GEN_REWRITE_TAC : ((conv -> conv) -> rewrites -> thm list -> tactic)

- STRUCTURE
- SYNOPSIS
- Rewrites a goal, selecting terms according to a user-specified strategy.
- DESCRIPTION
- Distinct rewriting tactics differ in the search strategies used in finding subterms on which to apply substitutions, and the built-in theorems used in rewriting. In the case of REWRITE_TAC, this is a recursive traversal starting from the body of the goal’s conclusion part, while in the case of ONCE_REWRITE_TAC, for example, the search stops as soon as a term on which a substitution is possible is found. GEN_REWRITE_TAC allows a user to specify a more complex strategy for rewriting.
The basis of pattern-matching for rewriting is the notion of conversions, through the application of REWR_CONV. Conversions are rules for mapping terms with theorems equating the given terms to other semantically equivalent ones.

When attempting to rewrite subterms recursively, the use of conversions (and therefore rewrites) can be automated further by using functions which take a conversion and search for instances at which they are applicable. Examples of these functions are ONCE_DEPTH_CONV and RAND_CONV. The first argument to GEN_REWRITE_TAC is such a function, which specifies a search strategy; i.e. it specifies how subterms (on which substitutions are allowed) should be searched for.

The second and third arguments are lists of theorems used for rewriting. The order in which these are used is not specified. The theorems need not be in equational form: negated terms, say "~ t", are transformed into the equivalent equational form "t = F", while other non-equational theorems with conclusion of form "t" are cast as the corresponding equations "t = T". Conjunctions are separated into the individual components, which are used as distinct rewrites.

- FAILURE
- GEN_REWRITE_TAC fails if the search strategy fails. It may also cause a non-terminating sequence of rewrites, depending on the search strategy used. The resulting tactic is invalid when a theorem which matches the goal (and which is thus used for rewriting it with) has a hypothesis which is not alpha-convertible to any of the assumptions of the goal. Applying such an invalid tactic may result in a proof of a theorem which does not correspond to the original goal.
- USES
- Detailed control of rewriting strategy, allowing a user to specify a search strategy.
- EXAMPLE
- Given a goal such as:we may want to rewrite only one side of it with a theorem, say ADD_SYM. Rewriting tactics which operate recursively result in divergence; the tactic ONCE_REWRITE_TAC [ADD_SYM] rewrites on both sides to produce the following goal:
?- a - (b + c) = a - (c + b)

as ADD_SYM matches at two positions. To rewrite on only one side of the equation, the following tactic can be used:?- a - (c + b) = a - (b + c)

which produces the desired goal:GEN_REWRITE_TAC (RAND_CONV o ONCE_DEPTH_CONV) empty_rewrites [ADD_SYM]

?- a - (c + b) = a - (c + b)

As another example, one can write a tactic which will behave similarly to REWRITE_TAC but will also include ADD_CLAUSES in the set of theorems to use always:

val ADD_REWRITE_TAC = GEN_REWRITE_TAC TOP_DEPTH_CONV (add_rewrites (implicit_rewrites ()) [ADD_CLAUSES])

- SEEALSO

HOL Kananaskis-14