GEN_REWRITE_RULE : ((conv -> conv) -> rewrites -> thm list -> thm -> thm)

- STRUCTURE
- SYNOPSIS
- Rewrites a theorem, selecting terms according to a user-specified strategy.
- DESCRIPTION
- Rewriting in HOL is based on the use of equational theorems as left-to-right replacements on the subterms of an object theorem. This replacement is mediated by the use of REWR_CONV, which finds matches between left-hand sides of given equations in a term and applies the substitution.
Equations used in rewriting are obtained from the theorem lists given as arguments to the function. These are at first transformed into a form suitable for rewriting. Conjunctions are separated into individual rewrites. Theorems with conclusions of the form "~t" are transformed into the corresponding equations "t = F". Theorems "t" which are not equations are cast as equations of form "t = T".

If a theorem is used to rewrite the object theorem, its assumptions are added to the assumptions of the returned theorem, unless they are alpha-convertible to existing assumptions. The matching involved uses variable instantiation. Thus, all free variables are generalized, and terms are instantiated before substitution. Theorems may have universally quantified variables.

The theorems with which rewriting is done are divided into two groups, to facilitate implementing other rewriting tools. However, they are considered in an order-independent fashion. (That is, the ordering is an implementation detail which is not specified.)

The search strategy for finding matching subterms is the first argument to the rule. Matching and substitution may occur at any level of the term, according to the specified search strategy: the whole term, or starting from any subterm. The search strategy also specifies the depth of the search: recursively up to an arbitrary depth until no matches occur, once over the selected subterm, or any more complex scheme.

- FAILURE
- GEN_REWRITE_RULE fails if the search strategy fails. It may also cause a non-terminating sequence of rewrites, depending on the search strategy used.
- USES
- This rule is used in the system to implement all other rewriting rules, and may provide a user with a method to fine-tune rewriting of theorems.
- EXAMPLE
- Suppose we have a theorem of the form:and we would like to rewrite the left-hand side with the theorem ADD_SYM without changing the right hand side. This can be done by using:
thm = |- (1 + 2) + 3 = (3 + 1) + 2

Other rules, such as ONCE_REWRITE_RULE, would match and substitute on both sides, which would not be the desirable result.GEN_REWRITE_RULE (RATOR_CONV o ONCE_DEPTH_CONV) empty_rewrites [ADD_SYM] mythm

As another example, REWRITE_RULE could be implemented as

which specifies that matches should be searched recursively starting from the whole term of the theorem, and implicit_rewrites must be added to the user defined set of theorems employed in rewriting.GEN_REWRITE_RULE TOP_DEPTH_CONV (implicit_rewrites())

- SEEALSO

HOL Kananaskis-14