SUBS

`SUBS`

Drule.SUBS : (thm list -> thm -> thm)

Makes simple term substitutions in a theorem using a given list of theorems.

Term substitution in HOL is performed by replacing free subterms according to the transformations specified by a list of equational theorems. Given a list of theorems A1|-t1=v1,...,An|-tn=vn and a theorem A|-t, SUBS simultaneously replaces each free occurrence of ti in t with vi:

          A1|-t1=v1 ... An|-tn=vn    A|-t
   ---------------------------------------------  SUBS[A1|-t1=v1;...;An|-tn=vn]
    A1 u ... u An u A |- t[v1,...,vn/t1,...,tn]       (A|-t)

No matching is involved; the occurrence of each ti being substituted for must be a free in t (see SUBST_MATCH). An occurrence which is not free can be substituted by using rewriting rules such as REWRITE_RULE, PURE_REWRITE_RULE and ONCE_REWRITE_RULE.

Failure

SUBS [th1,...,thn] (A|-t) fails if the conclusion of each theorem in the list is not an equation. No change is made to the theorem A |- t if no occurrence of any left-hand side of the supplied equations appears in t.

Example

Substitutions are made with the theorems

   - val thm1 = SPECL [Term`m:num`, Term`n:num`] arithmeticTheory.ADD_SYM
     val thm2 = CONJUNCT1 arithmeticTheory.ADD_CLAUSES;
   > val thm1 = |- m + n = n + m : thm
     val thm2 = |- 0 + m = m : thm

depending on the occurrence of free subterms

   - SUBS [thm1, thm2] (ASSUME (Term `(n + 0) + (0 + m) = m + n`));
   > val it =  [.] |- n + 0 + m = n + m : thm

   - SUBS [thm1, thm2] (ASSUME (Term `!n. (n + 0) + (0 + m) = m + n`));
   > val it =  [.] |- !n. n + 0 + m = m + n : thm

SUBS can sometimes be used when rewriting (for example, with REWRITE_RULE) would diverge and term instantiation is not needed. Moreover, applying the substitution rules is often much faster than using the rewriting rules.

Keyboard shortcuts

HOL: Reference

Failure

Example

See also

`SUBS`