, type and verification errors, code navigation, counter-example display and gutter highlights.
- There is a Dafny mode for Emacs.
- An old Visual Studio plugin is no longer supported
Information about installing IDE extensions for Dafny is found on the Dafny INSTALL page in the wiki.
More information about using VSCode IDE is here.
13.3. Dafny Programs and Files
A Dafny program is a set of modules.
Modules can refer to other modules, such as through import
declarations
or refines
clauses.
A Dafny program consists of all the modules needed so that all module
references are resolved.
Dafny programs are contained in files that have a .dfy
suffix.
Such files each hold
some number of top-level declarations. Thus a full program may be
distributed among multiple files.
To apply the dafny
tool to a Dafny program, the dafny
tool must be
given all the files making up a complete program (or, possibly, more than
one program at a time). This can be effected either by listing all of the files
by name on the command-line or by using include
directives within a file
to stipulate what other files contain modules that the given files need.
Thus the complete set of modules are all the modules in all the files listed
on the command-line or referenced, recursively, by include
directives
within those files. It does not matter if files are repeated either as
includes or on the command-line.15
All files recursively included are always parsed and type-checked. However, which files are verified, built, run, or processed by other dafny commands depends on the individual command. These commands are described in Section 13.5.1.
For the purpose of detecting duplicates, file names are considered equal if they have the same absolute path, compared as case-sensitive strings (regardless of whether the underlying file-system is case sensitive). Using symbolic links may make the same file have a different absolute path; this will generally cause duplicate declaration errors.
13.3.1. Dafny Build Artifacts: the Library Backend and .doo Files
As of Dafny 4.1, dafny
now supports outputting a single file containing
a fully-verified program along with metadata about how it was verified.
Such files use the extension .doo
, for Dafny Output Object,
and can be used as input anywhere a .dfy
file can be.
.doo
files are produced by an additional backend called the “Dafny Library” backend,
identified with the name lib
on the command line. For example, to build multiple
Dafny files into a single build artifact for shared reuse, the command would look something like:
dafny build -t:lib A.dfy B.dfy C.dfy --output:MyLib.doo
The Dafny code contained in a .doo
file is not re-verified when passed back to the dafny
tool,
just as included files and those passed with the --library
option are not.
Using .doo
files provides a guarantee that the Dafny code was in fact verified,
however, and therefore offers protection against build system mistakes.
.doo
files are therefore ideal for sharing Dafny libraries between projects.
.doo
files also contain metadata about the version of Dafny used to verify them
and the values of relevant options that affect the sound separate verification and
compilation of Dafny code, such as --unicode-char
.
This means attempting to use a library that was built with options
that are not compatible with the currently executing command options
will lead to errors.
This also includes attempting to use a .doo
file built with a different version of Dafny,
although this restriction may be lifted in the future.
A .doo
file is a compressed archive of multiple files, similar to the .jar
file format for Java packages.
The exact file format is internal and may evolve over time to support additional features.
Note that the library backend only supports the newer command-style CLI interface.
13.4. Dafny Code Style
There are coding style conventions for Dafny code, recorded here. Most significantly, code is written without tabs and with a 2 space indentation. Following code style conventions improves readability but does not alter program semantics.
13.5. Using Dafny From the Command Line
dafny
is a conventional command-line tool, operating just like other
command-line tools in Windows and Unix-like systems.
In general, the format of a command-line is determined by the shell program that is executing the command-line
(.e.g., bash, the windows shell, COMMAND, etc.),
but is expected to be a series of space-separated “words”, each representing a command, option, option argument, file, or folder.
13.5.1. dafny commands
As of v3.9.0, dafny
uses a command-style command-line (like git
for example); prior to v3.9.0, the
command line consisted only of options and files.
It is expected that additional commands will be added in the future.
Each command may have its own subcommands and its own options, in addition to generally applicable options.
Thus the format of the command-line is
a command name, followed by options and files:
dafny <command> <options> <files>
;
the command-name must be the first command-line argument.
The command-line dafny --help
or dafny -h
lists all the available commands.
The command-line dafny <command> --help
(or -h
or -?
) gives help information for that particular <command>, including the list of options.
Some options for a particular command are intended only for internal tool development; those are shown using the --help-internal
option instead of --help
.
Also, the command-style command-line has modernized the syntax of options; they are now POSIX-compliant.
Like many other tools, options now typically begin with a double hyphen,
with some options having a single-hyphen short form, such as --help
and -h
.
If no <command> is given, then the command-line is presumed to use old-style syntax, so any previously written command-line will still be valid.
dafny
recognizes the commands described in the following subsections. The most commonly used
are dafny verify
, dafny build
, and dafny run
.
The command-line also expects the following:
- Files are designated by absolute paths or paths relative to the current working directory. A command-line argument not matching a known option is considered a filepath, and likely one with an unsupported suffix, provoking an error message.
- Files containing dafny code must have a
.dfy
suffix. - There must be at least one
.dfy
file (except when using--stdin
or in the case ofdafny format
, see the Dafny format section) - The command-line may contain other kinds of files appropriate to the language that the Dafny files are being compiled to. The kind of file is determined by its suffix.
- Escape characters are determined by the shell executing the command-line.
- Per POSIX convention, the option
--
means that all subsequent command-line arguments are not options to the dafny tool; they are either files or arguments to thedafny run
command. - If an option is repeated (e.g., with a different argument), then the later instance on the command-line supersedes the earlier instance, with just a few options accumulating arguments.
- If an option takes an argument, the option name is followed by a
:
or=
or whitespace and then by the argument value; if the argument itself contains white space, the argument must be enclosed in quotes. It is recommended to use the:
or=
style to avoid misinterpretation or separation of a value from its option. - Boolean options can take the values
true
andfalse
(or any case-insensitive version of those words). For example, the value of--no-verify
is by defaultfalse
(that is, do verification). It can be explicitly set to true (no verification) using--no-verify
,--no-verify:true
,--no-verify=true
,--noverify true
; it can be explicitly set false (do verification) using--no-verify:false
or--no-verify=false
or--no-verify false
. - There is a potential ambiguity when the form
--option value
is used if the value is optional (such as for boolean values). In such a case an argument afer an option (that does not have an argument given with:
or=
) is interpreted as the value if it is indeed a valid value for that option. However, better style advises always using a ‘:’ or ‘=’ to set option values. No valid option values in dafny look like filenames or begin with--
.
13.5.1.1. Options that are not associated with a command
A few options are not part of a command. In these cases any single-hyphen spelling also permits a spelling beginning with ‘/’.
dafny --help
ordafny -h
lists all the available commandsdafny -?
ordafny -help
list all legacy optionsdafny --version
(or-version
) prints out the number of the version this build of dafny implements
13.5.1.2. dafny resolve
The dafny resolve
command checks the command-line and then parses and typechecks the given files and any included files.
The set of files considered by dafny
are those listed on the command-line,
including those named in a --library
option, and all files that are
named, recursively, in include
directives in files in the set being considered by the tool.
The set of files presented to an invocation of the dafny
tool must
contain all the declarations needed to resolve all names and types,
else name or type resolution errors will be emitted.
dafny
can parse and verify sets of files that do not form a
complete program because they are missing the implementations of
some constructs such as functions, lemmas, and loop bodies.16
However, dafny
will need all implementations in order to compile a working executable.
declaration and implementation of methods, functions and types in separate files, nor, for that matter,
separation of specification and declaration. Implementations can be
omitted simply by leaving them out of the declaration (or a lemma, for example).
However, a combination of traits
and
classes
can achieve a separation of interface
and specification from
implementation.
The options relevant to this command are
- those relevant to the command-line itself
--warn-as-errors
— turn all warnings into errors, which alters dafny’s exit code
- those that affect dafny` as a whole, such as
--cores
— set the number of cores dafny should use--show-snippets
— emit a line or so of source code along with an error message--library
— include this file in the program, but do not verify or compile it (multiple such library files can be listed using multiple instances of the--library
option)--stdin
– read from standard input
- those that affect the syntax of Dafny, such as
--prelude
--unicode-char
--function-syntax
--quantifier-syntax
--track-print-effects
--warn-shadowing
--warn-missing-constructor-parentheses
13.5.1.3. dafny verify
The dafny verify
command performs the dafny resolve
checks and then attempts to verify each declaration in the program.
A guide to controlling and aiding the verification process is given in a later section.
To be considered verified all the methods in all the files in a program must be verified, with consistent sets of options,
and with no unproven assumptions (see dafny audit
for a tool to help identify such assumptions).
Dafny works modularly, meaning that each method is considered by itself, using only the specifications of other methods. So, when using the dafny tool, you can verify the program all at once or one file at a time or groups of files at a time. On a large program, verifying all files at once can take quite a while, with little feedback as to progress, though it does save a small amount of work by parsing all files just once. But, one way or another, to have a complete verification, all implementations of all methods and functions must eventually be verified.
- By default, only those files listed on the command-line are verified in a given invocation of the
dafny
tool. - The option
--verify-included-files
(-verifyAllModules
in legacy mode) forces the contents of all non-library files to be verified, whether they are listed on the command-line or recursively included by files on the command-line. - The
--library
option marks files that are excluded from--verify-included-files
. Such a file may also, but need not, be the target of aninclude
directive in some file of the program; in any case, such files are included in the program but not in the set of files verified (or compiled). The intent of this option is to mark files that should be considered as libraries that are independently verified prior to being released for shared use. - Verifying files individually is equivalent to verifying them in groups, presuming no other changes.
It is also permitted to verify completely disjoint files or
programs together in a single run of
dafny
.
Various options control the verification process, in addition to all those described for dafny resolve
.
- What is verified
--verify-included-files
(when enabled, all included files are verified, except library files, otherwise just those files on the command-line)--relax-definite-assignment
--track-print-effects
--disable-nonlinear-arithmetic
- Control of the proof engine
--manual-lemma-induction
--verification-time-limit
--boogie
--boogie-filter
--solver-path
13.5.1.4. dafny translate <language>
The dafny translate
command translates Dafny source code to source code for another target programming language.
The command always performs the actions of dafny resolve
and, unless the --no-verify
option is specified, does the actions of dafny verify
.
The language is designated by a subcommand argument, rather than an option, and is required.
The current set of supported target languages is
- cs (C#)
- java (Java)
- js (JavaScript)
- py (Python)
- go (Go)
- cpp (C++ – but only limited support)
In addition to generating the target source code, dafny
may generate build artifacts to assist in compiling the generated code.
The specific files generated depend on the target programming language.
More detail is given in the section on compilation.
The dafny
tool intends that the compiled program in the target language be a semantically faithful rendering of the
(verified) Dafny program. However, resource and language limitations make this not always possible.
For example, though Dafny can express and reason about arrays of unbounded size,
not all target programming languages can represent arrays larger than the maximum signed 32-bit integer.
Various options control the translation process, in addition to all those described for dafny resolve
and dafny verify
.
- General options:
--no-verify
— turns off all attempts to verify the program--verbose
— print information about generated files
- The translation results
--output
(or-o
) — location of the generated file(s) (this specifies a file path and name; a folder location for artifacts is derived from this name)--include-runtime
— include the Dafny runtime for the target language in the generated artifacts--optimize-erasable-datatype-wrapper
--enforce-determinism
--test-assumptions
— (experimental) inserts runtime checks for unverified assumptions when they are compilable
13.5.1.5. dafny build
The dafny build
command runs dafny translate
and then compiles the result into an executable artifact for the target platform,
such as a .exe
or .dll
or executable .jar
, or just the source code for an interpreted language.
If the Dafny program does not have a Main entry point, then the build command creates a library, such as a .dll
or .jar
.
As with dafny translate
, all the previous phases are also executed, including verification (unless --no-verify
is a command-line option).
By default, the generated file is in the same directory and has the same name with a different extension as the first
.dfy file on the command line. This location and name can be set by the --output
option.
The location of the Main
entry point is described [here](#sec-user-guide-main}.
There are no additional options for dafny build
beyond those for dafny translate
and the previous compiler phases.
Note that dafny build
may do optimizations that dafny run
does not.
Details for specific target platforms are described in Section 25.7.
13.5.1.6. dafny run
The dafny run
command compiles the Dafny program and then runs the resulting executable.
Note that dafny run
is engineered to quickly compile and launch the program;
dafny build
may take more time to do optimizations of the build artifacts.
The form of the dafny run
command-line is slightly different than for other commands.
- It permits just one
.dfy
file, which must be the file containing theMain
entry point; the location of theMain
entry point is described [here](#sec-user-guide-main}. - Other files are included in the program either by
include
directives within that one file or by the--input
option on the command-line. - Anything that is not an option and is not that one dfy file is an argument to the program being run (and not to dafny itself).
- If the
--
option is used, then anything after that option is a command-line argument to the program being run.
If more complex build configurations are required, then use dafny build
and then execute the compiled program, as two separate steps.
dafny run
is primarily intended as a convenient way to run relatively simple Dafny programs.
Here are some examples:
dafny run A.dfy
– builds and runs the Main program inA.dfy
with no command-line argumentsdafny run A.dfy --no-verify
– builds the Main program inA.dfy
using the--no-verify
option, and then runs the program with no command-line argumentsdafny run A.dfy -- --no-verify
– builds the Main program inA.dfy
(not using the--no-verify
option), and then runs the program with one command-line argument, namely--no-verify
dafny run A.dfy 1 2 3 B.dfy
– builds the Main program inA.dfy
and then runs it with the four command-line arguments1 2 3 B.dfy
dafny run A.dfy 1 2 3 --input B.dfy
– builds the Main program inA.dfy
andB.dfy
, and then runs it with the three command-line arguments1 2 3
dafny run A.dfy 1 2 -- 3 -quiet
– builds the Main program inA.dfy
and then runs it with the four command-line arguments1 2 3 -quiet
Note: dafny run
will typically produce the same results as the executables produced by dafny build
. The only expected differences are these:
- performance —
dafny run
may not optimize as much asdafny build
- target-language-specific configuration issues — e.g. encoding issues:
dafny run
sets language-specific flags to request UTF-8 output for theprint
statement in all languages, whereasdafny build
leaves language-specific runtime configuration to the user.
13.5.1.7. dafny server
The dafny server
command starts the Dafny Language Server, which is an LSP-compliant implementation of Dafny.
The Dafny VSCode extension uses this LSP implementation, which in turn uses the same core Dafny implementation as the command-line tool.
The Dafny Language Server is described in more detail here.
13.5.1.8. dafny audit
The dafny audit
command reports issues in the Dafny code that might limit the soundness claims of verification.
This command is under development.
The command executes the dafny resolve
phase (accepting its options) and has the following additional options:
--report-file:<report-file>
— spcifies the path where the audit report file will be stored. Without this option, the report will be issued as standard warnings, written to standard-out.--report-format:<format>
— specifies the file format to use for the audit report. Supported options include:- ‘txt, ‘text’: plain text in the format of warnings
- ‘html’: standalone HTML (‘html’)
- ‘md’, ‘markdown’, ‘md-table’, ‘markdown-table’: a Markdown table
- ‘md-ietf’, ‘markdown-ietf’: an IETF-language document in Markdown format
- The default is to infer the format from the filename extension
--compare-report
— compare the report that would have been generated with the existing file given by –report-file, and fail if they differ.
The command emits exit codes of
- 1 for command-line errors
- 2 for parsing, type-checking or serious errors in running the auditor (e.g. failure to write a report or when report comparison fails)
- 0 for normal operation, including operation that identifies audit findings
It also takes the --verbose
option, which then gives information about the files being formatted.
The dafny audit
command currently reports the following:
-
Any declaration marked with the
{:axiom}
attribute. This is typically used to mark that a lemma with no body (and is therefore assumed to always be true) is intended as an axiom. The key purpose of theaudit
command is to ensure that all assumptions are intentional and acknowledged. To improve assurance, however, try to provide a proof. -
Any declaration marked with the
{:verify false}
attribute, which tells the verifier to skip verifying this declaration. Removing the attribute and providing a proof will improve assurance. -
Any declaration marked with the
{:extern}
attribute that has at least onerequires
orensures
clause. If code implemented externally, and called from Dafny, has anensures
clause, Dafny assumes that it satisfies that clause. Since Dafny cannot prove properties about code written in other languages, adding tests to provide evidence that anyensures
clauses do hold can improve assurance. The same considerations apply torequires
clauses on Dafny code intended to be called from external code. -
Any declaration marked with the
{:concurrent}
attribute. This is intended to indicate that the code is safe for use in a concurrent setting, but Dafny currently cannot prove that form of safety. The addition ofreads
clauses to methods will be a step toward being able to prove safety in this case, but until it exists, careful inspection and testing are appropriate. -
Any definition with an
assume
statement in its body. To improve assurance, attempt to convert it to anassert
statement and prove that it holds. Such a definition will not be compilable unless the statement is also marked with{:axiom}
. Alternatively, converting it to anexpect
statement will cause it to be checked at runtime. -
Any method marked with
decreases *
. Such a method may not terminate. Although this cannot cause an unsound proof, in the logic of Dafny, it’s generally important that any non-termination be intentional. -
Any
forall
statement without a body. This is equivalent to an assumption of its conclusion. To improve assurance, provide a body that proves the conclusion. -
Any loop without a body. This is equivalent to an assumption of any loop invariants in the code after the loop. To improve assurance, provide a body that establishes any stated invariants.
-
Any declaration with no body and at least one
ensures
clause. Any code that calls this declaration will assume that allensures
clauses are true after it returns. To improve assurance, provide a body that proves that anyensures
clauses hold.
13.5.1.9. dafny format
Dafny supports a formatter, which for now only changes the indentation of lines in a Dafny file, so that it conforms to the idiomatic Dafny code formatting style. For the formatter to work, the file should be parsed correctly by Dafny.
There are four ways to use the formatter:
dafny format <one or more .dfy files or folders>
formats the given Dafny files and the Dafny files in the folders, recursively, altering the files in place. For example,dafny format .
formats all the Dafny files recursively in the current folder.dafny format --print <files and/or folders>
formats each file but instead of altering the files, output the formatted content to stdoutdafny format --check <files and/or folders>
does not alter files. It will print a message concerning which files need formatting and return a non-zero exit code if any files would be changed by formatting.
You can also use --stdin
instead of providing a file, to format a full Dafny file from the standard input.
Input files can be named along with --stdin
, in which case both the files and the content of the stdin are formatted.
Each version of dafny format
returns a non-zero return code if there are any command-line or parsing
errors or if –check is stipulated and at least one file is not the same as its formatted version.
dafny format
does not necessarily report name or type resolution errors and does not attempt verification.
13.5.1.10. dafny test
This command (verifies and compiles the program and) runs every method in the program that is annotated with the {:test}
attribute.
Verification can be disabled using the --no-verify
option. dafny test
also accepts all other options of the dafny build
command.
In particular, it accepts the --target
option that specifies the programming language used in the build and execution phases.
dafny test
also accepts these options:
-spill-translation
- (default disabled) when enabled the compilation artifacts are retained--output
- gives the folder and filename root for compilation artifacts--methods-to-test
- the value is a (.NET) regular expression that is matched against the fully qualified name of the method; only those methods that match are tested
The order in which the tests are run is not specified.
For example, this code (as the file t.dfy
)
method {:test} m() {
mm();
print "Hi!\n";
}
method mm() {
print "mm\n";
}
module M {
method {:test} q() {
print 42, "\n";
}
}
class A {
static method {:test} t() { print "T\n"; }
}
and this command-line
dafny test --no-verify t.dfy
produce this output text:
M.q: 42
PASSED
A.t: T
PASSED
m: mm
Hi!
PASSED
and this command-line
dafny test --no-verify --methods-to-test='m' t.dfy
produces this output text:
m: mm
Hi!
PASSED
13.5.1.11. dafny doc
[Experimental]
The dafny doc
command generates HTML documentation pages describing the contents of each
module in a set of files, using the documentation comments in the source files.
This command is experimental; user feedback and contributor PRs on the layout of information and the navigation are welcome.
- The format of the documentation comments is described here.
- The
dafny doc
command accepts either files or folders as command-line arguments. A folder represents all the.dfy
files contained recursively in that folder. A file that is a.toml
project file represents all the files and options listed in the project file. - The command first parses and resolves the given files; it only proceeds to produce documentation
if type resolution is successful (on all files). All the command-line options relevant to
dafny resolve
are available fordafny doc
. - The value of the
--output
option is a folder in which all the generated files will be placed. The default location is./docs
. The folder is created if it does not already exist. Any existing content of the folder is overwritten. - If
--verbose
is enabled, a list of the generated files is emitted to stdout. - The output files contain information stating the source .dfy file in which the module is
declared. The
--file-name
option controls the form of the filename in that information:- –file-name:none – no source file information is emitted
- –file-name:name – (default) just the file name is emitted (e.g.,
Test.dfy
) - –file-name:absolute – an absolute full path is emitted
- –file-name:relative=
-- a file name relative to the given prefix is emitted
- If
--modify-time
is enabled, then the generated files contain information stating the last modified time of the source of the module being documented. - The
--program-name
option states text that will be included in the heading of the TOC and index pages
The output files are HTML files, all contained in the given folder, one per module plus an
index.html
file giving an overall table of contents and a nameindex.html
file containing
an alphabetical by name list of all the declarations in all the modules.
The documentation for the root module is in _.html
.
13.5.1.12. dafny generate-tests
This experimental command generates unit test code (as Dafny source code) that provides complete coverage of a method indicated with the –target-method option.
This command enforces determinism by default and disables the functionality provided by the –snow-snippets option.
This command is under development and not yet fully functional.
13.5.1.13. dafny find-dead-code
This experimental command finds dead code in a program, that is, code branches within a method that are not reachable by any inputs that satisfy the method’s preconditions.
This command is under development and not yet functional.
13.5.1.14. dafny measure-complexity
This experimental command reports complexity metrics of a program.
This command is under development and not yet functional.
13.5.1.15. Plugins
This execution mode is not a command, per se, but rather a command-line option that enables executing plugins to the dafny tool. Plugins may be either standalone tools or be additions to existing commands.
The form of the command-line is dafny --plugin:<path-to-one-assembly[,argument]*>
or dafny <command> --plugin:<path-to-one-assembly[,argument]*>
where the argument to --plugin
gives the path to the compiled assembly of the plugin and the arguments to be provided to the plugin.
More on writing and building plugins can be found in this section.
13.5.1.16. Legacy operation
Prior to implementing the command-based CLI, the dafny
command-line simply took files and options and the arguments to options.
That legacy mode of operation is still supported, though discouraged. The command dafny -?
produces the list of legacy options.
In particular, the common commands like dafny verify
and dafny build
are accomplished with combinations of
options like -compile
, -compileTarget
and -spillTargetCode
.
Users are encouraged to migrate to the command-based style of command-lines and the double-hyphen options.
13.5.2. In-tool help
As is typical for command-line tools, dafny
provides in-tool help through the -h
and --help
options:
dafny -h
,dafny --help
list the commands available in thedafny
tooldafny -?
lists all the (legacy) options implemented indafny
dafny <command> -h
,dafny <command> --help
,dafny <command> -?
list the options available for that command
13.5.3. dafny exit codes
The basic resolve, verify, translate, build, run and commands of dafny terminate with these exit codes.
- 0 – success
- 1 – invalid command-line arguments
- 2 – syntax, parse, or name or type resolution errors
- 3 – compilation errors
- 4 – verification errors
Errors in earlier phases of processing typically hide later errors. For example, if a program has parsing errors, verification or compilation will not be attempted.
Other dafny commands may have their own conventions for exit codes. However in all cases, an exit code of 0 indicates successful completion of the command’s task and small positive integer values indicate errors of some sort.
13.5.4. dafny output
Most output from dafny
is directed to the standard output of the shell invoking the tool, though some goes to standard error.
- Command-line errors: these are produced by the dotnet CommandLineOptions package are directed to standard-error
- Other errors: parsing, typechecking, verification and compilation errors are directed to standard-out
- Non-error progress information also is output to standard-out
- Dafny
print
statements, when executed, send output to standard-out - Dafny
expect
statements (when they fail) send a message to standard-out. - Dafny I/O libraries send output explicitly to either standard-out or standard-error
13.5.5. Project files
Commands on the Dafny CLI that can be passed a Dafny file can also be passed a Dafny project file. Such a project file may define which Dafny files the project contains and which Dafny options it uses. The project file must be a TOML file named dfyconfig.toml
for it to work on both the CLI and in the Dafny IDE, although the CLI will accept any .toml
file.
Here’s an example of a Dafny project file:
includes = ["src/**/*.dfy"]
excludes = ["**/ignore.dfy"]
[options]
enforce-determinism = true
warn-shadowing = true
- At most one
.toml
file may be named on the command-line; when using the command-line no.toml
file is used by default. - In the
includes
andexcludes
lists, the file paths may have wildcards, including**
to mean any number of directory levels; filepaths are relative to the location of the.toml
file in which they are named. - Dafny will process the union of (a) the files on the command-line and (b) the files designated in the
.toml
file, which are those specified by theincludes
, omitting those specified by theexcludes
. Theexcludes
does not remove any files that are listed explicitly on the command-line. - Under the section
[options]
, any options from the Dafny CLI can be specified using the option’s name without the--
prefix. - When executing a
dafny
command using a project file, any options specified in the file that can be applied to the command, will be. Options that can’t be applied are ignored; options that are invalid for any dafny command trigger warnings. - Options specified on the command-line take precedence over any specified in the project file, no matter the order of items on the command-line.
- When using a Dafny IDE based on the
dafny server
command, the IDE will search for project files by traversing up the file tree looking for the closestdfyconfig.toml
file to the dfy being parsed that it can find. Options from the project file will override options passed todafny server
.
It’s not possible to use Dafny project files in combination with the legacy CLI UI.
13.6. Verification
In this section, we suggest a methodology to figure out why a single assertion might not hold, we propose techniques to deal with assertions that slow a proof down, we explain how to verify assertions in parallel or in a focused way, and we also give some more examples of useful options and attributes to control verification.
13.6.1. Verification debugging when verification fails
Let’s assume one assertion is failing (“assertion might not hold” or “postcondition might not hold”). What should you do next?
The following section is textual description of the animation below, which illustrates the principle of debugging an assertion by computing the weakest precondition:
13.6.1.1. Failing postconditions
Let’s look at an example of a failing postcondition.
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
if b {
return j;
}//^^^^^^^ a postcondition might not hold on this return path.
i := 2;
}
One first thing you can do is replace the statement return j;
by two statements i := j; return;
to better understand what is wrong:
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
if b {
i := j;
return;
}//^^^^^^^ a postcondition might not hold on this return path.
i := 2;
}
Now, you can assert the postcondition just before the return:
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
if b {
i := j;
assert 2 <= i; // This assertion might not hold
return;
}
i := 2;
}
That’s it! Now the postcondition is not failing anymore, but the assert
contains the error!
you can now move to the next section to find out how to debug this assert
.
13.6.1.2. Failing asserts
In the previous section, we arrived at the point where we have a failing assertion:
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
if b {
i := j;
assert 2 <= i; // This assertion might not hold
return;
}
i := 2;
}
To debug why this assert might not hold, we need to move this assert up, which is similar to computing the weakest precondition.
For example, if we have x := Y; assert F;
and the assert F;
might not hold, the weakest precondition for it to hold before x := Y;
can be written as the assertion assert F[x:= Y];
, where we replace every occurence of x
in F
into Y
.
Let’s do it in our example:
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
if b {
assert 2 <= j; // This assertion might not hold
i := j;
assert 2 <= i;
return;
}
i := 2;
}
Yay! The assertion assert 2 <= i;
is not proven wrong, which means that if we manage to prove assert 2 <= j;
, it will work.
Now, this assert should hold only if we are in this branch, so to move the assert up, we need to guard it.
Just before the if
, we can add the weakest precondition assert b ==> (2 <= j)
:
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
var j := if !b then 3 else 1;
assert b ==> 2 <= j; // This assertion might not hold
if b {
assert 2 <= j;
i := j;
assert 2 <= i;
return;
}
i := 2;
}
Again, now the error is only on the topmost assert, which means that we are making progress.
Now, either the error is obvious, or we can one more time replace j
by its value and create the assert assert b ==> ((if !b then 3 else 1) >= 2);
method FailingPostcondition(b: bool) returns (i: int)
ensures 2 <= i
{
assert b ==> 2 <= (if !b then 3 else 1); // This assertion might not hold
var j := if !b then 3 else 1;
assert b ==> 2 <= j;
if b {
assert 2 <= j;
i := j;
assert 2 <= i;
return;
}
i := 2;
}
At this point, this is pure logic. We can simplify the assumption:
b ==> 2 <= (if !b then 3 else 1)
!b || (if !b then 2 <= 3 else 2 <= 1)
!b || (if !b then true else false)
!b || !b;
!b;
Now we can understand what went wrong: When b is true, all of these formulas above are false, this is why the dafny
verifier was not able to prove them.
In the next section, we will explain how to “move asserts up” in certain useful patterns.
13.6.1.3. Failing asserts cases
This list is not exhaustive but can definitely be useful to provide the next step to figure out why Dafny could not prove an assertion.
Failing assert | Suggested rewriting |
---|---|
x := Y; assert P; |
assert P[x := Y]; x := Y; assert P; |
if B { assert P; ... } |
assert B ==> P; if B { assert P; ... } |
if B { ... } else { assert P; ... } |
assert !B ==> P; if B { ... } else { assert P; ... } |
if X { ... } else { ... } assert A; |
if X { ... assert A; } else { ... assert A; } assert A; |
assert forall x :: Q(x); |
forall x ensures Q(x) { assert Q(x); }; assert forall x :: Q(x); |
assert forall x :: P(x) ==> Q(x); |
forall x | P(x) ensures Q(x) { assert Q(x); }; assert forall x :: P(x) ==> Q(x); |
assert exists x | P(x) :: Q(x); assert exists x | P(x) :: Q'(x); |
if x :| P(x) { assert Q(x); assert Q'(x); } else { assert false; } |
assert exists x :: P(x); |
assert P(x0); assert exists x :: P(x); for a given expression x0 . |
ensures exists i :: P(i); |
returns (j: int) ensures P(j) ensures exists i :: P(i) in a lemma, so that the j can be computed explicitly. |
assert A == B; callLemma(x); assert B == C; |
calc == { A; B; { callLemma(x); } C; }; assert A == B; where the calc statement can be used to make intermediate computation steps explicit. Works with < , > , <= , >= , ==> , <== and <==> for example. |
assert A ==> B; |
if A { assert B; }; assert A ==> B; |
assert A && B; |
assert A; assert B; assert A && B; |
assert P(x); where P is an {:opaque} predicate |
reveal P(); assert P(x); |
assert P(x); where P is an {:opaque} predicate |
assert P(x) by { reveal P(); } |
assert P(x); where P is not an {:opaque} predicate with a lot of && in its body and is assumed |
Make P {:opaque} so that if it’s assumed, it can be proven more easily. You can always reveal it when needed. |
ensures P ==> Q on a lemma |
requires P ensures Q to avoid accidentally calling the lemma on inputs that do not satisfy P |
seq(size, i => P) |
seq(size, i requires 0 <= i < size => P); |
assert forall x :: G(i) ==> R(i); |
assert G(i0); assert R(i0); assert forall i :: G(i) ==> R(i); with a guess of the i0 that makes the second assert to fail. |
assert forall i | 0 < i <= m :: P(i); |
assert forall i | 0 < i < m :: P(i); assert forall i | i == m :: P(i); assert forall i | 0 < i <= m :: P(i); |
assert forall i | i == m :: P(m); |
assert P(m); assert forall i | i == m :: P(i); |
method m(i) returns (j: T) requires A(i) ensures B(i, j) { ... } method n() { ... var x := m(a); assert P(x); |
method m(i) returns (j: T) requires A(i) ensures B(i, j) { ... } method n() { ... assert A(k); assert forall x :: B(k, x) ==> P(x); var x := m(k); assert P(x); |
method m_mod(i) returns (j: T) requires A(i) modifies this, i ensures B(i, j) { ... } method n_mod() { ... var x: m_mod(a); assert P(x); |
method m_mod(i) returns (j: T) requires A(i) modifies this, i ensures B(i, j) { ... } method n_mod() { ... assert A(k); modify this, i; // Temporarily var x := T; // Temporarily assume B(k, x); // var x := m_mod(k); assert P(x); |
modify x, y; assert P(x, y, z); |
assert x != z && y != z; modify x, y; assert P(x, y, z); |
13.6.2. Verification debugging when verification is slow
In this section, we describe techniques to apply in the case when verification is slower than expected, does not terminate, or times out.
Additional detail is available in the verification optimization guide.
13.6.2.1. assume false;
Assuming false
is an empirical way to short-circuit the verifier and usually stop verification at a given point,17 and since the final compilation steps do not accept this command, it is safe to use it during development.
Another similar command, assert false;
, would also short-circuit the verifier, but it would still make the verifier try to prove false
, which can also lead to timeouts.
Thus, let us say a program of this shape takes forever to verify.
method NotTerminating(b: bool) {
assert X;
if b {
assert Y;
} else {
assert Z;
assert P;
}
}
What we can first do is add an assume false
at the beginning of the method:
method NotTerminating() {
assume false; // Will never compile, but everything verifies instantly
assert X;
if b {
assert Y;
} else {
assert Z;
assert P;
}
assert W;
}
This verifies instantly. This gives use a strategy to bisect, or do binary search to find the assertion that slows everything down.
Now, we move the assume false;
below the next assertion:
method NotTerminating() {
assert X;
assume false;
if b {
assert Y;
} else {
assert Z;
assert P;
}
assert W;
}
If verification is slow again, we can use techniques seen before to decompose the assertion and find which component is slow to prove.
If verification is fast, that’s the sign that X
is probably not the problem,. We now move the assume false;
after the if/then block:
method NotTerminating() {
assert X;
if b {
assert Y;
} else {
assert Z;
assert P;
}
assume false;
assert W;
}
Now, if verification is fast, we know that assert W;
is the problem. If it is slow, we know that one of the two branches of the if
is the problem.
The next step is to put an assume false;
at the end of the then
branch, and an assume false
at the beginning of the else branch:
method NotTerminating() {
assert X;
if b {
assert Y;
assume false;
} else {
assume false;
assert Z;
assert P;
}
assert W;
}
Now, if verification is slow, it means that assert Y;
is the problem.
If verification is fast, it means that the problem lies in the else
branch.
One trick to ensure we measure the verification time of the else
branch and not the then branch is to move the first assume false;
to the top of the then branch, along with a comment indicating that we are short-circuiting it for now.
Then, we can move the second assume false;
down and identify which of the two assertions makes the verifier slow.
method NotTerminating() {
assert X;
if b {
assume false; // Short-circuit because this branch is verified anyway
assert Y;
} else {
assert Z;
assume false;
assert P;
}
assert W;
}
If verification is fast, which of the two assertions assert Z;
or assert P;
causes the slowdown?18
We now hope you know enough of assume false;
to locate assertions that make verification slow.
Next, we will describe some other strategies at the assertion level to figure out what happens and perhaps fix it.
13.6.2.2. assert ... by {}
If an assertion assert X;
is slow, it is possible that calling a lemma or invoking other assertions can help to prove it: The postcondition of this lemma, or the added assertions, could help the dafny
verifier figure out faster how to prove the result.
assert SOMETHING_HELPING_TO_PROVE_LEMMA_PRECONDITION;
LEMMA();
assert X;
...
lemma ()
requires LEMMA_PRECONDITION
ensures X { ... }
However, this approach has the problem that it exposes the asserted expressions and lemma postconditions not only for the assertion we want to prove faster,
but also for every assertion that appears afterwards. This can result in slowdowns19.
A good practice consists of wrapping the intermediate verification steps in an assert ... by {}
, like this:
assert X by {
assert SOMETHING_HELPING_TO_PROVE_LEMMA_PRECONDITION;
LEMMA();
}
Now, only X
is available for the dafny
verifier to prove the rest of the method.
13.6.2.3. Labeling and revealing assertions
Another way to prevent assertions or preconditions from cluttering the verifier19 is to label and reveal them. Labeling an assertion has the effect of “hiding” its result, until there is a “reveal” calling that label.
The example of the previous section could be written like this.
assert p: SOMETHING_HELPING_TO_PROVE_LEMMA_PRECONDITION;
// p is not available here.
assert X by {
reveal p;
LEMMA();
}
Similarly, if a precondition is only needed to prove a specific result in a method, one can label and reveal the precondition, like this:
method Slow(i: int, j: int)
requires greater: i > j {
assert i >= j by {
reveal greater;
}
}
Labelled assert statements are available both in expressions and statements. Assertion labels are not accessible outside of the block which the assert statement is in. If you need to access an assertion label outside of the enclosing expression or statement, you need to lift the labelled statement at the right place manually, e.g. rewrite
ghost predicate P(i: int)
method TestMethod(x: bool)
requires r: x <==> P(1)
{
if x {
assert a: P(1) by { reveal r; }
}
assert x ==> P(1) by { reveal a; } // Error, a is not accessible
}
to
ghost predicate P(i: int)
method TestMethod(x: bool)
requires r: x <==> P(1)
{
assert a: x ==> P(1) by {
if x {
assert P(1) by { reveal r; } // Proved without revealing the precondition
}
}
assert x ==> P(1) by { reveal a; } // Now a is accessible
}
To lift assertions, please refer to the techniques described in Verification Debugging.
13.6.2.4. Non-opaque function method
Functions are normally used for specifications, but their functional syntax is sometimes also desirable to write application code.
However, doing so naively results in the body of a function method Fun()
be available for every caller, which can cause the verifier to time out or get extremely slow19.
A solution for that is to add the attribute {:opaque}
right between function method
and Fun()
, and use reveal Fun();
in the calling functions or methods when needed.
13.6.2.5. Conversion to and from bitvectors
Bitvectors and natural integers are very similar, but they are not treated the same by the dafny
verifier. As such, conversion from bv8
to an int
and vice-versa is not straightforward, and can result in slowdowns.
There are two solutions to this for now. First, one can define a subset type instead of using the built-in type bv8
:
type byte = x | 0 <= x < 256
One of the problems of this approach is that additions, substractions and multiplications do not enforce the result to be in the same bounds, so it would have to be checked, and possibly truncated with modulos. For example:
type byte = x | 0 <= x < 256
method m() {
var a: byte := 250;
var b: byte := 200;
var c := b - a; // inferred to be an 'int', its value will be 50.
var d := a + b; // inferred to be an 'int', its value will be 450.
var e := (a + b) % 256; // still inferred to be an 'int'...
var f: byte := (a + b) % 256; // OK
}
A better solution consists of creating a newtype that will have the ability to check bounds of arithmetic expressions, and can actually be compiled to bitvectors as well.
newtype {:nativeType "short"} byte = x | 0 <= x < 256
method m() {
var a: byte := 250;
var b: byte := 200;
var c := b - a; // OK, inferred to be a byte
var d := a + b; // Error: cannot prove that the result of a + b is of type `byte`.
var f := ((a as int + b as int) % 256) as byte; // OK
}
One might consider refactoring this code into separate functions if used over and over.
13.6.2.6. Nested loops
In the case of nested loops, the verifier might timeout sometimes because of inadequate or too much available information19. One way to mitigate this problem, when it happens, is to isolate the inner loop by refactoring it into a separate method, with suitable pre and postconditions that will usually assume and prove the invariant again. For example,
while X
invariant Y
{
while X'
invariant Y'
{
}
}
could be refactored as this:
`while X
invariant Y
{
innerLoop();
}
...
method innerLoop()
require Y'
ensures Y'
In the next section, when everything can be proven in a timely manner, we explain another strategy to decrease proof time by parallelizing it if needed, and making the verifier focus on certain parts.
13.6.3. Assertion batches
To understand how to control verification,
it is first useful to understand how dafny
verifies functions and methods.
For every method (or function, constructor, etc.), dafny
extracts assertions. Here is a non-exhaustive list of such extracted assertions:
Integer assertions:
- Every division yields an assertion that the divisor is never zero.
- Every bounded number operation yields an assertion that the result will be within the same bounds (no overflow, no underflows).
- Every conversion yields an assertion that conversion is compatible.
- Every bitvector shift yields an assertion that the shift amount is never negative, and that the shift amount is within the width of the value.
Object assertions:
- Every object property access yields an assertion that the object is not null.
- Every assignment
o.f := E;
yields an assertion thato
is among the set of objects of themodifies
clause of the enclosing loop or method. - Every read
o.f
yields an assertion thato
is among the set of objects of thereads
clause of the enclosing function or predicate; or themodifies
clause of the enclosing method. - Every array access
a[x]
yields the assertion that0 <= x < a.Length
. - Every sequence access
a[x]
yields an assertion, that0 <= x < |a|
, because sequences are never null. - Every datatype update expression and datatype destruction yields an assertion that the object has the given property.
- Every method overriding a
trait
yields an assertion that any postcondition it provides implies the postcondition of its parent trait, and an assertion that any precondition it provides is implied by the precondition of its parent trait.
Other implicit assertions:
- Every value whose type is assigned to a subset type yields an assertion that it satisfies the subset type constraint.
- Every non-empty subset type yields an assertion that its witness satisfies the constraint.
- Every Assign-such-that operator
x :| P(x)
yields an assertion thatexists x :: P(x)
. - Every recursive function yields an assertion that it terminates.
- Every match expression or alternative if statement yields an assertion that all cases are covered.
Explicit assertions:
- Any explicit
assert
statement is an assertion20. - A consecutive pair of lines in a
calc
statement forms an assertion that the expressions are related according to the common operator. - Every call to a function or method with a
requires
clause yields one assertion per requires clause20 (special cases such as sequence indexing come with a specialrequires
clause that the index is within bounds). - Every
ensures
clause yields an assertion at the end of the method and on every return, and onforall
statements. - Every
invariant
clause yields an assertion that it holds before the loop and an assertion that it holds at the end of the loop. - Every
decreases
clause yields an assertion at either a call site or at the end of a while loop. - Every
yield ensures
clause on an iterator yields assertions that the clause holds at every yielding point. - Every
yield requires
clause on an iterator yields assertions that the clause holds at every point when the iterator is called.
It is useful to mentally visualize all these assertions as a list that roughly follows the order in the code,
except for ensures
or decreases
that generate assertions that seem earlier in the code but, for verification purposes, would be placed later.
In this list, each assertion depends on other assertions, statements and expressions that appear earlier in the control flow21.
The fundamental unit of verification in dafny
is an assertion batch, which consists of one or more assertions from this “list”, along with all the remaining assertions turned into assumptions. To reduce overhead, by default dafny
collects all the assertions in the body of a given method into a single assertion batch that it sends to the verifier, which tries to prove it correct.
- If the verifier says it is correct,22 it means that all the assertions hold.
- If the verifier returns a counterexample, this counterexample is used to determine both the failing assertion and the failing path.
In order to retrieve additional failing assertions,
dafny
will again query the verifier after turning previously failed assertions into assumptions.23 24 - If the verifier returns
unknown
or times out, or even preemptively for difficult assertions or to reduce the chance that the verifier will ‘be confused’ by the many assertions in a large batch,dafny
may partition the assertions into smaller batches25. An extreme case is the use of the/vcsSplitOnEveryAssert
command-line option or the{:vcs_split_on_every_assert}
attribute, which causesdafny
to make one batch for each assertion.
13.6.3.1. Controlling assertion batches
Here is how you can control how dafny
partitions assertions into batches.
{:focus}
on an assert generates a separate assertion batch for the assertions of the enclosing block.{:split_here}
on an assert generates a separate assertion batch for assertions after this point.{:vcs_split_on_every_assert}
on a function or a method generates one assertion batch per assertion
We discourage the use of the following heuristics attributes to partition assertions into batches. The effect of these attributes may vary, because they are low-level attributes and tune low-level heuristics, and will result in splits that could be manually controlled anyway.
{:vcs_max_cost N}
on a function or method enables splitting the assertion batch until the “cost” of each batch is below N. Usually, you would set{:vcs_max_cost 0}
and{:vcs_max_splits N}
to ensure it generates N assertion batches.{:vcs_max_keep_going_splits N}
where N > 1 on a method dynamically splits the initial assertion batch up to N components if the verifier is stuck the first time.
13.6.4. Command-line options and other attributes to control verification
There are many great options that control various aspects of verifying dafny programs. Here we mention only a few:
- Control of output:
/dprint
,/rprint
,/stats
,/compileVerbose
- Whether to print warnings:
/proverWarnings
- Control of time:
/timeLimit
- Control of resources:
/rLimit
and{:rlimit}
- Control of the prover used:
/prover
- Control of how many times to unroll functions:
{:fuel}
You can search for them in this file as some of them are still documented in raw text format.
13.6.5. Debugging variable verification
When evolving a Dafny codebase, it can sometimes occur that a proof obligation succeeds at first only for the prover to time out or report a potential error after minor, valid changes. This is ultimately due to decidability limitations in the form of automated reasoning that Dafny uses. The Z3 SMT solver that Dafny depends on attempts to efficiently search for proofs, but does so using both incomplete heuristics and a degree of randomness, with the result that it can sometimes fail to find a proof even when one exists (or continue searching forever).
Dafny provides some features to mitigate this issue, primarily focused on early detection. The philosophy is that, if Dafny programmers are alerted to proofs that show early signs of variability, before they are obviously so, they can refactor the proofs to make them less variable before further development becomes difficult.
The mechanism for early detection focuses on measuring the resources used to complete a proof (either using duration or a more deterministic “resource count” metric available from Z3). Dafny can re-run a given proof attempt multiple times after automatically making minor changes to the structure of the input or to the random choices made by the solver. If the resources used during these attempts (or the ability to find a proof at all) vary widely, we say that the verification of the relevant properties is highly variable.
13.6.5.1. Measuring proof variability
To measure the variability of your proofs, start by using the
-randomSeedIterations:N
flag to instruct Dafny to attempt each proof
goal N
times, using a different random seed each time. The random seed
used for each attempt is derived from the global random seed S
specified with -randomSeed:S
, which defaults to 0
(which means use
an arbitrary – e.g. clock-based – seed).
For most use cases, it also makes sense to specify the
-verificationLogger:csv
flag, to log verification cost statistics to a
CSV file. By default, the resulting CSV files will be created in the
TestResults
folder of the current directory.
Once Dafny has completed, the
dafny-reportgenerator
tool is a convenient way to process the output. It allows you to specify
several limits on statistics computed from the elapsed time or solver
resource use of each proof goal, returning an error code when it detects
violations of these limits. You can find documentation on the full set
of options for dafny-reportgenerator
in its
README.md
file.
In general, we recommend something like the following:
dafny-reportgenerator --max-resource-cv-pct 10 TestResults/*.csv
This bounds the coefficient of variation of the solver resource count at 10% (0.10). We recommend a limit of less than 20%, perhaps even as low as 5%. However, when beginning to analyze a new project, it may be necessary to set limits as high as a few hundred percent and incrementally ratchet down the limit over time.
When first analyzing proof variability, you may also find that certain proof
goals succeed on some iterations and fail on others. If your aim is
first to ensure that variability doesn’t worsen and then to start
improving it, integrating dafny-reportgenerator
into CI and using the
--allow-different-outcomes
flag may be appropriate. Then, once you’ve
improved variability sufficiently, you can likely remove that flag (and
likely have significantly lower limits on other variability metrics).
13.6.5.2. Improving proof variability
Improving proof variability is typically closely related to improving performance overall. As such, techniques for debugging slow verification are typically useful for debugging highly variable verification, as well. See also the verification optimization guide.
13.7. Compilation
The dafny
tool can compile a Dafny program to one of several target languages. Details and idiosyncrasies of each
of these are described in the following subsections. In general note the following:
- The compiled code originating from
dafny
can be combined with other source and binary code, but only thedafny
-originated code is verified. - Output file or folder names can be set using
--output
. - Code generated by
dafny
requires a Dafny-specific runtime library. By default the runtime is included in the generated code. However fordafny translate
it is not included by default and must be explicitly requested using--include-runtime
. All runtime libraries are part of the Binary (./DafnyRuntime.*
) and Source (./Source/DafnyRuntime/DafnyRuntime.*
) releases. - Names in Dafny are written out as names in the target language. In some cases this can result in naming conflicts. Thus if a Dafny program is intended to be compiled to a target language X, you should avoid using Dafny identifiers that are not legal identifiers in X or that conflict with reserved words in X.
To be compilable to an executable program, a Dafny program must contain a Main
entry point, as described here.
13.7.1. extern
declarations
A Dafny declaration can be marked with the {:extern}
attribute to
indicate that it refers to an external definition that is already
present in the language that the Dafny code will be compiled to (or will
be present by the time the final target-language program is compiled or
run).
Because the {:extern}
attribute controls interaction with code written
in one of many languages, it has some language-specific behavior,
documented in the following sections. However, some aspects are
target-language independent and documented here.
The attribute can also take several forms, each defining a different
relationship between a Dafny name and a target language name. In the
form {:extern}
, the name of the external definition is
assumed to be the name of the Dafny declaration after some
target-specific name mangling. However, because naming conventions (and
the set of allowed identifiers) vary between languages, Dafny allows
additional forms for the {:extern}
attribute.
The form {:extern <s1>}
instructs dafny
to compile references to most
declarations using the name s1
instead of the Dafny name. For opaque
types, however, s1
is sometimes used as a hint as
to how to declare that type when compiling. This hint is interpreted
differently by each compiler.
Finally, the form {:extern <s1>, <s2>}
instructs dafny
to use s2
as
the direct name of the declaration. dafny
will typically use a
combination of s1
and s2
, such as s1.s2
, to reference the
declaration. It may also be the case that one of the arguments is simply
ignored, depending on the target language.
The recommended style is to prefer {:extern}
when possible, and use
similar names across languages. This is usually feasible because
existing external code is expected to have the same interface as the
code that dafny
would generate for a declaration of that form. Because
many Dafny types compile down to custom types defined in the Dafny
runtime library, it’s typically necessary to write wrappers by hand that
encapsulate existing external code using a compatible interface, and
those wrappers can have names chosen for compatibility. For example,
retrieving the list of command line arguments when compiling to C#
requires a wrapper such as the following:
using icharseq = Dafny.ISequence<char>;
using charseq = Dafny.Sequence<char>;
namespace Externs_Compile {
public partial class __default {
public static Dafny.ISequence<icharseq> GetCommandLineArgs() {
var dafnyArgs = Environment
.GetCommandLineArgs()
.Select(charseq.FromString);
return Dafny.Sequence<icharseq>.FromArray(dafnyArgs.ToArray());
}
}
This serves as an example of implementing an extern,
but was only necessary to retrieve command line arguments historically,
as dafny
now supports capturing these arguments via a main method
that accepts a seq<string>
(see the section on the Main method).
Note that dafny
does not check the arguments to {:extern}
, so it is
the user’s responsibility to ensure that the provided names result in
code that is well-formed in the target language.
Also note that the interface the external code needs to implement
may be affected by compilation flags. In this case, if --unicode-char:true
is provided, dafny
will compile its char
type to the Dafny.Rune
C# type instead, so the references to the C# type char
above
would need to be changed accordingly. The reference to charseq.FromString
would in turn need to be changed to charseq.UnicodeFromString
to
return the correct type.
Most declarations, including those for modules, classes, traits, member
variables, constructors, methods, function methods, and abstract types,
can be marked with {:extern}
.
Marking a module with {:extern}
indicates that the declarations
contained within can be found within the given module, namespace, package, or
similar construct within the target language. Some members of the Dafny
module may contain definitions, in which case code for those definitions
will be generated. Whether this results in valid target code may depend
on some target language support for something resembling “partial”
modules, where different subsets of the contents are defined in
different places.
The story for classes is similar. Code for a class will be generated
if any of its members are not {:extern}
. Depending on the target
language, making either all or none of the members {:extern}
may be
the only options that result in valid target code. Traits with
{:extern}
can refer to existing traits or interfaces in the target
language, or can refer to the interfaces of existing classes.
Member variables marked with {:extern}
refer to fields or properties
in existing target-language code. Constructors, methods, and functions
refer to the equivalent concepts in the target language. They
can have contracts, which are then assumed to hold for the existing
target-language code. They can also have bodies, but the bodies will not
be compiled in the presence of the {:extern}
attribute. Bodies can
still be used for reasoning, however, so may be valuable in some cases,
especially for function methods.
Types marked with {:extern}
must be opaque. The name argument, if any,
usually refers to the type name in the target language, but some
compilers treat it differently.
Detailed description of the dafny build
and dafny run
commands and
the --input
option (needed when dafny run
has more than one input file)
is contained in the section on command-line structure.
13.7.2. C#
For a simple Dafny-only program, the translation step converts a A.dfy
file into A.cs
;
the build step then produces a A.dll
, which can be used as a library or as an executable (run using dotnet A.dll
).
It is also possible to run the dafny files as part of a csproj
project, with these steps:
- create a dotnet project file with the command
dotnet new console
- delete the
Program.cs
file - build the dafny program:
dafny build A.dfy
- run the built program
dotnet A.dll
The last two steps can be combined:
dafny run A.dfy
Note that all input .dfy
files and any needed runtime library code are combined into a single .cs
file,
which is then compiled by dotnet
to a .dll
.
Examples of how to integrate C# libraries and source code with Dafny source code are contained in this separate document.
13.7.3. Java
The Dafny-to-Java compiler translation phase writes out the translated files of a file A.dfy
to a directory A-java
.
The build phase writes out a library or executable jar file.
The --output
option (-out
in the legacy CLI) can be used to choose a
different jar file path and name and correspondingly different directory for .java and .class files.
The compiler produces a single wrapper method that then calls classes in
relevant other .java
files. Because Java files must be named consistent
with the class they contain, but Dafny files do not, there may be no relation
between the Java file names and the Dafny file names.
However, the wrapper class that contains the Java main
method is named for
the first .dfy
file on the command-line.
The step of compiling Java files (using javac
) requires the Dafny runtime library.
That library is automatically included if dafny is doing the compilation,
but not if dafny is only doing translation.
Examples of how to integrate Java source code and libraries with Dafny source are contained in this separate document.
13.7.4. Javascript
The Dafny-to-Javascript compiler translates all the given .dfy
files into a single .js
file,
which can then be run using node
. (Javascript has no compilation step).
The build and run steps are simply
dafny build --target:js A.dfy
node A.js
Or, in one step,
dafny run A.dfy
Examples of how to integrate Javascript libraries and source code with Dafny source are contained in this separate document.
13.7.5. Go
The Dafny-to-Go compiler translates all the given .dfy
files into a single
.go
file in A-go/src/A.go
; the output folder can be specified with the
-out
option. For an input file A.dfy
the default output folder is A-go
.
Then, Dafny compiles this program and creates an A.exe
executable in the same folder as A.dfy
.
Some system runtime code is also placed in A-go/src
.
The build and run steps are
dafny build --target:go A.dfy
./A
The uncompiled code can be compiled and run by go
itself using
(cd A-go; GO111MODULE=auto GOPATH=`pwd` go run src/A.go)
The one-step process is
dafny run --target:go A.dfy
The GO111MODULE
variable is used because Dafny translates to pre-module Go code.
When the implementation changes to current Go, the above command-line will
change, though the ./A
alternative will still be supported.
Examples of how to integrate Go source code and libraries with Dafny source are contained in this separate document.
13.7.6. Python
The Dafny-to-Python compiler is still under development. However, simple
Dafny programs can be built and run as follows. The Dafny-to-Python
compiler translates the .dfy
files into a single .py
file along with
supporting runtime library code, all placed in the output location
(A-py
for an input file A.dfy, by default).
The build and run steps are
dafny build --target:py A.dfy
python A-py/A.py
In one step:
dafny run --target:py A.dfy
Examples of how to integrate Python libraries and source code with Dafny source are contained in this separate document.
13.7.7. C++
The C++ backend was written assuming that it would primarily support writing C/C++ style code in Dafny, which leads to some limitations in the current implementation.
- The C++ compiler does not support BigIntegers, so do not use
int
, or raw instances ofarr.Length
, or sequence length, etc. in executable code. You can however, usearr.Length as uint64
if you can prove your array is an appropriate size. The compiler will report inappropriate integer use. - The C++ compiler does not support advanced Dafny features like traits or coinductive types.
- There is very limited support for higher order functions even for array initialization. Use
extern definitions like newArrayFill (see
extern.dfy) or
similar. See also the example in [
functions.dfy
] (https://github.com/dafny-lang/dafny/blob/master/Test/c++/functions.dfy). - The current backend also assumes the use of C++17 in order to cleanly and performantly implement datatypes.
13.7.8. Supported features by target language
Some Dafny features are not supported by every target language. The table below shows which features are supported by each backend. An empty cell indicates that a feature is not supported, while an X indicates that it is.
13.8. Dafny Command Line Options
There are many command-line options to the dafny
tool.
The most current documentation of the options is within the tool itself,
using the -?
or --help
or -h
options.
Remember that options are typically stated with either a leading --
.
Legacy options begin with either ‘-‘ or ‘/’; however they are being
migrated to the POSIX-compliant --
form as needed.
13.8.1. Help and version information
These options emit general information about commands, options and attributes. When present, the dafny program will terminates after emitting the requested information but without processig any files.
-
--help
,-h
- shows the various commands (which have help information under them asdafny <command> -h
-
--version
- show the version of the build
Legacy options:
-
-?
- print out the legacy list of command-line options and terminate. All of these options are also described in this and the following sections. -
-attrHelp
- print out the current list of supported attribute declarations and terminate. -
-env:<n>
- print the command-line arguments supplied to the program. The value of<n>
can be one of the following.-
0
- never print command-line arguments. -
1
(default) - print them to Boogie (.bpl
) files and prover logs. -
2
- operate like with option1
but also print to standard output.
-
-
-wait
- wait for the user to pressEnter
before terminating after a successful execution.
13.8.2. Controlling input
These options control how Dafny processes its input.
-
-stdin
- read standard input and treat it as Dafny source code, instead of reading from a file. -
--library:<files>
- treat the given files as library code, namely, skip these files (and any files recursively included) during verification; the value may be a comma-separated-list of files or folders; folders are expanded into a list of all .dfy files contained, recursively, in those folders -
--prelude:<file>
(was-dprelude
) - select an alternative Dafny prelude file. This file contains Boogie definitions (including many axioms) required by the translator from Dafny to Boogie. Using an alternative prelude is primarily useful if you’re extending the Dafny language or changing how Dafny constructs are modeled. The default prelude is here.
13.8.3. Controlling plugins
Dafny has a plugin capability. A plugin has access to an AST of the dafny input files after all parsing and resolution are performed (but not verification) and also to the command-line options.
This facility is still experimental and very much in flux, particularly
the form of the AST. The best guides to writing a new plugin are
(a) the documentation in the section of this manual on plugins
and (b) example plugins in the
src/Tools
folder of the dafny-lang/compiler-bootstrap
repo.
The value of the option --plugin
is a path to a dotnet dll that contains
the compiled plugin.
13.8.4. Controlling output
These options instruct Dafny to print various information about your program during processing, including variations of the original source code (which can be helpful for debugging).
-
--use-basename-for-filename
- when enabled, just the filename without the directory path is used in error messages; this make error message shorter and not tied to the local environment (which is a help in testing) -
--output
,-o
- location of output files [translate, build] -
--show-snippets
- include with an error message some of the source code text in the neighborhood of the error; the error location (file, line, column) is always given -
--solver-log <file>
- [verification only] the file in which to place the SMT text sent to the solver -
--log-format <configuration>
- [verification only] (was-verificationLogger:<configuration string>
) log verification results to the given test result logger. The currently supported loggers aretrx
,csv
, andtext
. These are the XML-based formats commonly used for test results for .NET languages, a custom CSV schema, and a textual format meant for human consumption, respectively. You can provide configuration using the same string format as when using the--logger
option for dotnet test, such as:-verificationLogger:trx;LogFileName=<...>
The exact mapping of verification concepts to these formats is experimental and subject to change!
The
trx
andcsv
loggers automatically choose an output file name by default, and print the name of this file to the console. Thetext
logger prints its output to the console by default, but can send output to a file given theLogFileName
option.The
text
logger also includes a more detailed breakdown of what assertions appear in each assertion batch. When combined with the-vcsSplitOnEveryAssert
option, it will provide approximate time and resource use costs for each assertion, allowing identification of especially expensive assertions.
Legacy options:
-
-stats
- print various statistics about the Dafny files supplied on the command line. The statistics include the number of total functions, recursive functions, total methods, ghost methods, classes, and modules. They also include the maximum call graph width and the maximum module height. -
-dprint:<file>
- print the Dafny program after parsing (use-
for<file>
to print to the console). -
-rprint:<file>
- print the Dafny program after type resolution (use-
for<file>
to print to the console). -
-printMode:<Everything|DllEmbed|NoIncludes|NoGhost>
- select what to include in the output requested by-dprint
or-rprint
. The argument can be one of the following.-
Everything
(default) - include everything. -
DllEmbed
- print the source that will be included in a compiled DLL. -
NoIncludes
- disable printing of methods incorporated via the include mechanism that have the{:verify false}
attribute, as well as datatypes and fields included from other files. -
NoGhost
- disables printing of functions, ghost methods, and proof statements in implementation methods. Also disable anythingNoIncludes
disables.
-
-
-printIncludes:<None|Immediate|Transitive>
- select what information from included files to incorporate into the output selected by-dprint
or-rprint
. The argument can be one of the following.-
None
(default) - don’t print anything from included files. -
Immediate
- print files directly included by files specified on the command line. Exit after printing. -
Transitive
- print files transitively included by files specified on the command line. Exit after printing.
-
-
-view:<view1, view2>
- this option limits what is printed by /rprint for a module to the names that are part of the given export set; the option argument is a comma-separated list of fully-qualified export set names. -
-funcCallGraph
- print out the function call graph. Each line has the formatfunc,mod=callee*
, wherefunc
is the name of a function,mod
is the name of its containing module, andcallee*
is a space-separated list of the functions thatfunc
calls. -
--show-snippets
(was-showSnippets:<n>
) - show a source code snippet for each Dafny message. The legacy option was-showSnippets
with values 0 and 1 for false and true. -
-printTooltips
- dump additional positional information (displayed as mouse-over tooltips by LSP clients) to standard output asInfo
messages. -
-pmtrace
- print debugging information from the pattern-match compiler. -
-titrace
- print debugging information during the type inference process. -
-diagnosticsFormat:<text|json>
- control how to report errors, warnings, and info messages.<fmt>
may be one of the following:text
(default): Report diagnostics in human-readable format.-
json
: Report diagnostics in JSON format, one object per diagnostic, one diagnostic per line. Info-level messages are only included with-printTooltips
. End positions are only included with-showSnippets:1
. Diagnostics are the following format (but without newlines):{ "location": { "filename": "xyz.dfy", "range": { // Start and (optional) end of diagnostic "start": { "pos": 83, // 0-based character offset in input "line": 6, // 1-based line number "character": 0 // 0-based column number }, "end": { "pos": 86, "line": 6, "character": 3 } } }, "severity": 2, // 1: error; 2: warning; 4: info "message": "module-level const declarations are always non-instance ...", "source": "Parser", "relatedInformation": [ // Additional messages, if any { "location": { ... }, // Like above "message": "...", } ] }
13.8.5. Controlling language features
These options allow some Dafny language features to be enabled or disabled. Some of these options exist for backward compatibility with older versions of Dafny.
-
--function-syntax
(value ‘3’ or ‘4’) - permits a choice of using the Dafny 3 syntax (function
andfunction method
) or the Dafny 4 syntax (ghost function
andfunction
) -
--quantifier-syntax
(value ‘3’ or ‘4’) - permits a choice between the Dafny 3 and Dafny 4 syntax for quantifiers -
--unicode-char
- if false, thechar
type represents any UTF-16 code unit, that is, any 16-bit value, including surrogate code points and allows\uXXXX
escapes in string and character literals. If true,char
represnts any Unicode scalar value, that is, any Unicode code point excluding surrogates and allows\U{X..X}
escapes in string and character literals. The default is false for Dafny version 3 and true for version 4. The legacy option was-unicodeChar:<n>
with values 0 and 1 for false and true above.
Legacy options:
-
-noIncludes
- ignoreinclude
directives in the program. -
-noExterns
- ignoreextern
attributes in the program.
-
--function-syntax:<version>
(was-functionSyntax:<version>
) - select what function syntax to recognize. The syntax for functions is changing from Dafny version 3 to version 4. This switch gives early access to the new syntax, and also provides a mode to help with migration. The valid arguments include the following.-
3
(default) - compiled functions are writtenfunction method
andpredicate method
. Ghost functions are writtenfunction
andpredicate
. -
4
- compiled functions are writtenfunction
andpredicate
. Ghost functions are writtenghost function
andghost predicate
. -
migration3to4
- compiled functions are writtenfunction method
andpredicate method
. Ghost functions are writtenghost function
andghost predicate
. To migrate from version 3 to version 4, use this flag on your version 3 program to flag all occurrences offunction
andpredicate
as parsing errors. These are ghost functions, so change those into the new syntaxghost function
andghost predicate
. Then, start using
-functionSyntax:4
. This will flag all occurrences offunction method
andpredicate method
as parsing errors. So, change those to justfunction
andpredicate
. As a result, your program will use version 4 syntax and have the same meaning as your previous version 3 program. -
experimentalDefaultGhost
- likemigration3to4
, but allowfunction
andpredicate
as alternatives to declaring ghost functions and predicates, respectively -
experimentalDefaultCompiled
- likemigration3to4
, but allowfunction
andpredicate
as alternatives to declaring compiled functions and predicates, respectively -
experimentalPredicateAlwaysGhost
- compiled functions are writtenfunction
. Ghost functions are writtenghost function
. Predicates are always ghost and are writtenpredicate
.
This option can also be set locally (at the module level) using the
:options
attribute: -
module {:options "--function-syntax:4"} M {
predicate CompiledPredicate() { true }
}
-
--quantifier-syntax:<version>
(was-quantifierSyntax:<version>
) - select what quantifier syntax to recognize. The syntax for quantification domains is changing from Dafny version 3 to version 4, more specifically where quantifier ranges (| <Range>
) are allowed. This switch gives early access to the new syntax.3
(default) - Ranges are only allowed after all quantified variables are declared. (e.g.set x, y | 0 <= x < |s| && y in s[x] && 0 <= y :: y
)4
- Ranges are allowed after each quantified variable declaration. (e.g.set x | 0 <= x < |s|, y <- s[x] | 0 <= y :: y
)
Note that quantifier variable domains (
<- <Domain>
) are available in both syntax versions. -
-disableScopes
- treat all export sets asexport reveal *
to never hide function bodies or type definitions during translation. -
-allowsGlobals
- allow the implicit class_default
to contain fields, instance functions, and instance methods. These class members are declared at the module scope, outside of explicit classes. This command-line option is provided to simplify a transition from the behavior in the language prior to version 1.9.3, from which point onward all functions and methods declared at the module scope are implicitly static and field declarations are not allowed at the module scope.
13.8.6. Controlling warnings
These options control what warnings Dafny produces, and whether to treat warnings as errors.
-
--warn-as-errors
(was-warningsAsErrors
) - treat warnings as errors. -
--warn-shadowing
(was-warnShadowing
) - emit a warning if the name of a declared variable caused another variable to be shadowed. -
--warn-missing-constructor-parentheses
- warn if a constructor name in a pattern might be misinterpreted
Legacy options
-
-deprecation:<n>
- control warnings about deprecated features. The value of<n>
can be any of the following.-
0
- don’t issue any warnings. -
1
(default) - issue warnings. -
2
- issue warnings and advise about alternate syntax.
-
13.8.7. Controlling verification
These options control how Dafny verifies the input program, including how much it verifies, what techniques it uses to perform verification, and what information it produces about the verification process.
-
--no-verify
- turns off verification (for translate, build, run commands) -
--verify-included-files
(was-verifyAllModules
) - verify modules that come from include directives.By default, Dafny only verifies files explicitly listed on the command line: if
a.dfy
includesb.dfy
, a call toDafny a.dfy
will detect and report verification errors froma.dfy
but not fromb.dfy
.With this option, Dafny will instead verify everything: all input modules and all their transitive dependencies. This way
Dafny a.dfy
will verifya.dfy
and all files that it includes (hereb.dfy
), as well all files that these files include, etc.Running Dafny with this option on the file containing your main result is a good way to ensure that all its dependencies verify.
-
--track-print-effects
- If true, a compiled method, constructor, or iterator is allowed to have print effects only if it is marked with . (default false) The legacy option was-trackPrintEffects:<n>
) with values 0 or 1 for false and true. --relax-definite-assignment
- control the rules governing definite assignment, the property that every variable is eventually assigned a value before it is used.- if false (default), enforce definite-assignment for all non-yield-parameter variables and fields, regardless of their types
- if false and
--enforce-determinism
is true, then also performs checks in the compiler that no nondeterministic statements are used - if true, enforce definite-assignment rules for compiled variables and fields whose types do not support auto-initialization and for ghost variables and fields whose type is possibly empty.
-
--disable-nonlinear-arithmetic
(was-noNLarith
) - reduce Z3’s knowledge of non-linear arithmetic (the operators*
,/
, and%
). Enabling this option will typically require more manual work to complete proofs (by explicitly applying lemmas about non-linear operators), but will also result in more predictable behavior, since Z3 can sometimes get stuck down an unproductive path while attempting to prove things about those operators. (This option will perhaps be replaced by-arith
in the future. For now, it takes precedence over-arith
.) -
--manual-lemma-induction
- diables automatic inducntion for lemmas --isolate-assertions
- verify assertions individually
Controlling the proof engine:
--cores:<n>
- sets the number or percent of the available cores to be used for verification--verification-time-limit <seconds>
- imposes a time limit on each verification attempt--verification-error-limit <number>
- limits the number of verification errors reported (0 is no limit)--resource-limit
- states a resource limit (to be used by the backend solver)
Legacy options:
-
-dafnyVerify:<n>
[discouraged] - turn verification of the program on or off. The value of<n>
can be any of the following.-
0
- stop after type checking. -
1
- continue on to verification and compilation.
-
-
-separateModuleOutput
- output verification results for each module separately, rather than aggregating them after they are all finished. -
-mimicVerificationOf:<dafny version>
- letdafny
attempt to mimic the verification behavior of a previous version ofdafny
. This can be useful during migration to a newer version ofdafny
when a Dafny program has proofs, such as methods or lemmas, that are highly variable in the sense that their verification may become slower or fail altogether after logically irrelevant changes are made in the verification input.Accepted versions are:
3.3
. Note that falling back on the behavior of version 3.3 turns off features that prevent certain classes of verification variability. -
-noCheating:<n>
- control whether certain assumptions are allowed. The value of<n>
can be one of the following.-
0
(default) - allowassume
statements and free invariants. -
1
- treat all assumptions asassert
statements, and drop free invariants.
-
-
-induction:<n>
- control the behavior of induction. The value of<n>
can be one of the following.-
0
- never do induction, not even when attributes request it. -
1
- apply induction only when attributes request it. -
2
- apply induction as requested (by attributes) and also for heuristically chosen quantifiers. -
3
- apply induction as requested, and for heuristically chosen quantifiers and lemmas. -
4
(default) - apply induction as requested, and for all lemmas.
-
-
-inductionHeuristic:<n>
- control the heuristics used for induction. The value of<n>
can be one of the following.-
0
- use the least discriminating induction heuristic (that is, lean toward applying induction more often). -
1
,2
,3
,4
,5
- use an intermediate heuristic, ordered as follows as far as how discriminating they are: 0 < 1 < 2 < (3,4) < 5 < 6. -
6
(default) - use the most discriminating induction heuristic.
-
-
-allocated:<n>
- specify defaults for where Dafny should assert and assumeallocated(x)
for various parametersx
, local variablesx
, bound variablesx
, etc. Lower<n>
may require more manualallocated(x)
annotations and thus may be more difficult to use. The value of<n>
can be one of the following.-
0
- never assume or assertallocated(x)
by default. -
1
- assumeallocated(x)
only for non-ghost variables and fields. (These assumptions are free, since non-ghost variables always contain allocated values at run-time.) This option may speed up verification relative to-allocated:2
. -
2
- assert/assumeallocated(x)
on all variables, even bound variables in quantifiers. This option is the easiest to use for code that uses the heap heavily. -
3
- (default) make frugal use of heap parameters. -
4
- like3
but addallocated
antecedents when ranges don’t imply allocatedness.
Warning: this option should be chosen consistently across an entire project; it would be unsound to use different defaults for different files or modules within a project. Furthermore, modes
-allocated:0
and-allocated:1
let functions depend on the allocation state, which is not sound in general. -
-
-noAutoReq
- ignoreautoReq
attributes, and therefore do not automatically generaterequires
clauses. -
-autoReqPrint:<file>
- print the requires clauses that were automatically generated byautoReq
to the given<file>
. -
-arith:<n>
- control how arithmetic is modeled during verification. This is an experimental switch, and its options may change. The value of<n>
can be one of the following.-
0
- use Boogie/Z3 built-ins for all arithmetic operations. -
1
(default) - like0
, but introduce symbolic synonyms for*
,/
, and%
, and allow these operators to be used in triggers. -
2
- like1
, but also introduce symbolic synonyms for+
and-
. -
3
- turn off non-linear arithmetic in the SMT solver. Still use Boogie/Z3 built-in symbols for all arithmetic operations. -
4
- like3
, but introduce symbolic synonyms for*
,/
, and%
, and allow these operators to be used in triggers. -
5
- like4
, but also introduce symbolic synonyms for+
and-
. -
6
- like5
, and introduce axioms that distribute+
over*
. -
7
- like6
, and introduce facts about the associativity of literal arguments over*
. -
8
- like7
, and introduce axioms for the connection between*
,/
, and%
. -
9
- like8
, and introduce axioms for sign of multiplication. -
10
- like9
, and introduce axioms for commutativity and associativity of*
.
-
-
-autoTriggers:<n>
- control automatic generation of{:trigger}
annotations. See triggers. The value of<n>
can be one of the following.-
0
- do not generate{:trigger}
annotations for user-level quantifiers. -
1
(default) - add a{:trigger}
annotation to each user-level quantifier. Existing annotations are preserved.
-
-
-rewriteFocalPredicates:<n>
- control rewriting of predicates in the body of prefix lemmas. See the section about nicer extreme proofs. The value of<n>
can be one of the following.-
0
- don’t rewrite predicates in the body of prefix lemmas. -
1
(default) - in the body of prefix lemmas, rewrite any use of a focal predicateP
toP#[_k-1]
.
-
-
-extractCounterexample
- control generation of counterexamples. If verification fails, report a detailed counterexample for the first failing assertion. Requires specifying the-mv
option, to specify where to write the counterexample, as well as the-proverOpt:O:model_compress=false
and-proverOpt:O:model.completion=true
options.
13.8.8. Controlling compilation
These options control what code gets compiled, what target language is used, how compilation proceeds, and whether the compiled program is immediately executed.
-
--target:<s>
or-t:<s>
(was-compileTarget:<s>
) - set the target programming language for the compiler. The value of<s>
can be one of the following.-
cs
- C# . Produces a .dll file that can be run usingdotnet
. For example,dafny Hello.dfy
will produceHello.dll
andHello.runtimeconfig.json
. The dll can be run usingdotnet Hello.dll
. -
go
- Go. The default output ofdafny Hello.dfy -compileTarget:go
is in theHello-go
folder. It is run usingGOPATH=`pwd`/Hello-go/ GO111MODULE=auto go run Hello-go/src/Hello.go
-
js
- Javascript. The default output ofdafny Hello.dfy -compileTarget:js
is the fileHello.js
, which can be run usingnode Hello.js
. (You must havebignumber.js
installed.) -
java
- Java. The default output ofdafny Hello.dfy -compileTarget:java
is in theHello-java
folder. The compiled program can be run usingjava -cp Hello-java:Hello-java/DafnyRuntime.jar Hello
. -
py
- Python. The default output ofdafny Hello.dfy -compileTarget:py
is in theHello-py
folder. The compiled program can be run usingpython Hello-py/Hello.py
, wherepython
is Python version 3. -
cpp
- C++. The default output ofdafny Hello.dfy -compileTarget:cpp
isHello.exe
and other files written to the current folder. The compiled program can be run using./Hello.exe
.
-
-
--input <file>
- designates files to be include in the compilation in addition to the main file indafny run
; these may be non-.dfy files; this option may be specified more than once -
--output:<file>
or-o:<file>
(was-out:<file>
) - set the name to use for compiled code files.
By default, dafny
reuses the name of the Dafny file being compiled.
Compilers that generate a single file use the file name as-is (e.g. the
C# backend will generate <file>.dll
and optionally <file>.cs
with
-spillTargetCode
). Compilers that generate multiple files use the file
name as a directory name (e.g. the Java backend will generate files in
directory <file>-java/
). Any file extension is ignored, so
-out:<file>
is the same as -out:<file>.<ext>
if <file>
contains no
periods.
--include-runtime
- include the runtime library for the target language in the generated artifacts. This is true by default for build and run, but false by default for translate. The legacy option-useRuntimeLib
had the opposite effect: when enabled, the compiled assembly referred to the pre-builtDafnyRuntime.dll
in the compiled assembly rather than includingDafnyRuntime.cs
in the build process.
Legacy options:
-
-compile:<n>
- [obsolete - usedafny build
ordafny run
] control whether compilation happens. The value of<n>
can be one of the following. Note that if the program is compiled, it will be compiled to the target language determined by the-compileTarget
option, which is C# by default.-
0
- do not compile the program -
1
(default) - upon successful verification, compile the program to the target language. -
2
- always compile, regardless of verification success. -
3
- if verification is successful, compile the program (like option1
), and then if there is aMain
method, attempt to run the program. -
4
- always compile (like option2
), and then if there is aMain
method, attempt to run the program.
-
-
-spillTargetCode:<n>
- [obsolete - usedafny translate
) control whether to write out compiled code in the target language (instead of just holding it in internal temporary memory). The value of<n>
can be one of the following.-
0
(default) - don’t make any extra effort to write the textual target program (but still compile it, if-compile
indicates to do so). -
1
- write it out to the target language, if it is being compiled. -
2
- write the compiled program if it passes verification, regardless of the-compile
setting. -
3
- write the compiled program regardless of verification success and the-compile
setting.
-
Note that some compiler targets may (always or in some situations) write
out the textual target program as part of compilation, in which case
-spillTargetCode:0
behaves the same way as -spillTargetCode:1
.
-
-Main:<name>
- specify the (fully-qualified) name of the method to use as the executable entry point. The default is the method with the{:main}
attribute, or else the method namedMain
. -
-compileVerbose:<n>
- control whether to write out compilation progress information. The value of<n>
can be one of the following.-
0
- do not print any information (silent mode) -
1
(default) - print information such as the files being created by the compiler
-
-
-coverage:<file>
- emit branch-coverage calls and outputs into<file>
, including a legend that gives a description of each source-location identifier used in the branch-coverage calls. (Use-
as<file>
to print to the console.) -
-optimize
- produce optimized C# code by passing the/optimize
flag to thecsc
executable. -
-optimizeResolution:<n>
- control optimization of method target resolution. The value of<n>
can be one of the following.-
0
- resolve and translate all methods. -
1
- translate methods only in the call graph of the current verification target. -
2
(default) - as in1
, but resolve only methods that are defined in the current verification target file, not in included files.
-
-
-testContracts:<mode>
- test certain function and method contracts at runtime. This works by generating a wrapper for each function or method to be tested that includes a sequence ofexpect
statements for each requires clause, a call to the original, and sequence ofexpect
statements for eachensures
clause. This is particularly useful for code marked with the{:extern}
attribute and implemented in the target language instead of Dafny. Having runtime checks of the contracts on such code makes it possible to gather evidence that the target-language code satisfies the assumptions made of it during Dafny verification through mechanisms ranging from manual tests through fuzzing to full verification. For the latter two use cases, having checks forrequires
clauses can be helpful, even if the Dafny calling code will never violate them.The
<mode>
parameter can currently be one of the following.-
Externs
- insert dynamic checks when calling any function or method marked with the{:extern}
attribute, wherever the call occurs. -
TestedExterns
- insert dynamic checks when calling any function or method marked with the{:extern}
attribute directly from a function or method marked with the{:test}
attribute.
-
13.8.9. Controlling Boogie
Dafny builds on top of Boogie, a general-purpose intermediate language for verification. Options supported by Boogie on its own are also supported by Dafny. Some of the Boogie options most relevant to Dafny users include the following. We use the term “procedure” below to refer to a Dafny function, lemma, method, or predicate, following Boogie terminology.
-
--solver-path
- specifies a custom SMT solver to use -
--solver-plugin
- specifies a plugin to use as the SMT solver, instead of an external pdafny translaterocess -
--boogie-filter
- restricts the set of verification tasks (for debugging) -
--boogie
- arguments to send to boogie
Legacy options:
-
-proc:<name>
- verify only the procedure named<name>
. The name can include*
to indicate arbitrary sequences of characters. -
-trace
- print extra information during verification, including timing, resource use, and outcome for each procedure incrementally, as verification finishes. -
-randomSeed:<n>
- turn on randomization of the input that Boogie passes to the SMT solver and turn on randomization in the SMT solver itself.Certain Boogie inputs cause proof variability in the sense that changes to the input that preserve its meaning may cause the output to change. The
-randomSeed
option simulates meaning-preserving changes to the input without requiring the user to actually make those changes.The
-randomSeed
option is implemented by renaming variables and reordering declarations in the input, and by setting solver options that have similar effects. -
-randomSeedIterations:<n>
- attempt to prove each VC<n>
times with<n>
random seeds. If-randomSeed
has been provided, each proof attempt will use a new random seed derived from this original seed. If not, it will implicitly use-randomSeed:0
to ensure a difference between iterations. This option can be very useful for identifying input programs for which verification is highly variable. If the verification times or solver resource counts associated with each proof attempt vary widely for a given procedure, small changes to that procedure might be more likely to cause proofs to fail in the future. -
-vcsSplitOnEveryAssert
- prove each (explicit or implicit) assertion in each procedure separately. See also the attribute{:vcs_split_on_every_assert}
for restricting this option on specific procedures. By default, Boogie attempts to prove that every assertion in a given procedure holds all at once, in a single query to an SMT solver. This usually performs well, but sometimes causes the solver to take longer. If a proof that you believe should succeed is timing out, using this option can sometimes help. -
-timeLimit:<n>
- spend at most<n>
seconds attempting to prove any single SMT query. This setting can also be set per method using the attribute{:timeLimit n}
. -
-rlimit:<n>
- set the maximum solver resource count to use while proving a single SMT query. This can be a more deterministic approach than setting a time limit. To choose an appropriate value, please refer to the documentation of the attribute{:rlimit}
that can be applied per procedure. -
-print:<file>
- print the translation of the Dafny file to a Boogie file.
If you have Boogie installed locally, you can run the printed Boogie file with the following script:
DOTNET=$(which dotnet)
BOOGIE_ROOT="path/to/boogie/Source"
BOOGIE="$BOOGIE_ROOT/BoogieDriver/bin/Debug/net6.0/BoogieDriver.dll"
if [[ ! -x "$DOTNET" ]]; then
echo "Error: Dafny requires .NET Core to run on non-Windows systems."
exit 1
fi
#Uncomment if you prefer to use the executable instead of the DLL
#BOOGIE=$(which boogie)
BOOGIE_OPTIONS="/infer:j"
PROVER_OPTIONS="\
/proverOpt:O:auto_config=false \
/proverOpt:O:type_check=true \
/proverOpt:O:smt.case_split=3 \
/proverOpt:O:smt.qi.eager_threshold=100 \
/proverOpt:O:smt.delay_units=true \
/proverOpt:O:smt.arith.solver=2 \
"
"$DOTNET" "$BOOGIE" $BOOGIE_OPTIONS $PROVER_OPTIONS "$@"
#Uncomment if you want to use the executable instead of the DLL
#"$BOOGIE" $BOOGIE_OPTIONS $PROVER_OPTIONS "$@"
13.8.10. Controlling the prover
Much of controlling the prover is accomplished by controlling verification condition generation (25.9.7) or Boogie (Section 13.8.9). The following options are also commonly used:
-
--verification-error-limit:<n>
- limits the number of verification errors reported per procedure. Default is 5; 0 means as many as possible; a small positive number runs faster but a large positive number reports more errors per run -
--verification-time-limit:<n>
(was-timeLimit:<n>
) - limits the number of seconds spent trying to verify each procedure.
13.8.11. Controlling test generation
Dafny is capable of generating unit (runtime) tests. It does so by asking the prover to solve for values of inputs to a method that cause the program to execute specific blocks or paths. A detailed description of how to do this is given in a separate document.
14. Dafny VSCode extension and the Dafny Language Server
14.1. Dafny functionality within VSCode
There is a language server for Dafny, which implements the Language Server Protocol. This server is used by the Dafny VSCode Extension; it currently offers the following features:
- Quick syntax highlighting
- As-you-type parsing, resolution and verification diagnostics
- Support for Dafny plugins
- Expanded explanations (in addition to the error message) for selected errors (and more being added), shown by hovering
- Quick fixes for selected errors (and more being added)
- Limited support for symbol completion
- Limited support for code navigation
- Counter-example display
- Highlighting of ghost statements
- Gutter highlights
- A variety of Preference settings
Most of the Dafny functionality is simply there when editing a .dfy file with VSCode that has the Dafny extension installed. Some actions are available through added menu items. The Dafny functionality within VSCode can be found in these locations:
- The preferences are under the menu Code->Preferences->Settings->Dafny extension configuration. There are two sections of settings.
- A hover over an error location will bring up a hover popup, which will show expanded error information and any quick fix options that are available.
- Within a .dfy editor, a right-click brings up a context menu, which has a menu item ‘Dafny’. Under it are actions to Build or Run a program, to turn on or off counterexample display, find definitions, and the like.
14.2. Gutter highlights
Feedback on a program is show visually as underlining with squiggles within the text and as various markings in various colors in the gutter down the left side of an editor window.
The first time a file is loaded, the gutter will highlight in a transparent squiggly green line all the methods that need to be verified, like this:
When the file is saved (in verification on save), or whenever the Dafny verifier is ready (in verification on change), it will start to verify methods. That line will turn into a thin green rectangle on methods that have been verified, and display an animated less transparent green squiggly line on methods that are being actively verified:
When the verification finishes, if a method, a function, a constant with default initialization or a subset type with a witness has some verification errors in it, the editor will display two yellow vertical rails indicating an error context.
Inside this context, if there is a failing assertion on a line, it will fill the gap between the vertical yellow bars with a red rectangle, even if there might be other assertions that are verified on the line. If there is no error on a line, but there is at least one assertion that verified, it will display a green disk with a white checkmark on it, which can be used to check progress in a proof search.
As soon as a line is changed, the gutter icons turn transparent and squiggly, to indicate their obsolescence.
The red error rectangles occupy only half the horizontal space, to visualise their possible obsolescence.
When the file is saved (in verification on save), or as soon as possible otherwise, these squiggly icons will be animated while the Dafny verifier inspect the area.
If the method was verifying before a change, instead of two yellow vertical bars with a red squiggly line, the gutter icons display an animated squiggly but more firm green line, thereby indicating that the method used to verify, but Dafny is still re-verifying it.
If there is a parse or resolution error, the previous gutter icons turn gray and a red triangle indicates the position of the parse or resolution error.
14.3. The Dafny Server
Before Dafny implemented the official Language Server Protocol, it implemented its own protocol for Emacs, which resulted in a project called DafnyServer.
The Dafny Server has integration tests that serve as the basis of the documentation.
The server is essentially a REPL, which produces output in the same format as the Dafny CLI; clients thus do not need to understand the internals of Dafny’s caching. A typical editing session proceeds as follows:
- When a new Dafny file is opened, the editor starts a new instance of the Dafny server. The cache is blank at that point.
- The editor sends a copy of the buffer for initial verification. This takes some time, after which the server returns a list of errors.
- The user makes modifications; the editor periodically sends a new copy of the buffer’s contents to the Dafny server, which quickly returns an updated list of errors.
The client-server protocol is sequential, uses JSON, and works over ASCII pipes by base64-encoding utf-8 queries. It defines one type of query, and two types of responses:
Queries are of the following form:
verify
<base64 encoded JSON payload>
[[DAFNY-CLIENT: EOM]]
Responses are of the following form:
<list of errors and usual output, as produced by the Dafny CLI>
[SUCCESS] [[DAFNY-SERVER: EOM]]
or
<error message>
[FAILURE] [[DAFNY-SERVER: EOM]]
The JSON payload is an utf-8 encoded string resulting of the serialization of a dictionary with 4 fields:
- args: An array of Dafny arguments, as passed to the Dafny CLI
- source: A Dafny program, or the path to a Dafny source file.
- sourceIsFile: A boolean indicating whether the ‘source’ argument is a Dafny program or the path to one.
- filename: The name of the original source file, to be used in error messages
For small files, embedding the Dafny source directly into a message is convenient; for larger files, however, it is generally better for performance to write the source snapshot to a separate file, and to pass that to Dafny by setting the ‘sourceIsFile’ flag to true.
For example, if you compile and run DafnyServer.exe
, you could paste the following command:
verify
eyJhcmdzIjpbIi9jb21waWxlOjAiLCIvcHJpbnRUb29sdGlwcyIsIi90aW1lTGltaXQ6MjAiXSwi
ZmlsZW5hbWUiOiJ0cmFuc2NyaXB0Iiwic291cmNlIjoibWV0aG9kIEEoYTppbnQpIHJldHVybnMg
KGI6IGludCkge1xuICBiIDo9IGE7XG4gIGFzc2VydCBmYWxzZTtcbn1cbiIsInNvdXJjZUlzRmls
ZSI6ZmFsc2V9
[[DAFNY-CLIENT: EOM]]
The interpreter sees the command verify
, and then starts reading every line until it sees [[DAFNY-CLIENT: EOM]]
The payload is a base64 encoded string that you could encode or decode using JavaScript’s atob
and btoa
function.
For example, the payload above was generated using the following code:
btoa(JSON.stringify({
"args": [
"/compile:0",
"/printTooltips",
"/timeLimit:20"
],
"filename":"transcript",
"source":
`method A(a:int) returns (b: int) {
b := a;
assert false;
}
`,"sourceIsFile": false}))
=== "eyJhcmdzIjpbIi9jb21waWxlOjAiLCIvcHJpbnRUb29sdGlwcyIsIi90aW1lTGltaXQ6MjAiXSwiZmlsZW5hbWUiOiJ0cmFuc2NyaXB0Iiwic291cmNlIjoibWV0aG9kIEEoYTppbnQpIHJldHVybnMgKGI6IGludCkge1xuICBiIDo9IGE7XG4gIGFzc2VydCBmYWxzZTtcbn1cbiIsInNvdXJjZUlzRmlsZSI6ZmFsc2V9"
Thus to decode such output, you’d manually use JSON.parse(atob(payload))
.
The Dafny Server is still supported, but we recommend using the Language Server instead. It is supported in VSCode only. The Language Server also provides features such as ghost highlighting or symbol hovering.
15. Plugins to Dafny
Dafny has a plugin architecture that permits users to build tools for the Dafny language without having to replicate parsing and name/type resolution of Dafny programs. Such a tool might just do some analysis on the Dafny program, without concern for verifying or compiling the program. Or it might modify the program (actually, modify the program’s AST) and then continue on with verification and compilation with the core Dafny tool. A user plugin might also be used in the Language Server and thereby be available in the VSCode (or other) IDE.
_ This is an experimental aspect of Dafny. The plugin API directly exposes the Dafny AST, which is constantly evolving. Hence, always recompile your plugin against the binary of Dafny that will be importing your plugin._
Plugins are libraries linked to a Dafny.dll
of the same version as the language server.
A plugin typically defines:
-
Zero or one class extending
Microsoft.Dafny.Plugins.PluginConfiguration
, which receives plugins arguments in its methodParseArguments
, and 1) Can return a list ofMicrosoft.Dafny.Plugins.Rewriter
s when its methodGetRewriters()
is called by Dafny, 2) Can return a list ofMicrosoft.Dafny.Plugins.Compiler
s when its methodGetCompilers()
is called by Dafny, 3) If the configuration extends the subclassMicrosoft.Dafny.LanguageServer.Plugins.PluginConfiguration
, then it can return a list ofMicrosoft.Dafny.LanguageServer.Plugins.DafnyCodeActionProvider
s when its methodGetDafnyCodeActionProviders()
is called by the Dafny Language Server. - Zero or more classes extending
Microsoft.Dafny.Plugins.Rewriter
. If a configuration class is provided, it is responsible for instantiating them and returning them inGetRewriters()
. If no configuration class is provided, an automatic configuration will load every definedRewriter
automatically. - Zero or more classes extending
Microsoft.Dafny.Plugins.Compiler
. If a configuration class is provided, it is responsible for instantiating them and returning them inGetCompilers()
. If no configuration class is provided, an automatic configuration will load every definedCompiler
automatically. - Zero or more classes extending
Microsoft.Dafny.LanguageServer.Plugins.DafnyCodeActionProvider
. Only a configuration class of typeMicrosoft.Dafny.LanguageServer.Plugins.PluginConfiguration
can be responsible for instantiating them and returning them inGetDafnyCodeActionProviders()
.
The most important methods of the class Rewriter
that plugins override are
- (experimental)
PreResolve(ModuleDefinition)
: Here you can optionally modify the AST before it is resolved. PostResolve(ModuleDefinition)
: This method is repeatedly called with every resolved and type-checked module, before verification. Plugins override this method typically to report additional diagnostics.PostResolve(Program)
: This method is called once after allPostResolve(ModuleDefinition)
have been called.
Plugins are typically used to report additional diagnostics such as unsupported constructs for specific compilers (through the methods Èrror(...)
and Warning(...)
of the field Reporter
of the class Rewriter
)
Note that all plugin errors should use the original program’s expressions’ token and NOT Token.NoToken
, else no error will be displayed in the IDE.
15.1. Code actions plugin tutorial
In this section, we will create a plugin to provide more code actions to Dafny. The code actions will be simple: Add a dummy comment in front of the first method name, if the selection is on the line of the method.
Assuming the Dafny source code is installed in the folder dafny/
start by creating an empty folder next to it, e.g. PluginTutorial/
mkdir PluginTutorial
cd PluginTutorial
Then, create a dotnet class project
dotnet new classlib
It will create a file Class1.cs
that you can rename
mv Class1.cs PluginAddComment.cs
Open the newly created file PluginTutorial.csproj
, and add the following after </PropertyGroup>
:
<ItemGroup>
<ProjectReference Include="../dafny/source/DafnyLanguageServer/DafnyLanguageServer.csproj" />
</ItemGroup>
Then, open the file PluginAddComment.cs
, remove everything, and write the imports and a namespace:
using Microsoft.Dafny;
using Microsoft.Dafny.LanguageServer.Plugins;
using Microsoft.Boogie;
using Microsoft.Dafny.LanguageServer.Language;
using System.Linq;
using Range = OmniSharp.Extensions.LanguageServer.Protocol.Models.Range;
namespace PluginAddComment;
After that, add a PluginConfiguration
that will expose all the quickfixers of your plugin.
This class will be discovered and instantiated automatically by Dafny.
public class TestConfiguration : PluginConfiguration {
public override DafnyCodeActionProvider[] GetDafnyCodeActionProviders() {
return new DafnyCodeActionProvider[] { new AddCommentDafnyCodeActionProvider() };
}
}
Note that you could also override the methods GetRewriters()
and GetCompilers()
for other purposes, but this is out of scope for this tutorial.
Then, we need to create the quickFixer AddCommentDafnyCodeActionProvider
itself:
public class AddCommentDafnyCodeActionProvider : DafnyCodeActionProvider {
public override IEnumerable<DafnyCodeAction> GetDafnyCodeActions(IDafnyCodeActionInput input, Range selection) {
return new DafnyCodeAction[] { };
}
}
For now, this quick fixer returns nothing. input
is the program state, and selection
is where the caret is.
We replace the return statement with a conditional that tests whether the selection is on the first line:
var firstTokenRange = input.Program?.GetFirstTopLevelToken()?.GetLspRange();
if(firstTokenRange != null && firstTokenRange.Start.Line == selection.Start.Line) {
return new DafnyCodeAction[] {
// TODO
};
} else {
return new DafnyCodeAction[] { };
}
Every quick fix consists of a title (provided immediately), and zero or more DafnyCodeActionEdit
(computed lazily).
A DafnyCodeActionEdit
has a Range
to remove and some string
to insert instead. All DafnyCodeActionEdit
s
of the same DafnyCodeAction
are applied at the same time if selected.
To create a DafnyCodeAction
, we can either use the easy-to-use InstantDafnyCodeAction
, which accepts a title and an array of edits:
return new DafnyCodeAction[] {
new InstantDafnyCodeAction("Insert comment", new DafnyCodeActionEdit[] {
new DafnyCodeActionEdit(firstTokenRange.GetStartRange(), "/*First comment*/")
})
};
or we can implement our custom inherited class of DafnyCodeAction
:
public class CustomDafnyCodeAction: DafnyCodeAction {
public Range whereToInsert;
public CustomDafnyCodeAction(Range whereToInsert): base("Insert comment") {
this.whereToInsert = whereToInsert;
}
public override DafnyCodeActionEdit[] GetEdits() {
return new DafnyCodeActionEdit[] {
new DafnyCodeActionEdit(whereToInsert.GetStartRange(), "/*A comment*/")
};
}
}
In that case, we could return:
return new DafnyCodeAction[] {
new CustomDafnyCodeAction(firstTokenRange)
};
That’s it! Now, build your library while inside your folder:
> dotnet build
This will create the file PluginTutorial/bin/Debug/net6.0/PluginTutorial.dll
.
Now, open VSCode, open Dafny settings, and enter the absolute path to this DLL in the plugins section.
Restart VSCode, and it should work!
16. Full list of legacy command-line options {#sec-full-command-line-options}
For the on-line version only, the output of dafny -?
follows. Note that with the advent of dafny commands, many options are only applicable to some (if any) commandsi, some are renamed, and some are obsolete and will eventually be removed.
Use 'dafny --help' to see help for a newer Dafny CLI format.
Usage: dafny [ option ... ] [ filename ... ]
---- General options -------------------------------------------------------
/version print the dafny version number
/help print this message
/attrHelp print a message about supported declaration attributes
/env:<n> print command line arguments
0 - never, 1 (default) - during BPL print and prover log,
2 - like 1 and also to standard output
/printVerifiedProceduresCount:<n>
0 - no
1 (default) - yes
/wait await Enter from keyboard before terminating program
/xml:<file> also produce output in XML format to <file>
All the .dfy files supplied on the command line along with files recursively
included by 'include' directives are considered a single Dafny program;
however only those files listed on the command line are verified.
Exit code: 0 -- success; 1 -- invalid command-line; 2 -- parse or type errors;
3 -- compilation errors; 4 -- verification errors
---- Input configuration ---------------------------------------------------
/dprelude:<file>
Choose the Dafny prelude file.
/stdin
Read standard input and treat it as an input .dfy file.
---- Plugins ---------------------------------------------------------------
/plugin:<path-to-one-assembly[,argument]*>
(experimental) One path to an assembly that contains at least one
instantiatable class extending Microsoft.Dafny.Plugin.Rewriter. It
can also extend Microsoft.Dafny.Plugins.PluginConfiguration to
receive arguments. More information about what plugins do and how
to define them:
https://github.com/dafny-lang/dafny/blob/master/Source/DafnyLanguageServer/README.md#about-plugins
---- Overall reporting and printing ----------------------------------------
/dprint:<file>
Print Dafny program after parsing it.
(Use - as <file> to print to console.)
/printMode:<Everything|DllEmbed|NoIncludes|NoGhost>
Everything (default) - Print everything listed below.
DllEmbed - print the source that will be included in a compiled dll.
NoIncludes - disable printing of {:verify false} methods
incorporated via the include mechanism, as well as datatypes and
fields included from other files.
NoGhost - disable printing of functions, ghost methods, and proof
statements in implementation methods. It also disables anything
NoIncludes disables.
/rprint:<file>
Print Dafny program after resolving it.
(use - as <file> to print to console.)
/showSnippets:<value>
0 (default) - Don't show source code snippets for Dafny messages.
1 - Show a source code snippet for each Dafny message.
/stats
Print interesting statistics about the Dafny files supplied.
/printIncludes:<None|Immediate|Transitive>
None (default) - Print nothing.
Immediate - Print files included by files listed on the command line.
Transitive - Recurses on the files printed by Immediate.
Immediate and Transitive will exit after printing.
/view:<view1, view2>
Print the filtered views of a module after it is resolved (/rprint).
If print before the module is resolved (/dprint), then everything in
the module is printed. If no view is specified, then everything in
the module is printed.
/funcCallGraph
Print out the function call graph. Format is: func,mod=callee*
/pmtrace
Print pattern-match compiler debug info.
/printTooltips
Dump additional positional information (displayed as mouse-over
tooltips by the VS Code plugin) to stdout as 'Info' messages.
/diagnosticsFormat:<text|json>
Choose how to report errors, warnings, and info messages.
text (default) - Use human readable output
json - Print each message as a JSON object, one per line.
---- Language feature selection --------------------------------------------
/generalTraits:<value>
0 (default) - Every trait implicitly extends 'object', and thus is a reference type. Only traits and classes can extend traits.
1 - A trait is a reference type iff it or one of its ancestor traits is 'object'. Any type with members can extend traits.
/ntitrace:<value>
0 (default) - Don't print debug information for the new type system.
1 - Print debug information for the new type system.
/titrace:<value>
0 (default) - Don't print type-inference debug information.
1 - Print type-inference debug information.
/typeSystemRefresh:<value>
0 (default) - The type-inference engine and supported types are those of Dafny 4.0.
1 - Use an updated type-inference engine. Warning: This mode is under construction and probably won't work at this time.
/unicodeChar:<value>
0 - The char type represents any UTF-16 code unit.
1 (default) - The char type represents any Unicode scalar value.
/noIncludes
Ignore include directives.
/noExterns
Ignore extern attributes.
/functionSyntax:<version>
The syntax for functions is changing from Dafny version 3 to version
4. This switch gives early access to the new syntax, and also
provides a mode to help with migration.
3 - Compiled functions are written `function method` and
`predicate method`. Ghost functions are written `function` and
`predicate`.
4 (default) - Compiled functions are written `function` and `predicate`. Ghost
functions are written `ghost function` and `ghost predicate`.
migration3to4 - Compiled functions are written `function method` and
`predicate method`. Ghost functions are written `ghost function`
and `ghost predicate`. To migrate from version 3 to version 4,
use this flag on your version 3 program. This will give flag all
occurrences of `function` and `predicate` as parsing errors.
These are ghost functions, so change those into the new syntax
`ghost function` and `ghost predicate`. Then, start using
/functionSyntax:4. This will flag all occurrences of `function
method` and `predicate method` as parsing errors. So, change
those to just `function` and `predicate`. Now, your program uses
version 4 syntax and has the exact same meaning as your previous
version 3 program.
experimentalDefaultGhost - Like migration3to4, but allow `function`
and `predicate` as alternatives to declaring ghost functions and
predicates, respectively.
experimentalDefaultCompiled - Like migration3to4, but allow
`function` and `predicate` as alternatives to declaring compiled
functions and predicates, respectively.
experimentalPredicateAlwaysGhost - Compiled functions are written
`function`. Ghost functions are written `ghost function`.
Predicates are always ghost and are written `predicate`.
/quantifierSyntax:<version>
The syntax for quantification domains is changing from Dafny version
3 to version 4, more specifically where quantifier ranges (|
<Range>) are allowed. This switch gives early access to the new
syntax.
3 - Ranges are only allowed after all quantified variables
are declared. (e.g. set x, y | 0 <= x < |s| && y in s[x] && 0 <=
y :: y)
4 (default) - Ranges are allowed after each quantified variable declaration.
(e.g. set x | 0 <= x < |s|, y <- s[x] | 0 <= y :: y)
Note that quantifier variable domains (<- <Domain>) are available in
both syntax versions.
/disableScopes
Treat all export sets as 'export reveal *'. i.e. don't hide function
bodies or type definitions during translation.
---- Warning selection -----------------------------------------------------
/warnShadowing
Emits a warning if the name of a declared variable caused another
variable to be shadowed.
/warnMissingConstructorParenthesis
Emits a warning when a constructor name in a case pattern is not
followed by parentheses.
/deprecation:<n>
0 - Don't give any warnings about deprecated features.
1 (default) - Show warnings about deprecated features.
/warningsAsErrors
Treat warnings as errors.
---- Verification options -------------------------------------------------
/dafnyVerify:<n>
0 - Stop after resolution and typechecking.
1 - Continue on to verification and compilation.
/verifyAllModules
Verify modules that come from an include directive.
/separateModuleOutput
Output verification results for each module separately, rather than
aggregating them after they are all finished.
/verificationLogger:<configuration string>
Logs verification results to the given test result logger. The
currently supported loggers are `trx`, `csv`, and `text`. These are
the XML-based format commonly used for test results for .NET
languages, a custom CSV schema, and a textual format meant for human
consumption. You can provide configuration using the same string
format as when using the --logger option for dotnet test, such as:
/verificationLogger:trx;LogFileName=<...>.
The exact mapping of verification concepts to these formats is
experimental and subject to change!
The `trx` and `csv` loggers automatically choose an output file name
by default, and print the name of this file to the console. The
`text` logger prints its output to the console by default, but can
send output to a file given the `LogFileName` option.
The `text` logger also includes a more detailed breakdown of what
assertions appear in each assertion batch. When combined with the
`/vcsSplitOnEveryAssert` option, it will provide approximate time
and resource use costs for each assertion, allowing identification
of especially expensive assertions.
/noCheating:<n>
0 (default) - Allow assume statements and free invariants.
1 - Treat all assumptions as asserts, and drop free.
/induction:<n>
0 - Never do induction, not even when attributes request it.
1 - Only apply induction when attributes request it.
2 - Apply induction as requested (by attributes) and also for
heuristically chosen quantifiers.
3 - Apply induction as requested, and for heuristically chosen
quantifiers and lemmas.
4 (default) - Apply induction as requested, and for lemmas.
/inductionHeuristic:<n>
0 - Least discriminating induction heuristic (that is, lean toward
applying induction more often).
1,2,3,4,5 - Levels in between, ordered as follows as far as how
discriminating they are: 0 < 1 < 2 < (3,4) < 5 < 6.
6 (default) - Most discriminating.
/trackPrintEffects:<n>
0 (default) - Every compiled method, constructor, and iterator,
whether or not it bears a {:print} attribute, may have print
effects.
1 - A compiled method, constructor, or iterator is allowed to have
print effects only if it is marked with {:print}.
/definiteAssignment:<n>
0 - Ignores definite-assignment rules. This mode is for testing
only--it is not sound.
1 (default) - Enforces definite-assignment rules for compiled
variables and fields whose types do not support
auto-initialization, and for ghost variables and fields whose
type is possibly empty.
2 - Enforces definite-assignment for all non-yield-parameter
variables and fields, regardless of their types.
3 - Like 2, but also performs checks in the compiler that no
nondeterministic statements are used; thus, a program that
passes at this level 3 is one that the language guarantees that
values seen during execution will be the same in every run of
the program.
4 - Like 1, but enforces definite assignment for all local variables
and out-parameters, regardless of their types. (Whether or not
fields and new arrays are subject to definite assignments depends
on their types.)
/noAutoReq
Ignore autoReq attributes.
/autoReqPrint:<file>
Print out requirements that were automatically generated by autoReq.
/noNLarith
Reduce Z3's knowledge of non-linear arithmetic (*,/,%).
Results in more manual work, but also produces more predictable
behavior. (This switch will perhaps be replaced by /arith in the
future. For now, it takes precedence of /arith.)
/arith:<n>
(experimental) Adjust how Dafny interprets arithmetic operations.
0 - Use Boogie/Z3 built-ins for all arithmetic operations.
1 (default) - Like 0, but introduce symbolic synonyms for *,/,%, and
allow these operators to be used in triggers.
2 - Like 1, but introduce symbolic synonyms also for +,-.
3 - Turn off non-linear arithmetic in the SMT solver. Still, use
Boogie/Z3 built-in symbols for all arithmetic operations.
4 - Like 3, but introduce symbolic synonyms for *,/,%, and allow
these operators to be used in triggers.
5 - Like 4, but introduce symbolic synonyms also for +,-.
6 - Like 5, and introduce axioms that distribute + over *.
7 - like 6, and introduce facts that associate literals arguments of *.
8 - Like 7, and introduce axiom for the connection between *,/,%.
9 - Like 8, and introduce axioms for sign of multiplication.
10 - Like 9, and introduce axioms for commutativity and
associativity of *.
/autoTriggers:<n>
0 - Do not generate {:trigger} annotations for user-level
quantifiers.
1 (default) - Add a {:trigger} to each user-level quantifier.
Existing annotations are preserved.
/rewriteFocalPredicates:<n>
0 - Don't rewrite predicates in the body of prefix lemmas.
1 (default) - In the body of prefix lemmas, rewrite any use of a
focal predicate P to P#[_k-1].
/extractCounterexample
If verification fails, report a detailed counterexample for the
first failing assertion. Requires specifying the /mv:<file> option as well
as /proverOpt:O:model_compress=false (for z3 version < 4.8.7) or
/proverOpt:O:model.compact=false (for z3 version >= 4.8.7), and
/proverOpt:O:model.completion=true.
---- Test generation options -----------------------------------------------
/warnDeadCode
Use counterexample generation to warn about potential dead code.
/generateTestMode:<None|Block|Path>
None (default) - Has no effect.
Block - Prints block-coverage tests for the given program.
Path - Prints path-coverage tests for the given program.
Using /definiteAssignment:3, /generateTestNoPrune,
/generateTestSeqLengthLimit, and /loopUnroll is highly recommended
when generating tests.
/generateTestSeqLengthLimit:<n>
Add an axiom that sets the length of all sequences to be no greater
than <n>. 0 (default) indicates no limit.
/generateTestTargetMethod:<methodName>
If specified, only this method will be tested.
/generateTestPrintBpl:<fileName>
Print the Boogie code used during test generation.
/generateTestNoPrune
Disable axiom pruning that Dafny uses to speed up verification.
---- Compilation options ---------------------------------------------------
/compileSuffix:<value>
Add the suffix _Compile to module names without :extern
/compileTarget:<language>
cs (default) - Compile to .NET via C#.
go - Compile to Go.
js - Compile to JavaScript.
java - Compile to Java.
py - Compile to Python.
cpp - Compile to C++.
dfy - Compile to Dafny.
Note that the C++ backend has various limitations (see
Docs/Compilation/Cpp.md). This includes lack of support for
BigIntegers (aka int), most higher order functions, and advanced
features like traits or co-inductive types.
/library:<value>
The contents of this file and any files it includes can be referenced from other files as if they were included.
However, these contents are skipped during code generation and verification.
This option is useful in a diamond dependency situation,
to prevent code from the bottom dependency from being generated more than once.
The value may be a comma-separated list of files and folders.
/optimizeErasableDatatypeWrapper:<value>
0 - Include all non-ghost datatype constructors in the compiled code
1 (default) - In the compiled target code, transform any non-extern
datatype with a single non-ghost constructor that has a single
non-ghost parameter into just that parameter. For example, the type
datatype Record = Record(x: int)
is transformed into just 'int' in the target code.
/out:<file>
Specify the filename and location for the generated target language files.
/compile:<n>
0 - Do not compile Dafny program.
1 (default) - Upon successful verification of the Dafny program,
compile it to the designated target language. (/noVerify
automatically counts as a failed verification.)
2 - Always attempt to compile Dafny program to the target language,
regardless of verification outcome.
3 - If there is a Main method and there are no verification errors
and /noVerify is not used, compiles program in memory (i.e.,
does not write an output file) and runs it.
4 - Like (3), but attempts to compile and run regardless of
verification outcome.
/Main:<name>
Specify the (fully-qualified) name of the method to use as the executable entry point.
Default is the method with the {:main} attribute, or else the method named 'Main'.
A Main method can have at most one (non-ghost) argument of type `seq<string>`
--args <arg1> <arg2> ...
When running a Dafny file through /compile:3 or /compile:4, '--args' provides
all arguments after it to the Main function, at index starting at 1.
Index 0 is used to store the executable's name if it exists.
/runAllTests:<n> (experimental)
0 (default) - Annotates compiled methods with the {:test}
attribute such that they can be tested using a testing framework
in the target language (e.g. xUnit for C#).
1 - Emits a main method in the target language that will execute
every method in the program with the {:test} attribute. Cannot
be used if the program already contains a main method. Note that
/compile:3 or 4 must be provided as well to actually execute
this main method!
/compileVerbose:<n>
0 - Don't print status of compilation to the console.
1 (default) - Print information such as files being written by the
compiler to the console.
/spillTargetCode:<n>
Explicitly writes the code in the target language to one or more files.
This is not necessary to run a Dafny program, but may be of interest when
building multi-language programs or for debugging.
0 (default) - Don't make any extra effort to write the textual
target program (but still compile it, if /compile indicates to
do so).
1 - Write the textual target program, if it is being compiled.
2 - Write the textual target program, provided it passes the
verifier (and /noVerify is NOT used), regardless of /compile
setting.
3 - Write the textual target program, regardless of verification
outcome and /compile setting.
Note, some compiler targets may (always or in some situations) write
out the textual target program as part of compilation, in which case
/spillTargetCode:0 behaves the same way as /spillTargetCode:1.
/coverage:<file>
The compiler emits branch-coverage calls and outputs into <file> a
legend that gives a description of each source-location identifier
used in the branch-coverage calls. (Use - as <file> to print to the
console.)
/optimize
Produce optimized C# code by passing the /optimize flag to csc.exe.
/optimizeResolution:<n>
0 - Resolve and translate all methods.
1 - Translate methods only in the call graph of current verification
target.
2 (default) - As in 1, but only resolve method bodies in
non-included Dafny sources.
/useRuntimeLib
Refer to a pre-built DafnyRuntime.dll in the compiled assembly
rather than including DafnyRuntime.cs verbatim.
/testContracts:<Externs|TestedExterns>
Enable run-time testing of the compilable portions of certain function
or method contracts, at their call sites. The current implementation
focuses on {:extern} code but may support other code in the future.
Externs - Check contracts on every call to a function or method marked
with the {:extern} attribute, regardless of where it occurs.
TestedExterns - Check contracts on every call to a function or method
marked with the {:extern} attribute when it occurs in a method
with the {:test} attribute, and warn if no corresponding test
exists for a given external declaration.
----------------------------------------------------------------------------
Dafny generally accepts Boogie options and passes these on to Boogie.
However, some Boogie options, like /loopUnroll, may not be sound for
Dafny or may not have the same meaning for a Dafny program as it would
for a similar Boogie program.
---- Boogie options --------------------------------------------------------
Multiple .bpl files supplied on the command line are concatenated into one
Boogie program.
/lib:<name> : Include definitions in library <name>. The file <name>.bpl
must be an included resource in Core.dll. Currently, the
following libraries are supported---base, node.
/proc:<p> : Only check procedures matched by pattern <p>. This option
may be specified multiple times to match multiple patterns.
The pattern <p> matches the whole procedure name and may
contain * wildcards which match any character zero or more
times.
/noProc:<p> : Do not check procedures matched by pattern <p>. Exclusions
with /noProc are applied after inclusions with /proc.
/noResolve : parse only
/noTypecheck : parse and resolve only
/print:<file> : print Boogie program after parsing it
(use - as <file> to print to console)
/pretty:<n>
0 - print each Boogie statement on one line (faster).
1 (default) - pretty-print with some line breaks.
/printWithUniqueIds : print augmented information that uniquely
identifies variables
/printUnstructured : with /print option, desugars all structured statements
/printDesugared : with /print option, desugars calls
/printLambdaLifting : with /print option, desugars lambda lifting
/freeVarLambdaLifting : Boogie's lambda lifting transforms the bodies of lambda
expressions into templates with holes. By default, holes
are maximally large subexpressions that do not contain
bound variables. This option performs a form of lambda
lifting in which holes are the lambda's free variables.
/overlookTypeErrors : skip any implementation with resolution or type
checking errors
/loopUnroll:<n>
unroll loops, following up to n back edges (and then some)
/soundLoopUnrolling
sound loop unrolling
/printModel:<n>
0 (default) - do not print Z3's error model
1 - print Z3's error model
/printModelToFile:<file>
print model to <file> instead of console
/mv:<file> Specify file to save the model with captured states
(see documentation for :captureState attribute)
/enhancedErrorMessages:<n>
0 (default) - no enhanced error messages
1 - Z3 error model enhanced error messages
/printCFG:<prefix> : print control flow graph of each implementation in
Graphviz format to files named:
<prefix>.<procedure name>.dot
/useBaseNameForFileName : When parsing use basename of file for tokens instead
of the path supplied on the command line
/emitDebugInformation:<n>
0 - do not emit debug information
1 (default) - emit the debug information :qid, :skolemid and set-info :boogie-vc-id
/normalizeNames:<n>
0 (default) - Keep Boogie program names when generating SMT commands
1 - Normalize Boogie program names when generating SMT commands.
This keeps SMT solver input, and thus output,
constant when renaming declarations in the input program.
/normalizeDeclarationOrder:<n>
0 - Keep order of top-level declarations when generating SMT commands.
1 (default) - Normalize order of top-level declarations when generating SMT commands.
This keeps SMT solver input, and thus output,
constant when reordering declarations in the input program.
---- Inference options -----------------------------------------------------
/infer:<flags>
use abstract interpretation to infer invariants
<flags> must specify exactly one of the following domains:
t = trivial bottom/top lattice
j = stronger intervals
together with any of the following options:
s = debug statistics
0..9 = number of iterations before applying a widen (default=0)
/checkInfer instrument inferred invariants as asserts to be checked by
theorem prover
/contractInfer
perform procedure contract inference
/instrumentInfer
h - instrument inferred invariants only at beginning of
loop headers (default)
e - instrument inferred invariants at beginning and end
of every block (this mode is intended for use in
debugging of abstract domains)
/printInstrumented
print Boogie program after it has been instrumented with
invariants
---- Debugging and general tracing options ---------------------------------
/silent print nothing at all
/quiet print nothing but warnings and errors
/trace blurt out various debug trace information
/traceTimes output timing information at certain points in the pipeline
/tracePOs output information about the number of proof obligations
(also included in the /trace output)
/break launch and break into debugger
---- Civl options ----------------------------------------------------------
/trustMoverTypes
do not verify mover type annotations on atomic action declarations
/trustNoninterference
do not perform noninterference checks
/trustRefinement
do not perform refinement checks
/trustLayersUpto:<n>
do not verify layers <n> and below
/trustLayersDownto:<n>
do not verify layers <n> and above
/trustSequentialization
do not perform sequentialization checks
/civlDesugaredFile:<file>
print plain Boogie program to <file>
---- Verification-condition generation options -----------------------------
/liveVariableAnalysis:<c>
0 = do not perform live variable analysis
1 = perform live variable analysis (default)
2 = perform interprocedural live variable analysis
/noVerify skip VC generation and invocation of the theorem prover
/verifySnapshots:<n>
verify several program snapshots (named <filename>.v0.bpl
to <filename>.vN.bpl) using verification result caching:
0 - do not use any verification result caching (default)
1 - use the basic verification result caching
2 - use the more advanced verification result caching
3 - use the more advanced caching and report errors according
to the new source locations for errors and their
related locations (but not /errorTrace and CaptureState
locations)
/traceCaching:<n>
0 (default) - none
1 - for testing
2 - for benchmarking
3 - for testing, benchmarking, and debugging
/verifySeparately
verify each input program separately
/removeEmptyBlocks:<c>
0 - do not remove empty blocks during VC generation
1 - remove empty blocks (default)
/coalesceBlocks:<c>
0 = do not coalesce blocks
1 = coalesce blocks (default)
/traceverify print debug output during verification condition generation
/subsumption:<c>
apply subsumption to asserted conditions:
0 - never, 1 - not for quantifiers, 2 (default) - always
/alwaysAssumeFreeLoopInvariants
usually, a free loop invariant (or assume
statement in that position) is ignored in checking contexts
(like other free things); this option includes these free
loop invariants as assumes in both contexts
/inline:<i> use inlining strategy <i> for procedures with the :inline
attribute, see /attrHelp for details:
none
assume (default)
assert
spec
/printInlined
print the implementation after inlining calls to
procedures with the :inline attribute (works with /inline)
/recursionBound:<n>
Set the recursion bound for stratified inlining to
be n (default 500)
/smoke Soundness Smoke Test: try to stick assert false; in some
places in the BPL and see if we can still prove it
/smokeTimeout:<n>
Timeout, in seconds, for a single theorem prover
invocation during smoke test, defaults to 10.
/causalImplies
Translate Boogie's A ==> B into prover's A ==> A && B.
/typeEncoding:<t>
Encoding of types when generating VC of a polymorphic program:
m = monomorphic (default)
p = predicates
a = arguments
Boogie automatically detects monomorphic programs and enables
monomorphic VC generation, thereby overriding the above option.
If the latter two options are used, then arrays are handled via axioms.
/useArrayAxioms
If monomorphic type encoding is used, arrays are handled by default with
the SMT theory of arrays. This option allows the use of axioms instead.
/reflectAdd In the VC, generate an auxiliary symbol, elsewhere defined
to be +, instead of +.
/prune
Turn on pruning. Pruning will remove any top-level Boogie declarations
that are not accessible by the implementation that is about to be verified.
Without pruning, due to the unstable nature of SMT solvers,
a change to any part of a Boogie program has the potential
to affect the verification of any other part of the program.
/printPruned:<file>
After pruning, print the Boogie program to the specified file.
/relaxFocus Process foci in a bottom-up fashion. This way only generates
a linear number of splits. The default way (top-down) is more
aggressive and it may create an exponential number of splits.
/randomSeed:<s>
Supply the random seed for /randomizeVcIterations option.
/randomizeVcIterations:<n>
Turn on randomization of the input that Boogie passes to the
SMT solver and turn on randomization in the SMT solver itself.
Attempt to randomize and prove each VC n times using the random
seed s provided by the option /randomSeed:<s>. If /randomSeed option
is not provided, s is chosen to be zero.
Certain Boogie inputs are unstable in the sense that changes to
the input that preserve its meaning may cause the output to change.
This option simulates meaning-preserving changes to the input
without requiring the user to actually make those changes.
This option is implemented by renaming variables and reordering
declarations in the input, and by setting solver options that have
similar effects.
---- Verification-condition splitting --------------------------------------
/vcsMaxCost:<f>
VC will not be split unless the cost of a VC exceeds this
number, defaults to 2000.0. This does NOT apply in the
keep-going mode after first round of splitting.
/vcsMaxSplits:<n>
Maximal number of VC generated per method. In keep
going mode only applies to the first round.
Defaults to 1.
/vcsMaxKeepGoingSplits:<n>
If set to more than 1, activates the keep
going mode, where after the first round of splitting,
VCs that timed out are split into <n> pieces and retried
until we succeed proving them, or there is only one
assertion on a single path and it timeouts (in which
case error is reported for that assertion).
Defaults to 1.
/vcsKeepGoingTimeout:<n>
Timeout in seconds for a single theorem prover
invocation in keep going mode, except for the final
single-assertion case. Defaults to 1s.
/vcsFinalAssertTimeout:<n>
Timeout in seconds for the single last
assertion in the keep going mode. Defaults to 30s.
/vcsPathJoinMult:<f>
If more than one path join at a block, by how much
multiply the number of paths in that block, to accomodate
for the fact that the prover will learn something on one
paths, before proceeding to another. Defaults to 0.8.
/vcsPathCostMult:<f1>
/vcsAssumeMult:<f2>
The cost of a block is
(<assert-cost> + <f2>*<assume-cost>) *
(1.0 + <f1>*<entering-paths>)
<f1> defaults to 1.0, <f2> defaults to 0.01.
The cost of a single assertion or assumption is
currently always 1.0.
/vcsPathSplitMult:<f>
If the best path split of a VC of cost A is into
VCs of cost B and C, then the split is applied if
A >= <f>*(B+C), otherwise assertion splitting will be
applied. Defaults to 0.5 (always do path splitting if
possible), set to more to do less path splitting
and more assertion splitting.
/vcsSplitOnEveryAssert
Splits every VC so that each assertion is isolated
into its own VC. May result in VCs without any assertions.
/vcsDumpSplits
For split #n dump split.n.dot and split.n.bpl.
Warning: Affects error reporting.
/vcsCores:<n>
Try to verify <n> VCs at once. Defaults to 1.
/vcsLoad:<f> Sets vcsCores to the machine's ProcessorCount * f,
rounded to the nearest integer (where 0.0 <= f <= 3.0),
but never to less than 1.
---- Prover options --------------------------------------------------------
/errorLimit:<num>
Limit the number of errors produced for each procedure
(default is 5, some provers may support only 1).
Set num to 0 to find as many assertion failures as possible.
/timeLimit:<num>
Limit the number of seconds spent trying to verify
each procedure
/rlimit:<num>
Limit the Z3 resource spent trying to verify each procedure
/errorTrace:<n>
0 - no Trace labels in the error output,
1 (default) - include useful Trace labels in error output,
2 - include all Trace labels in the error output
/vcBrackets:<b>
bracket odd-charactered identifier names with |'s. <b> is:
0 - no (default),
1 - yes
/proverDll:<tp>
use theorem prover <tp>, where <tp> is either the name of
a DLL containing the prover interface located in the
Boogie directory, or a full path to a DLL containing such
an interface. The default interface shipped is:
SMTLib (uses the SMTLib2 format and calls an SMT solver)
/proverOpt:KEY[=VALUE]
Provide a prover-specific option (short form /p).
/proverHelp Print prover-specific options supported by /proverOpt.
/proverLog:<file>
Log input for the theorem prover. Like filenames
supplied as arguments to other options, <file> can use the
following macros:
@TIME@ expands to the current time
@PREFIX@ expands to the concatenation of strings given
by /logPrefix options
@FILE@ expands to the last filename specified on the
command line
In addition, /proverLog can also use the macro '@PROC@',
which causes there to be one prover log file per
verification condition, and the macro then expands to the
name of the procedure that the verification condition is for.
/logPrefix:<str>
Defines the expansion of the macro '@PREFIX@', which can
be used in various filenames specified by other options.
/proverLogAppend
Append (not overwrite) the specified prover log file
/proverWarnings
0 (default) - don't print, 1 - print to stdout,
2 - print to stderr
/restartProver
Restart the prover after each query
17. Dafny Grammar
The Dafny grammar has a traditional structure: a scanner tokenizes the textual input into a sequence of tokens; the parser consumes the tokens to produce an AST. The AST is then passed on for name and type resolution and further processing.
Dafny uses the Coco/R lexer and parser generator for its lexer and parser
(http://www.ssw.uni-linz.ac.at/Research/Projects/Coco)[@Linz:Coco].
See the Coco/R Reference
manual
for details.
The Dafny input file to Coco/R is the Dafny.atg
file in the source tree.
The grammar is an attributed extended BNF grammar. The attributed adjective indicates that the BNF productions are parameterized by boolean parameters that control variations of the production rules, such as whether a particular alternative is permitted or not. Using such attributes allows combining non-terminals with quite similar production rules, making a simpler, more compact and more readable grammer.
The grammar rules presented here replicate those in the source code, but omit semantic actions, error recovery markers, and conflict resolution syntax. Some uses of the attribute parameters are described informally.
The names of character sets and tokens start with a lower case letter; the names of grammar non-terminals start with an upper-case letter.
17.1. Dafny Syntax
This section gives the definitions of Dafny tokens.
17.1.1. Classes of characters
These definitions define some names as representing subsets of the set of characters. Here,
- double quotes enclose the set of characters constituting the class,
- single quotes enclose a single character (perhaps an escaped representation using
\
), - the binary
+
indicates set union, - binary
-
indicates set difference, and ANY
indicates the set of all (unicode) characters.
letter = "ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz"
digit = "0123456789"
posDigit = "123456789"
posDigitFrom2 = "23456789"
hexdigit = "0123456789ABCDEFabcdef"
special = "'_?"
cr = '\r'
lf = '\n'
tab = '\t'
space = ' '
nondigitIdChar = letter + special
idchar = nondigitIdChar + digit
nonidchar = ANY - idchar
charChar = ANY - '\'' - '\\' - cr - lf
stringChar = ANY - '"' - '\\' - cr - lf
verbatimStringChar = ANY - '"'
A nonidchar
is any character except those that can be used in an identifier.
Here the scanner generator will interpret ANY
as any unicode character.
However, nonidchar
is used only to mark the end of the !in
token;
in this context any character other than whitespace or printable ASCII
will trigger a subsequent scanning or parsing error.
17.1.2. Definitions of tokens
These definitions use
- double-quotes to indicate a verbatim string (with no escaping of characters)
'"'
to indicate a literal double-quote character- vertical bar to indicate alternatives
- square brackets to indicate an optional part
- curly braces to indicate 0-or-more repetitions
- parentheses to indicate grouping
- a
-
sign to indicate set difference: any character sequence matched by the left operand except character sequences matched by the right operand - a sequence of any of the above to indicate concatenation without whitespace
reservedword =
"abstract" | "allocated" | "as" | "assert" | "assume" |
"bool" | "break" | "by" |
"calc" | "case" | "char" | "class" | "codatatype" |
"const" | "constructor" | "continue" |
"datatype" | "decreases" |
"else" | "ensures" | "exists" | "expect" | "export" | "extends" |
"false" | "for" | "forall" | "fresh" | "function" | "ghost" |
"if" | "imap" | "import" | "in" | "include" |
"int" | "invariant" | "is" | "iset" | "iterator" |
"label" | "lemma" | "map" | "match" | "method" |
"modifies" | "modify" | "module" | "multiset" |
"nameonly" | "nat" | "new" | "newtype" | "null" |
"object" | "object?" | "old" | "opaque" | "opened" | "ORDINAL"
"predicate" | "print" | "provides" |
"reads" | "real" | "refines" | "requires" | "return" |
"returns" | "reveal" | "reveals" |
"seq" | "set" | "static" | "string" |
"then" | "this" | "trait" | "true" | "twostate" | "type" |
"unchanged" | "var" | "while" | "witness" |
"yield" | "yields" |
arrayToken | bvToken
arrayToken = "array" [ posDigitFrom2 | posDigit digit { digit }]["?"]
bvToken = "bv" ( 0 | posDigit { digit } )
ident = nondigitIdChar { idchar } - charToken - reservedword
digits = digit {["_"] digit}
hexdigits = "0x" hexdigit {["_"] hexdigit}
decimaldigits = digit {["_"] digit} '.' digit {["_"] digit}
escapedChar =
( "\'" | "\"" | "\\" | "\0" | "\n" | "\r" | "\t"
| "\u" hexdigit hexdigit hexdigit hexdigit
| "\U{" hexdigit { hexdigit } "}"
)
charToken = "'" ( charChar | escapedChar ) "'"
stringToken =
'"' { stringChar | escapedChar } '"'
| "@" '"' { verbatimStringChar | '"' '"' } '"'
ellipsis = "..."
There are a few words that have a special meaning in certain contexts, but are not reserved words and can be used as identifiers outside of those contexts:
least
andgreatest
are recognized as adjectives to the keywordpredicate
(cf. Section 12.4).older
is a modifier for parameters of non-extreme predicates (cf. Section 6.4.6).
The \uXXXX
form of an escapedChar
is only used when the option --unicode-char=false
is set (which is the default for Dafny 3.x);
the \U{XXXXXX}
form of an escapedChar
is only used when the option --unicode-char=true
is set (which is the default for Dafny 4.x).
17.2. Dafny Grammar productions
The grammar productions are presented in the following Extended BNF syntax:
- identifiers starting with a lower case letter denote terminal symbols (tokens) as defined in the previous subsection
- identifiers starting with an upper case letter denote nonterminal symbols
- strings (a sequence of characters enclosed by double quote characters) denote the sequence of enclosed characters
=
separates the sides of a production, e.g.A = a b c
|
separates alternatives, e.g.a b | c | d e
meansa b
orc
ord e
(
)
groups alternatives, e.g.(a | b) c
meansa c
orb c
[ ]
option, e.g.[a] b
meansa b
orb
{ }
iteration (0 or more times), e.g.{a} b
meansb
ora b
ora a b
or …- We allow
|
inside[ ]
and{ }
. So[a | b]
is short for[(a | b)]
and{a | b}
is short for{(a | b)}
. //
in a line introduces a comment that extends to the end-of-the line, but does not terminate the production- The first production defines the name of the grammar, in this case
Dafny
.
In addition to the Coco rules, for the sake of readability we have adopted these additional conventions.
- We allow
-
to be used.a - b
means it matches if it matchesa
but notb
. - We omit the
.
that marks the end of a CoCo/R production. - we omit deprecated features.
To aid in explaining the grammar we have added some additional productions that are not present in the original grammar. We name these with a trailing underscore. Inlining these where they are referenced will reconstruct the original grammar.
17.2.1. Programs
Dafny = { IncludeDirective_ } { TopDecl(isTopLevel:true, isAbstract: false) } EOF
17.2.1.1. Include directives
IncludeDirective_ = "include" stringToken
17.2.1.2. Top-level declarations
TopDecl(isTopLevel, isAbstract) =
{ DeclModifier }
( SubModuleDecl(isTopLevel)
| ClassDecl
| DatatypeDecl
| NewtypeDecl
| SynonymTypeDecl // includes abstract types
| IteratorDecl
| TraitDecl
| ClassMemberDecl(allowConstructors: false, isValueType: true, moduleLevelDecl: true)
)
17.2.1.3. Declaration modifiers
DeclModifier = ( "abstract" | "ghost" | "static" | "opaque" )
17.2.2. Modules
SubModuleDecl(isTopLevel) = ( ModuleDefinition | ModuleImport | ModuleExport )
Module export declarations are not permitted if isTopLevel
is true.
17.2.2.1. Module Definitions
ModuleDefinition(isTopLevel) =
"module" { Attribute } ModuleQualifiedName
[ "refines" ModuleQualifiedName ]
"{" { TopDecl(isTopLevel:false, isAbstract) } "}"
The isAbstract
argument is true if the preceding DeclModifiers
include “abstract”.
17.2.2.2. Module Imports
ModuleImport =
"import"
[ "opened" ]
( QualifiedModuleExport
| ModuleName "=" QualifiedModuleExport
| ModuleName ":" QualifiedModuleExport
)
QualifiedModuleExport =
ModuleQualifiedName [ "`" ModuleExportSuffix ]
ModuleExportSuffix =
( ExportId
| "{" ExportId { "," ExportId } "}"
)
17.2.2.3. Module Export Definitions
ModuleExport =
"export"
[ ExportId ]
[ "..." ]
{
"extends" ExportId { "," ExportId }
| "provides" ( ExportSignature { "," ExportSignature } | "*" )
| "reveals" ( ExportSignature { "," ExportSignature } | "*" )
}
ExportSignature = TypeNameOrCtorSuffix [ "." TypeNameOrCtorSuffix ]
17.2.3. Types
Type = DomainType_ | ArrowType_
DomainType_ =
( BoolType_ | CharType_ | IntType_ | RealType_
| OrdinalType_ | BitVectorType_ | ObjectType_
| FiniteSetType_ | InfiniteSetType_
| MultisetType_
| FiniteMapType_ | InfiniteMapType_
| SequenceType_
| NatType_
| StringType_
| ArrayType_
| TupleType
| NamedType
)
NamedType = NameSegmentForTypeName { "." NameSegmentForTypeName }
NameSegmentForTypeName = Ident [ GenericInstantiation ]
17.2.3.1. Basic types
BoolType_ = "bool"
IntType_ = "int"
RealType_ = "real"
BitVectorType_ = bvToken
OrdinalType_ = "ORDINAL"
CharType_ = "char"
17.2.3.2. Generic instantiation
GenericInstantiation = "<" Type { "," Type } ">"
17.2.3.3. Type parameter
GenericParameters(allowVariance) =
"<" [ Variance ] TypeVariableName { TypeParameterCharacteristics }
{ "," [ Variance ] TypeVariableName { TypeParameterCharacteristics } }
">"
// The optional Variance indicator is permitted only if allowVariance is true
Variance = ( "*" | "+" | "!" | "-" )
TypeParameterCharacteristics = "(" TPCharOption { "," TPCharOption } ")"
TPCharOption = ( "==" | "0" | "00" | "!" "new" )
17.2.3.4. Collection types
FiniteSetType_ = "set" [ GenericInstantiation ]
InfiniteSetType_ = "iset" [ GenericInstantiation ]
MultisetType_ = "multiset" [ GenericInstantiation ]
SequenceType_ = "seq" [ GenericInstantiation ]
StringType_ = "string"
FiniteMapType_ = "map" [ GenericInstantiation ]
InfiniteMapType_ = "imap" [ GenericInstantiation ]
17.2.3.5. Type definitions
SynonymTypeDecl =
SynonymTypeDecl_ | OpaqueTypeDecl_ | SubsetTypeDecl_
SynonymTypeName = NoUSIdent
SynonymTypeDecl_ =
"type" { Attribute } SynonymTypeName
{ TypeParameterCharacteristics }
[ GenericParameters ]
"=" Type
OpaqueTypeDecl_ =
"type" { Attribute } SynonymTypeName
{ TypeParameterCharacteristics }
[ GenericParameters ]
[ TypeMembers ]
TypeMembers =
"{"
{
{ DeclModifier }
ClassMemberDecl(allowConstructors: false,
isValueType: true,
moduleLevelDecl: false,
isWithinAbstractModule: module.IsAbstract)
}
"}"
SubsetTypeDecl_ =
"type"
{ Attribute }
SynonymTypeName [ GenericParameters ]
"="
LocalIdentTypeOptional
"|"
Expression(allowLemma: false, allowLambda: true)
[ "ghost" "witness" Expression(allowLemma: false, allowLambda: true)
| "witness" Expression((allowLemma: false, allowLambda: true)
| "witness" "*"
]
NatType_ = "nat"
NewtypeDecl = "newtype" { Attribute } NewtypeName "="
[ ellipsis ]
( LocalIdentTypeOptional
"|"
Expression(allowLemma: false, allowLambda: true)
[ "ghost" "witness" Expression(allowLemma: false, allowLambda: true)
| "witness" Expression((allowLemma: false, allowLambda: true)
| "witness" "*"
]
| Type
)
[ TypeMembers ]
17.2.3.6. Class type
ClassDecl = "class" { Attribute } ClassName [ GenericParameters ]
["extends" Type {"," Type} | ellipsis ]
"{" { { DeclModifier }
ClassMemberDecl(modifiers,
allowConstructors: true,
isValueType: false,
moduleLevelDecl: false)
}
"}"
ClassMemberDecl(modifiers, allowConstructors, isValueType, moduleLevelDecl) =
( FieldDecl(isValueType) // allowed iff moduleLevelDecl is false
| ConstantFieldDecl(moduleLevelDecl)
| FunctionDecl(isWithinAbstractModule)
| MethodDecl(modifiers, allowConstructors)
)
17.2.3.7. Trait types
TraitDecl =
"trait" { Attribute } ClassName [ GenericParameters ]
[ "extends" Type { "," Type } | ellipsis ]
"{"
{ { DeclModifier } ClassMemberDecl(allowConstructors: true,
isValueType: false,
moduleLevelDecl: false,
isWithinAbstractModule: false) }
"}"
17.2.3.8. Object type
ObjectType_ = "object" | "object?"
17.2.3.9. Array types
ArrayType_ = arrayToken [ GenericInstantiation ]
17.2.3.10. Iterator types
IteratorDecl = "iterator" { Attribute } IteratorName
( [ GenericParameters ]
Formals(allowGhostKeyword: true, allowNewKeyword: false,
allowOlderKeyword: false)
[ "yields" Formals(allowGhostKeyword: true, allowNewKeyword: false,
allowOlderKeyword: false) ]
| ellipsis
)
IteratorSpec
[ BlockStmt ]
17.2.3.11. Arrow types
ArrowType_ = ( DomainType_ "~>" Type
| DomainType_ "-->" Type
| DomainType_ "->" Type
)
17.2.3.12. Algebraic datatypes
DatatypeDecl =
( "datatype" | "codatatype" )
{ Attribute }
DatatypeName [ GenericParameters ]
"="
[ ellipsis ]
[ "|" ] DatatypeMemberDecl
{ "|" DatatypeMemberDecl }
[ TypeMembers ]
DatatypeMemberDecl =
{ Attribute } DatatypeMemberName [ FormalsOptionalIds ]
17.2.4. Type member declarations
17.2.4.1. Fields
FieldDecl(isValueType) =
"var" { Attribute } FIdentType { "," FIdentType }
A FieldDecl
is not permitted if isValueType
is true.
17.2.4.2. Constant fields
ConstantFieldDecl(moduleLevelDecl) =
"const" { Attribute } CIdentType [ ellipsis ]
[ ":=" Expression(allowLemma: false, allowLambda:true) ]
If moduleLevelDecl
is true, then the static
modifier is not permitted
(the constant field is static implicitly).
17.2.4.3. Method declarations
MethodDecl(isGhost, allowConstructors, isWithinAbstractModule) =
MethodKeyword_ { Attribute } [ MethodFunctionName ]
( MethodSignature_(isGhost, isExtreme: true iff this is a least
or greatest lemma declaration)
| ellipsis
)
MethodSpec(isConstructor: true iff this is a constructor declaration)
[ BlockStmt ]
MethodKeyword_ = ( "method"
| "constructor"
| "lemma"
| "twostate" "lemma"
| "least" "lemma"
| "greatest" "lemma"
)
MethodSignature_(isGhost, isExtreme) =
[ GenericParameters ]
[ KType ] // permitted only if isExtreme == true
Formals(allowGhostKeyword: !isGhost, allowNewKeyword: isTwostateLemma,
allowOlderKeyword: false, allowDefault: true)
[ "returns" Formals(allowGhostKeyword: !isGhost, allowNewKeyword: false,
allowOlderKeyword: false, allowDefault: false) ]
KType = "[" ( "nat" | "ORDINAL" ) "]"
Formals(allowGhostKeyword, allowNewKeyword, allowOlderKeyword, allowDefault) =
"(" [ GIdentType(allowGhostKeyword, allowNewKeyword, allowOlderKeyword,
allowNameOnlyKeyword: true, allowDefault)
{ "," GIdentType(allowGhostKeyword, allowNewKeyword, allowOlderKeyword,
allowNameOnlyKeyword: true, allowDefault) }
]
")"
If isWithinAbstractModule
is false, then the method must have
a body for the program that contains the declaration to be compiled.
The KType
may be specified only for least and greatest lemmas.
17.2.4.4. Function declarations
FunctionDecl(isWithinAbstractModule) =
( [ "twostate" ] "function" [ "method" ] { Attribute }
MethodFunctionName
FunctionSignatureOrEllipsis_(allowGhostKeyword:
("method" present),
allowNewKeyword:
"twostate" present)
| "predicate" [ "method" ] { Attribute }
MethodFunctionName
PredicateSignatureOrEllipsis_(allowGhostKeyword:
("method" present),
allowNewKeyword:
"twostate" present,
allowOlderKeyword: true)
| ( "least" | "greatest" ) "predicate" { Attribute }
MethodFunctionName
PredicateSignatureOrEllipsis_(allowGhostKeyword: false,
allowNewKeyword: "twostate" present,
allowOlderKeyword: false))
)
FunctionSpec
[ FunctionBody ]
FunctionSignatureOrEllipsis_(allowGhostKeyword) =
FunctionSignature_(allowGhostKeyword) | ellipsis
FunctionSignature_(allowGhostKeyword, allowNewKeyword) =
[ GenericParameters ]
Formals(allowGhostKeyword, allowNewKeyword, allowOlderKeyword: true,
allowDefault: true)
":"
( Type
| "(" GIdentType(allowGhostKeyword: false,
allowNewKeyword: false,
allowOlderKeyword: false,
allowNameOnlyKeyword: false,
allowDefault: false)
")"
)
PredicateSignatureOrEllipsis_(allowGhostKeyword, allowNewKeyword,
allowOlderKeyword) =
PredicateSignature_(allowGhostKeyword, allowNewKeyword, allowOlderKeyword)
| ellipsis
PredicateSignature_(allowGhostKeyword, allowNewKeyword, allowOlderKeyword) =
[ GenericParameters ]
[ KType ]
Formals(allowGhostKeyword, allowNewKeyword, allowOlderKeyword,
allowDefault: true)
[
":"
( Type
| "(" Ident ":" "bool" ")"
)
]
FunctionBody = "{" Expression(allowLemma: true, allowLambda: true)
"}" [ "by" "method" BlockStmt ]
17.2.5. Specifications
17.2.5.1. Method specifications
MethodSpec =
{ ModifiesClause(allowLambda: false)
| RequiresClause(allowLabel: true)
| EnsuresClause(allowLambda: false)
| DecreasesClause(allowWildcard: true, allowLambda: false)
}
17.2.5.2. Function specifications
FunctionSpec =
{ RequiresClause(allowLabel: true)
| ReadsClause(allowLemma: false, allowLambda: false, allowWild: true)
| EnsuresClause(allowLambda: false)
| DecreasesClause(allowWildcard: false, allowLambda: false)
}
17.2.5.3. Lambda function specifications
LambdaSpec =
{ ReadsClause(allowLemma: true, allowLambda: false, allowWild: true)
| "requires" Expression(allowLemma: false, allowLambda: false)
}
17.2.5.4. Iterator specifications
IteratorSpec =
{ ReadsClause(allowLemma: false, allowLambda: false,
allowWild: false)
| ModifiesClause(allowLambda: false)
| [ "yield" ] RequiresClause(allowLabel: !isYield)
| [ "yield" ] EnsuresClause(allowLambda: false)
| DecreasesClause(allowWildcard: false, allowLambda: false)
}
17.2.5.5. Loop specifications
LoopSpec =
{ InvariantClause_
| DecreasesClause(allowWildcard: true, allowLambda: true)
| ModifiesClause(allowLambda: true)
}
17.2.5.6. Requires clauses
RequiresClause(allowLabel) =
"requires" { Attribute }
[ LabelName ":" ] // Label allowed only if allowLabel is true
Expression(allowLemma: false, allowLambda: false)
17.2.5.7. Ensures clauses
EnsuresClause(allowLambda) =
"ensures" { Attribute } Expression(allowLemma: false, allowLambda)
17.2.5.8. Decreases clauses
DecreasesClause(allowWildcard, allowLambda) =
"decreases" { Attribute } DecreasesList(allowWildcard, allowLambda)
DecreasesList(allowWildcard, allowLambda) =
PossiblyWildExpression(allowLambda, allowWildcard)
{ "," PossiblyWildExpression(allowLambda, allowWildcard) }
PossiblyWildExpression(allowLambda, allowWild) =
( "*" // if allowWild is false, using '*' provokes an error
| Expression(allowLemma: false, allowLambda)
)
17.2.5.9. Modifies clauses
ModifiesClause(allowLambda) =
"modifies" { Attribute }
FrameExpression(allowLemma: false, allowLambda)
{ "," FrameExpression(allowLemma: false, allowLambda) }
17.2.5.10. Invariant clauses
InvariantClause_ =
"invariant" { Attribute }
Expression(allowLemma: false, allowLambda: true)
17.2.5.11. Reads clauses
ReadsClause(allowLemma, allowLambda, allowWild) =
"reads"
{ Attribute }
PossiblyWildFrameExpression(allowLemma, allowLambda, allowWild)
{ "," PossiblyWildFrameExpression(allowLemma, allowLambda, allowWild) }
17.2.5.12. Frame expressions
FrameExpression(allowLemma, allowLambda) =
( Expression(allowLemma, allowLambda) [ FrameField ]
| FrameField
)
FrameField = "`" IdentOrDigits
PossiblyWildFrameExpression(allowLemma, allowLambda, allowWild) =
( "*" // error if !allowWild and '*'
| FrameExpression(allowLemma, allowLambda)
)
17.2.6. Statements
17.2.6.1. Labeled statement
Stmt = { "label" LabelName ":" } NonLabeledStmt
17.2.6.2. Non-Labeled statement
NonLabeledStmt =
( AssertStmt | AssumeStmt | BlockStmt | BreakStmt
| CalcStmt | ExpectStmt | ForallStmt | IfStmt
| MatchStmt | ModifyStmt
| PrintStmt | ReturnStmt | RevealStmt
| UpdateStmt | UpdateFailureStmt
| VarDeclStatement | WhileStmt | ForLoopStmt | YieldStmt
)
17.2.6.3. Break and continue statements
BreakStmt =
( "break" LabelName ";"
| "continue" LabelName ";"
| { "break" } "break" ";"
| { "break" } "continue" ";"
)
17.2.6.4. Block statement
BlockStmt = "{" { Stmt } "}"
17.2.6.5. Return statement
ReturnStmt = "return" [ Rhs { "," Rhs } ] ";"
17.2.6.6. Yield statement
YieldStmt = "yield" [ Rhs { "," Rhs } ] ";"
17.2.6.7. Update and call statement
UpdateStmt =
Lhs
( {Attribute} ";"
|
{ "," Lhs }
( ":=" Rhs { "," Rhs }
| ":|" [ "assume" ]
Expression(allowLemma: false, allowLambda: true)
)
";"
)
17.2.6.8. Update with failure statement
UpdateFailureStmt =
[ Lhs { "," Lhs } ]
":-"
[ "expect" | "assert" | "assume" ]
Expression(allowLemma: false, allowLambda: false)
{ "," Rhs }
";"
17.2.6.9. Variable declaration statement
VarDeclStatement =
[ "ghost" ] "var" { Attribute }
(
LocalIdentTypeOptional
{ "," { Attribute } LocalIdentTypeOptional }
[ ":="
Rhs { "," Rhs }
| ":-"
[ "expect" | "assert" | "assume" ]
Expression(allowLemma: false, allowLambda: false)
{ "," Rhs }
| { Attribute }
":|"
[ "assume" ] Expression(allowLemma: false, allowLambda: true)
]
|
CasePatternLocal
( ":=" | { Attribute } ":|" )
Expression(allowLemma: false, allowLambda: true)
)
";"
CasePatternLocal =
( [ Ident ] "(" CasePatternLocal { "," CasePatternLocal } ")"
| LocalIdentTypeOptional
)
17.2.6.10. Guards
Guard = ( "*"
| "(" "*" ")"
| Expression(allowLemma: true, allowLambda: true)
)
17.2.6.11. Binding guards
BindingGuard(allowLambda) =
IdentTypeOptional { "," IdentTypeOptional }
{ Attribute }
":|"
Expression(allowLemma: true, allowLambda)
17.2.6.12. If statement
IfStmt = "if"
( AlternativeBlock(allowBindingGuards: true)
|
( BindingGuard(allowLambda: true)
| Guard
)
BlockStmt [ "else" ( IfStmt | BlockStmt ) ]
)
AlternativeBlock(allowBindingGuards) =
( { AlternativeBlockCase(allowBindingGuards) }
| "{" { AlternativeBlockCase(allowBindingGuards) } "}"
)
AlternativeBlockCase(allowBindingGuards) =
{ "case"
(
BindingGuard(allowLambda: false) //permitted iff allowBindingGuards == true
| Expression(allowLemma: true, allowLambda: false)
) "=>" { Stmt }
}
17.2.6.13. While Statement
WhileStmt =
"while"
( LoopSpec
AlternativeBlock(allowBindingGuards: false)
| Guard
LoopSpec
( BlockStmt
| // no body
)
)
17.2.6.14. For statement
ForLoopStmt =
"for" IdentTypeOptional ":="
Expression(allowLemma: false, allowLambda: false)
( "to" | "downto" )
( "*" | Expression(allowLemma: false, allowLambda: false)
)
LoopSpec
( BlockStmt
| // no body
)
17.2.6.15. Match statement
MatchStmt =
"match"
Expression(allowLemma: true, allowLambda: true)
( "{" { CaseStmt } "}"
| { CaseStmt }
)
CaseStmt = "case" ExtendedPattern "=>" { Stmt }
17.2.6.16. Assert statement
AssertStmt =
"assert"
{ Attribute }
[ LabelName ":" ]
Expression(allowLemma: false, allowLambda: true)
( ";"
| "by" BlockStmt
)
17.2.6.17. Assume statement
AssumeStmt =
"assume"
{ Attribute }
Expression(allowLemma: false, allowLambda: true)
";"
17.2.6.18. Expect statement
ExpectStmt =
"expect"
{ Attribute }
Expression(allowLemma: false, allowLambda: true)
[ "," Expression(allowLemma: false, allowLambda: true) ]
";"
17.2.6.19. Print statement
PrintStmt =
"print"
Expression(allowLemma: false, allowLambda: true)
{ "," Expression(allowLemma: false, allowLambda: true) }
";"
17.2.6.20. Reveal statement
RevealStmt =
"reveal"
Expression(allowLemma: false, allowLambda: true)
{ "," Expression(allowLemma: false, allowLambda: true) }
";"
17.2.6.21. Forall statement
ForallStmt =
"forall"
( "(" [ QuantifierDomain ] ")"
| [ QuantifierDomain ]
)
{ EnsuresClause(allowLambda: true) }
[ BlockStmt ]
17.2.6.22. Modify statement
ModifyStmt =
"modify"
{ Attribute }
FrameExpression(allowLemma: false, allowLambda: true)
{ "," FrameExpression(allowLemma: false, allowLambda: true) }
";"
17.2.6.23. Calc statement
CalcStmt = "calc" { Attribute } [ CalcOp ] "{" CalcBody_ "}"
CalcBody_ = { CalcLine_ [ CalcOp ] Hints_ }
CalcLine_ = Expression(allowLemma: false, allowLambda: true) ";"
Hints_ = { ( BlockStmt | CalcStmt ) }
CalcOp =
( "==" [ "#" "["
Expression(allowLemma: true, allowLambda: true) "]" ]
| "<" | ">"
| "!=" | "<=" | ">="
| "<==>" | "==>" | "<=="
)
17.2.7. Expressions
17.2.7.1. Top-level expression
Expression(allowLemma, allowLambda, allowBitwiseOps = true) =
EquivExpression(allowLemma, allowLambda, allowBitwiseOps)
[ ";" Expression(allowLemma, allowLambda, allowBitwiseOps) ]
The “allowLemma” argument says whether or not the expression
to be parsed is allowed to have the form S;E
where S
is a call to a lemma.
“allowLemma” should be passed in as “false” whenever the expression to
be parsed sits in a context that itself is terminated by a semi-colon.
The “allowLambda” says whether or not the expression to be parsed is
allowed to be a lambda expression. More precisely, an identifier or
parenthesized, comma-delimited list of identifiers is allowed to
continue as a lambda expression (that is, continue with a reads
, requires
,
or =>
) only if “allowLambda” is true. This affects function/method/iterator
specifications, if/while statements with guarded alternatives, and expressions
in the specification of a lambda expression itself.
17.2.7.2. Equivalence expression
EquivExpression(allowLemma, allowLambda, allowBitwiseOps) =
ImpliesExpliesExpression(allowLemma, allowLambda, allowBitwiseOps)
{ "<==>" ImpliesExpliesExpression(allowLemma, allowLambda, allowBitwiseOps) }
17.2.7.3. Implies expression
ImpliesExpliesExpression(allowLemma, allowLambda, allowBitwiseOps) =
LogicalExpression(allowLemma, allowLambda)
[ ( "==>" ImpliesExpression(allowLemma, allowLambda, allowBitwiseOps)
| "<==" LogicalExpression(allowLemma, allowLambda, allowBitwiseOps)
{ "<==" LogicalExpression(allowLemma, allowLambda, allowBitwiseOps) }
)
]
ImpliesExpression(allowLemma, allowLambda, allowBitwiseOps) =
LogicalExpression(allowLemma, allowLambda, allowBitwiseOps)
[ "==>" ImpliesExpression(allowLemma, allowLambda, allowBitwiseOps) ]
17.2.7.4. Logical expression
LogicalExpression(allowLemma, allowLambda, allowBitwiseOps) =
[ "&&" | "||" ]
RelationalExpression(allowLemma, allowLambda, allowBitwiseOps)
{ ( "&&" | "||" )
RelationalExpression(allowLemma, allowLambda, allowBitwiseOps)
}
17.2.7.5. Relational expression
RelationalExpression(allowLemma, allowLambda, allowBitwiseOps) =
ShiftTerm(allowLemma, allowLambda, allowBitwiseOps)
{ RelOp ShiftTerm(allowLemma, allowLambda, allowBitwiseOps) }
RelOp =
( "=="
[ "#" "[" Expression(allowLemma: true, allowLambda: true) "]" ]
| "!="
[ "#" "[" Expression(allowLemma: true, allowLambda: true) "]" ]
| "<" | ">" | "<=" | ">="
| "in"
| "!in"
| "!!"
)
17.2.7.6. Bit-shift expression
ShiftTerm(allowLemma, allowLambda, allowBitwiseOps) =
Term(allowLemma, allowLambda, allowBitwiseOps)
{ ShiftOp Term(allowLemma, allowLambda, allowBitwiseOps) }
ShiftOp = ( "<<" | ">>" )
17.2.7.7. Term (addition operations)
Term(allowLemma, allowLambda, allowBitwiseOps) =
Factor(allowLemma, allowLambda, allowBitwiseOps)
{ AddOp Factor(allowLemma, allowLambda, allowBitwiseOps) }
AddOp = ( "+" | "-" )
17.2.7.8. Factor (multiplication operations)
Factor(allowLemma, allowLambda, allowBitwiseOps) =
BitvectorFactor(allowLemma, allowLambda, allowBitwiseOps)
{ MulOp BitvectorFactor(allowLemma, allowLambda, allowBitwiseOps) }
MulOp = ( "*" | "/" | "%" )
17.2.7.9. Bit-vector expression
BitvectorFactor(allowLemma, allowLambda, allowBitwiseOps) =
AsExpression(allowLemma, allowLambda, allowBitwiseOps)
{ BVOp AsExpression(allowLemma, allowLambda, allowBitwiseOps) }
BVOp = ( "|" | "&" | "^" )
If allowBitwiseOps
is false, it is an error to have a bitvector operation.
17.2.7.10. As/Is expression
AsExpression(allowLemma, allowLambda, allowBitwiseOps) =
UnaryExpression(allowLemma, allowLambda, allowBitwiseOps)
{ ( "as" | "is" ) Type }
17.2.7.11. Unary expression
UnaryExpression(allowLemma, allowLambda, allowBitwiseOps) =
( "-" UnaryExpression(allowLemma, allowLambda, allowBitwiseOps)
| "!" UnaryExpression(allowLemma, allowLambda, allowBitwiseOps)
| PrimaryExpression(allowLemma, allowLambda, allowBitwiseOps)
)
17.2.7.12. Primary expression
PrimaryExpression(allowLemma, allowLambda, allowBitwiseOps) =
( NameSegment { Suffix }
| LambdaExpression(allowLemma, allowBitwiseOps)
| MapDisplayExpr { Suffix }
| SeqDisplayExpr { Suffix }
| SetDisplayExpr { Suffix }
| EndlessExpression(allowLemma, allowLambda, allowBitwiseOps)
| ConstAtomExpression { Suffix }
)
17.2.7.13. Lambda expression
LambdaExpression(allowLemma, allowBitwiseOps) =
( WildIdent
| "(" [ IdentTypeOptional { "," IdentTypeOptional } ] ")"
)
LambdaSpec
"=>"
Expression(allowLemma, allowLambda: true, allowBitwiseOps)
17.2.7.14. Left-hand-side expression #g-lhs-expression}
(discussion) {
Lhs =
( NameSegment { Suffix }
| ConstAtomExpression Suffix { Suffix }
)
17.2.7.15. Right-hand-side expression
Rhs =
ArrayAllocation
| ObjectAllocation_
| Expression(allowLemma: false, allowLambda: true, allowBitwiseOps: true)
| HavocRhs_
)
{ Attribute }
17.2.7.16. Array allocation right-hand-side expression
ArrayAllocation_ =
"new" [ Type ] "[" [ Expressions ] "]"
[ "(" Expression(allowLemma: true, allowLambda: true) ")"
| "[" [ Expressions ] "]"
]
17.2.7.17. Object allocation right-hand-side expression
ObjectAllocation_ = "new" Type [ "." TypeNameOrCtorSuffix ]
[ "(" [ Bindings ] ")" ]
17.2.7.18. Havoc right-hand-side expression
HavocRhs_ = "*"
17.2.7.19. Atomic expressions
ConstAtomExpression =
( LiteralExpression
| ThisExpression_
| FreshExpression_
| AllocatedExpression_
| UnchangedExpression_
| OldExpression_
| CardinalityExpression_
| ParensExpression
)
17.2.7.20. Literal expressions
LiteralExpression =
( "false" | "true" | "null" | Nat | Dec |
charToken | stringToken )
Nat = ( digits | hexdigits )
Dec = decimaldigits
17.2.7.21. This expression
ThisExpression_ = "this"
17.2.7.22. Old and Old@ Expressions
OldExpression_ =
"old" [ "@" LabelName ]
"(" Expression(allowLemma: true, allowLambda: true) ")"
17.2.7.23. Fresh Expressions
FreshExpression_ =
"fresh" [ "@" LabelName ]
"(" Expression(allowLemma: true, allowLambda: true) ")"
17.2.7.24. Allocated Expressions
AllocatedExpression_ =
"allocated" "(" Expression(allowLemma: true, allowLambda: true) ")"
17.2.7.25. Unchanged Expressions
UnchangedExpression_ =
"unchanged" [ "@" LabelName ]
"(" FrameExpression(allowLemma: true, allowLambda: true)
{ "," FrameExpression(allowLemma: true, allowLambda: true) }
")"
17.2.7.26. Cardinality Expressions
CardinalityExpression_ =
"|" Expression(allowLemma: true, allowLambda: true) "|"
17.2.7.27. Parenthesized Expression
ParensExpression =
"(" [ TupleArgs ] ")"
TupleArgs =
[ "ghost" ]
ActualBinding(isGhost) // argument is true iff the ghost modifier is present
{ ","
[ "ghost" ]
ActualBinding(isGhost) // argument is true iff the ghost modifier is present
}
17.2.7.28. Sequence Display Expression
SeqDisplayExpr =
( "[" [ Expressions ] "]"
| "seq" [ GenericInstantiation ]
"(" Expression(allowLemma: true, allowLambda: true)
"," Expression(allowLemma: true, allowLambda: true)
")"
)
17.2.7.29. Set Display Expression
SetDisplayExpr =
( [ "iset" | "multiset" ] "{" [ Expressions ] "}"
| "multiset" "(" Expression(allowLemma: true,
allowLambda: true) ")"
)
17.2.7.30. Map Display Expression
MapDisplayExpr =
("map" | "imap" ) "[" [ MapLiteralExpressions ] "]"
MapLiteralExpressions =
Expression(allowLemma: true, allowLambda: true)
":="
Expression(allowLemma: true, allowLambda: true)
{ ","
Expression(allowLemma: true, allowLambda: true)
":="
Expression(allowLemma: true, allowLambda: true)
}
17.2.7.31. Endless Expression
EndlessExpression(allowLemma, allowLambda, allowBitwiseOps) =
( IfExpression(allowLemma, allowLambda, allowBitwiseOps)
| MatchExpression(allowLemma, allowLambda, allowBitwiseOps)
| QuantifierExpression(allowLemma, allowLambda)
| SetComprehensionExpr(allowLemma, allowLambda, allowBitwiseOps)
| StmtInExpr
Expression(allowLemma, allowLambda, allowBitwiseOps)
| LetExpression(allowLemma, allowLambda, allowBitwiseOps)
| MapComprehensionExpr(allowLemma, allowLambda, allowBitwiseOps)
)
17.2.7.32. If expression
IfExpression(allowLemma, allowLambda, allowBitwiseOps) =
"if" ( BindingGuard(allowLambda: true)
| Expression(allowLemma: true, allowLambda: true, allowBitwiseOps: true)
)
"then" Expression(allowLemma: true, allowLambda: true, allowBitwiseOps: true)
"else" Expression(allowLemma, allowLambda, allowBitwiseOps)
17.2.7.33. Match Expression
MatchExpression(allowLemma, allowLambda, allowBitwiseOps) =
"match"
Expression(allowLemma, allowLambda, allowBitwiseOps)
( "{" { CaseExpression(allowLemma: true, allowLambda, allowBitwiseOps: true) } "}"
| { CaseExpression(allowLemma, allowLambda, allowBitwiseOps) }
)
CaseExpression(allowLemma, allowLambda, allowBitwiseOps) =
"case" { Attribute } ExtendedPattern "=>" Expression(allowLemma, allowLambda, allowBitwiseOps)
17.2.7.34. Case and Extended Patterns
CasePattern =
( IdentTypeOptional
| [Ident] "(" [ CasePattern { "," CasePattern } ] ")"
)
SingleExtendedPattern =
( PossiblyNegatedLiteralExpression
| IdentTypeOptional
| [ Ident ] "(" [ SingleExtendedPattern { "," SingleExtendedPattern } ] ")"
)
ExtendedPattern =
( [ "|" ] SingleExtendedPattern { "|" SingleExtendedPattern } )
PossiblyNegatedLiteralExpression =
( "-" ( Nat | Dec )
| LiteralExpression
)
17.2.7.35. Quantifier expression
QuantifierExpression(allowLemma, allowLambda) =
( "forall" | "exists" ) QuantifierDomain "::"
Expression(allowLemma, allowLambda)
17.2.7.36. Set Comprehension Expressions
SetComprehensionExpr(allowLemma, allowLambda) =
[ "set" | "iset" ]
QuantifierDomain(allowLemma, allowLambda)
[ "::" Expression(allowLemma, allowLambda) ]
17.2.7.37. Map Comprehension Expression
MapComprehensionExpr(allowLemma, allowLambda) =
( "map" | "imap" )
QuantifierDomain(allowLemma, allowLambda)
"::"
Expression(allowLemma, allowLambda)
[ ":=" Expression(allowLemma, allowLambda) ]
17.2.7.38. Statements in an Expression
StmtInExpr = ( AssertStmt | AssumeStmt | ExpectStmt
| RevealStmt | CalcStmt
)
17.2.7.39. Let and Let or Fail Expression
LetExpression(allowLemma, allowLambda) =
(
[ "ghost" ] "var" CasePattern { "," CasePattern }
( ":=" | ":-" | { Attribute } ":|" )
Expression(allowLemma: false, allowLambda: true)
{ "," Expression(allowLemma: false, allowLambda: true) }
|
":-"
Expression(allowLemma: false, allowLambda: true)
)
";"
Expression(allowLemma, allowLambda)
17.2.7.40. Name Segment
NameSegment = Ident [ GenericInstantiation | HashCall ]
17.2.7.41. Hash Call
HashCall = "#" [ GenericInstantiation ]
"[" Expression(allowLemma: true, allowLambda: true) "]"
"(" [ Bindings ] ")"
17.2.7.42. Suffix
Suffix =
( AugmentedDotSuffix_
| DatatypeUpdateSuffix_
| SubsequenceSuffix_
| SlicesByLengthSuffix_
| SequenceUpdateSuffix_
| SelectionSuffix_
| ArgumentListSuffix_
)
17.2.7.43. Augmented Dot Suffix
AugmentedDotSuffix_ = "." DotSuffix
[ GenericInstantiation | HashCall ]
17.2.7.44. Datatype Update Suffix
DatatypeUpdateSuffix_ =
"." "(" MemberBindingUpdate { "," MemberBindingUpdate } ")"
MemberBindingUpdate =
( ident | digits )
":=" Expression(allowLemma: true, allowLambda: true)
17.2.7.45. Subsequence Suffix
SubsequenceSuffix_ =
"[" [ Expression(allowLemma: true, allowLambda: true) ]
".." [ Expression(allowLemma: true, allowLambda: true) ]
"]"
17.2.7.46. Subsequence Slices Suffix
SlicesByLengthSuffix_ =
"[" Expression(allowLemma: true, allowLambda: true) ":"
[
Expression(allowLemma: true, allowLambda: true)
{ ":" Expression(allowLemma: true, allowLambda: true) }
[ ":" ]
]
"]"
17.2.7.47. Sequence Update Suffix
SequenceUpdateSuffix_ =
"[" Expression(allowLemma: true, allowLambda: true)
":=" Expression(allowLemma: true, allowLambda: true)
"]"
17.2.7.48. Selection Suffix
SelectionSuffix_ =
"[" Expression(allowLemma: true, allowLambda: true)
{ "," Expression(allowLemma: true, allowLambda: true) }
"]"
17.2.7.49. Argument List Suffix
ArgumentListSuffix_ = "(" [ Expressions ] ")"
17.2.7.50. Expression Lists
Expressions =
Expression(allowLemma: true, allowLambda: true)
{ "," Expression(allowLemma: true, allowLambda: true) }
17.2.7.51. Parameter Bindings
ActualBindings =
ActualBinding
{ "," ActualBinding }
ActualBinding(isGhost = false) =
[ NoUSIdentOrDigits ":=" ]
Expression(allowLemma: true, allowLambda: true)
17.2.7.52. Quantifier domains
QuantifierDomain(allowLemma, allowLambda) =
QuantifierVarDecl(allowLemma, allowLambda)
{ "," QuantifierVarDecl(allowLemma, allowLambda) }
QuantifierVarDecl(allowLemma, allowLambda) =
IdentTypeOptional
[ "<-" Expression(allowLemma, allowLambda) ]
{ Attribute }
[ | Expression(allowLemma, allowLambda) ]
17.2.7.53. Basic name and type combinations
Ident = ident
DotSuffix = ( ident | digits | "requires" | "reads" )
NoUSIdent = ident - "_" { idchar }
WildIdent = NoUSIdent | "_"
IdentOrDigits = Ident | digits
NoUSIdentOrDigits = NoUSIdent | digits
ModuleName = NoUSIdent
ClassName = NoUSIdent // also traits
DatatypeName = NoUSIdent
DatatypeMemberName = NoUSIdentOrDigits
NewtypeName = NoUSIdent
SynonymTypeName = NoUSIdent
IteratorName = NoUSIdent
TypeVariableName = NoUSIdent
MethodFunctionName = NoUSIdentOrDigits
LabelName = NoUSIdentOrDigits
AttributeName = NoUSIdent
ExportId = NoUSIdentOrDigits
TypeNameOrCtorSuffix = NoUSIdentOrDigits
ModuleQualifiedName = ModuleName { "." ModuleName }
IdentType = WildIdent ":" Type
FIdentType = NoUSIdentOrDigits ":" Type
CIdentType = NoUSIdentOrDigits [ ":" Type ]
GIdentType(allowGhostKeyword, allowNewKeyword, allowOlderKeyword, allowNameOnlyKeyword, allowDefault) =
{ "ghost" | "new" | "nameonly" | "older" } IdentType
[ ":=" Expression(allowLemma: true, allowLambda: true) ]
LocalIdentTypeOptional = WildIdent [ ":" Type ]
IdentTypeOptional = WildIdent [ ":" Type ]
TypeIdentOptional =
{ "ghost" | "nameonly" } [ NoUSIdentOrDigits ":" ] Type
[ ":=" Expression(allowLemma: true, allowLambda: true) ]
FormalsOptionalIds = "(" [ TypeIdentOptional
{ "," TypeIdentOptional } ] ")"
18. Testing syntax rendering
Sample math B: $a \to b$ or
$$ a \to \pi $$
or ( a \top ) or [ a \to \pi ]
Colors
integer literal: 10
hex literal: 0xDEAD
real literal: 1.1
boolean literal: true false
char literal: 'c'
string literal: "abc"
verbatim string: @"abc"
ident: ijk
type: int
generic type: map<int,T>
operator: <=
punctuation: { }
keyword: while
spec: requires
comment: // comment
attribute {: name }
error: $
Syntax color tests:
integer: 0 00 20 01 0_1
float: .0 1.0 1. 0_1.1_0
bad: 0_
hex: 0x10_abcdefABCDEF
string: "string \n \t \r \0" "a\"b" "'" "\'" ""
string: "!@#$%^&*()_-+={}[]|:;\\<>,.?/~`"
string: "\u1234 "
string: " " : "\0\n\r\t\'\"\\"
notstring: "abcde
notstring: "\u123 " : "x\Zz" : "x\ux"
vstring: @"" @"a" @"""" @"'\" @"\u"
vstring: @"xx""y y""zz "
vstring: @" " @" "
vstring: @"x
x"
bad: @!
char: 'a' '\n' '\'' '"' '\"' ' ' '\\'
char: '\0' '\r' '\t' '\u1234'
badchar: $ `
ids: '\u123' '\Z' '\u' '\u2222Z'
ids: '\u123ZZZ' '\u2222Z'
ids: 'a : a' : 'ab' : 'a'b' : 'a''b'
ids: a_b _ab ab? _0
id-label: a@label
literal: true false null
op: - ! ~ x -!~x
op: a + b - c * d / e % f a+b-c*d/e%f
op: <= >= < > == != b&&c || ==> <==> <==
op: !=# !! in !in
op: !in∆ !iné
not op: !inx
punc: . , :: | :| := ( ) [ ] { }
types: int real string char bool nat ORDINAL
types: object object?
types: bv1 bv10 bv0
types: array array2 array20 array10
types: array? array2? array20? array10?
ids: array1 array0 array02 bv02 bv_1
ids: intx natx int0 int_ int? bv1_ bv1x array2x
types: seq<int> set < bool >
types: map<bool,bool> imap < bool , bool >
types: seq<Node> seq< Node >
types: seq<set< real> >
types: map<set<int>,seq<bool>>
types: G<A,int> G<G<A>,G<bool>>
types: seq map imap set iset multiset
ids: seqx mapx
no arg: seq < > seq < , > seq <bool , , bool > seq<bool,>
keywords: if while assert assume
spec: requires reads modifies
attribute: {: MyAttribute "asd", 34 }
attribute: {: MyAttribute }
comment: // comment
comment: /* comment */ after
comment: // comment /* asd */ dfg
comment: /* comment /* embedded */ tail */ after
comment: /* comment // embedded */ after
comment: /* comment
/* inner comment
*/
outer comment
*/ after
more after
19. References
-
The binding power of shift and bit-wise operations is different than in C-like languages. ↩
-
This is likely to change in the future to disallow multiple occurrences of the same key. ↩
-
This is likely to change in the future as follows: The
in
and!in
operations will no longer be supported on maps, withx in m
replaced byx in m.Keys
, and similarly for!in
. ↩ -
Traits are new to Dafny and are likely to evolve for a while. ↩
-
It is possible to conceive of a mechanism for disambiguating conflicting names, but this would add complexity to the language that does not appear to be needed, at least as yet. ↩
-
It would make sense to rename the special fields
_reads
and_modifies
to have the same names as the corresponding keywords,reads
andmodifies
, as is done for function values. Also, the various_decreases\(_i_\)
fields can be combined into one field nameddecreases
whose type is a n-tuple. These changes may be incorporated into a future version of Dafny. ↩ -
To be specific, Dafny has two forms of ↩
-
Note, two places where co-predicates and co-lemmas are not analogous are (a) co-predicates must not make recursive calls to their prefix predicates and (b) co-predicates cannot mention
_k
. ↩ -
The
:-
token is called the elephant symbol or operator. ↩ -
The semantics of
old
in Dafny differs from similar constructs in other specification languages like ACSL or JML. ↩ -
In order to be deterministic, the result of a function should only depend on the arguments and of the objects it reads, and Dafny does not provide a way to explicitly pass the entire heap as the argument to a function. See this post for more insights. ↩
-
This set of operations that are constant-folded may be enlarged in future versions of
dafny
. ↩ -
All entities that Dafny translates to Boogie have their attributes passed on to Boogie except for the
{:axiom}
attribute (which conflicts with Boogie usage) and the{:trigger}
attribute which is instead converted into a Boogie quantifier trigger. See Section 11 of [@Leino:Boogie2-RefMan]. ↩ -
Ghost inference has to be performed after type inference, at least because it is not possible to determine if a member access
a.b
refers to a ghost variable until the type ofa
is determined. ↩ -
Files may be included more than once or both included and listed on the command line. Duplicate inclusions are detected and each file processed only once. ↩
-
Unlike some languages, Dafny does not allow separation of ↩
-
assume false
tells thedafny
verifier “Assume everything is true from this point of the program”. The reason is that, ‘false’ proves anything. For example,false ==> A
is always true because it is equivalent to!false || A
, which reduces totrue || A
, which reduces totrue
. ↩ -
assert P;
. ↩ -
By default, the expression of an assertion or a precondition is added to the knowledge base of the
dafny
verifier for further assertions or postconditions. However, this is not always desirable, because if the verifier has too much knowledge, it might get lost trying to prove something in the wrong direction. ↩ ↩2 ↩3 ↩4 -
dafny
actually breaks things down further. For example, a preconditionrequires A && B
or an assert statementassert A && B;
turns into two assertions, more or less likerequires A requires B
andassert A; assert B;
. ↩ ↩2 -
All the complexities of the execution paths (if-then-else, loops, goto, break….) are, down the road and for verification purposes, cleverly encoded with variables recording the paths and guarding assumptions made on each path. In practice, a second clever encoding of variables enables grouping many assertions together, and recovers which assertion is failing based on the value of variables that the SMT solver returns. ↩
-
The formula sent to the underlying SMT solver is the negation of the formula that the verifier wants to prove - also called a VC or verification condition. Hence, if the SMT solver returns “unsat”, it means that the SMT formula is always false, meaning the verifier’s formula is always true. On the other side, if the SMT solver returns “sat”, it means that the SMT formula can be made true with a special variable assignment, which means that the verifier’s formula is false under that same variable assignment, meaning it’s a counter-example for the verifier. In practice and because of quantifiers, the SMT solver will usually return “unknown” instead of “sat”, but will still provide a variable assignment that it couldn’t prove that it does not make the formula true.
dafny
reports it as a “counter-example” but it might not be a real counter-example, only provide hints about whatdafny
knows. ↩ -
This post gives an overview of how assertions are turned into assumptions for verification purposes. ↩
-
Caveat about assertion and assumption: One big difference between an “assertion transformed in an assumption” and the original “assertion” is that the original “assertion” can unroll functions twice, whereas the “assumed assertion” can unroll them only once. Hence,
dafny
can still continue to analyze assertions after a failing assertion without automatically proving “false” (which would make all further assertions vacuous). ↩ -
To create a smaller batch,
dafny
duplicates the assertion batch, and arbitrarily transforms the clones of an assertion into assumptions except in exactly one batch, so that each assertion is verified only in one batch. This results in “easier” formulas for the verifier because it has less to prove, but it takes more overhead because every verification instance have a common set of axioms and there is no knowledge sharing between instances because they run independently. ↩ -
‘Sequentializing’ a
forall
statement refers to compiling it directly to a series of nested loops with the statement’s body directly inside. The alternative, default compilation strategy is to calculate the quantified variable bindings separately as a collection of tuples, and then execute the statement’s body for each tuple. Not allforall
statements can be sequentialized; See the implementation for details. ↩ -
This refers to an expression such as
['H', 'e', 'l', 'l', 'o']
, as opposed to a string literal such as"Hello"
. ↩ -
This refers to assign-such-that statements with multiple variables, and where at least one variable has potentially infinite bounds. For example, the implementation of the statement
var x: nat, y: nat :| 0 < x && 0 < y && x*x == y*y*y + 1;
needs to avoid the naive approach of iterating all possible values ofx
andy
in a nested loop. ↩ -
Sequence construction expressions often use a direct lambda expression, as in
seq(10, x => x * x)
, but they can also be used with arbitrary function values, as inseq(10, squareFn)
. ↩