4.5 Other Relations🔗ℹ

The contract, if present, is matched against every input to the metafunction and, if the match fails, an exception is raised. If a metavariable is repeated in a contract, it does not require the terms to be equal, unless there is an underscore subscript (i.e., the binding works like it does in define-language, not how it works in the patterns in the left-hand sides of the metafunction clauses).

If present, the term inside the maybe-pre-condition is evaluated after a successful match to the input pattern in the contract (with any variables from the input contract bound). If it returns #f, then the input contract is considered to not have matched and an error is also raised. When a metafunction returns, the expression in the maybe-post-condition is evaluated (if present), with any variables from the input or output contract bound.

The side-condition, hidden-side-condition, where, where/hidden, and where/error clauses behave as in the reduction-relation form.

The resulting metafunction raises an exception recognized by exn:fail:redex? if no clauses match or if one of the clauses matches multiple ways (and that leads to different results for the different matches).

The side-condition extra is evaluated after a successful match to the corresponding argument pattern. If it returns #f, the clause is considered not to have matched, and the next one is tried. The side-condition/hidden extra behaves the same, but is not typeset.

The where and where/hidden extra are like side-condition and side-condition/hidden, except the match guards the clause. The where/error extra is like where, except that the pattern must match.

The judgment-holds clause is like side-condition and where, except the given judgment or relation must hold for the clause to be taken.

The clause-name is used only when typesetting. See metafunction-cases.

The or clause is used to define piecewise conditional metafunctions. In particular, if any of the where or side-condition clauses fail, then evaluation continues after an or clause, treating the term that follows as the result (subject to any subsequent where clauses or side-conditions. This construction is equivalent to simply duplicating the left-hand side of the clause, once for each or expression, but signals to the typesetting library to use a large left curly brace to group the conditions in the or.

For example, here are two equivalent definitions of a biggest metafunction that typeset differently:

Note that metafunctions are assumed to always return the same results for the same inputs, and their results are cached, unless caching-enabled? is set to #f. Accordingly, if a metafunction is called with the same inputs twice, then its body is only evaluated a single time.

As an example, these metafunctions finds the free variables in an expression in the lc-lang above:

(define-metafunction lc-lang

free-vars : e -> (x ...)

[(free-vars (e_1 e_2 ...))

(∪ (free-vars e_1) (free-vars e_2) ...)]

[(free-vars x) (x)]

[(free-vars (λ (x ...) e))

(- (free-vars e) (x ...))])

The first argument to define-metafunction is the grammar (defined above). Following that are three cases, one for each variation of expressions (e in lc-lang). The free variables of an application are the free variables of each of the subterms; the free variables of a variable is just the variable itself, and the free variables of a λ expression are the free variables of the body, minus the bound parameters.

Here are the helper metafunctions used above.

(define-metafunction lc-lang

∪ : (x ...) ... -> (x ...)

[(∪ (x_1 ...) (x_2 ...) (x_3 ...) ...)

(∪ (x_1 ... x_2 ...) (x_3 ...) ...)]

[(∪ (x_1 ...))

(x_1 ...)]

[(∪) ()])

(define-metafunction lc-lang

- : (x ...) (x ...) -> (x ...)

[(- (x ...) ()) (x ...)]

[(- (x_1 ... x_2 x_3 ...) (x_2 x_4 ...))

(- (x_1 ... x_3 ...) (x_2 x_4 ...))

(side-condition (not (memq (term x_2) (term (x_3 ...)))))]

[(- (x_1 ...) (x_2 x_3 ...))

(- (x_1 ...) (x_3 ...))])

Note the side-condition in the second case of -. It ensures that there is a unique match for that case. Without it, (term (- (x x) x)) would lead to an ambiguous match.

Changed in version 1.4 of package redex-lib: Added #:post conditions.
Changed in version 1.5: Added or clauses.

For example, define-metafunction/extension may be used to extend the free-vars function above to the forms introduced by the language lc-num-lang.

(define-metafunction/extension free-vars lc-num-lang

free-vars-num : e -> (x ...)

[(free-vars-num number)

()]

[(free-vars-num (+ e_1 e_2))

(∪ (free-vars-num e_1)

(free-vars-num e_2))])

If a mode-spec appears, each rule must be such that its premises can be evaluated left-to-right without “guessing” values for any of their pattern variables. Redex checks this property using mode-spec declaration, which partitions positions into inputs I and outputs O. Output positions in conclusions and input positions in premises must be terms; input positions in conclusions and output positions in premises must be patterns. The rule-names are used by build-derivations and by render-judgment-form.

If a mode-spec is not present, Redex cannot compute a derivation for the judgment form, instead it can check that a given derivation is valid according to the rules.

When the optional contract-spec declaration is present, Redex dynamically checks that the terms flowing through these positions match the provided patterns, raising an exception recognized by exn:fail:redex? if not. The term in the optional invariant-spec is evaluated after the output positions have been computed and the contract has matched successfully, with variables (that have underscores) from the contract bound; a result of #f is considered to be a contract violation and an exception is raised.

The premises must be in the same order in the derivation struct’s subs field as they appear in the definition of the judgment form.

A rule’s side-condition and side-condition/hidden premises are similar to those in reduction-relation and define-metafunction, except that they do not implicitly unquote their right-hand sides. In other words, a premise of the form (side-condition term) is close to the premise (where #t term), except it does not typeset with the “#t = ”, as that would and it holds whenever the expression evaluates to any non #f value (not just #t).

Note that current-traced-metafunctions also traces judgment forms and is helpful when debugging.

The mode specification in this judgment form and the original must be the same.

The mode-spec, contract-spec, invariant-spec, and rules are as in define-judgment-form.

The mode specification in this judgment form and the original must be the same.

In its second form, produces a list of terms by instantiating the supplied term template with each satisfying assignment of pattern variables. In the second case, if a relation is supplied, there are no pattern variables, so the result is either a list with one element or the empty list.

In both of the first two forms, any given judgment form must have a mode.

In its third form, the judgment-form-id must not have a mode, and the derivation-expr must produce a derviation struct. The result of judgment-holds is #t when the derivation is valid, according to the rules of the judgment form, and #f otherwise. Note that the premises of the derivation must appear in the same order as the premises in the definition of the judgment form.

The term field holds an s-expression based rendering of the conclusion of the derivation, the name field holds the name of the clause with term as the conclusion, and subs contains the sub-derivations.

See also build-derivations.

The contract specification for a relation restricts the patterns that can be used as input to a relation. For each argument to the relation, there should be a single pattern, using x or × to separate the argument contracts.

The tracing looks just like the tracing done by the racket/trace library, except that the first column printed by each traced call indicate if this call to the metafunction is cached. Specifically, a c is printed in the first column if the result is just returned from the cache and a space is printed if the metafunction or judgment call is actually performed.

Defaults to '().