SUBST : (term, thm) subst -> term -> thm -> thm

SYNOPSIS

Makes a set of parallel substitutions in a theorem.

DESCRIBE

Implements the following rule of simultaneous substitution

    A1 |- t1 = u1 ,  ... , An |- tn = un ,    A |- t[t1,...,tn]
   -------------------------------------------------------------
                  A u A1 u ... u An |- t[ui]

Evaluating

   SUBST [x1 |-> (A1 |- t1=u1) ,..., xn |-> (An |- tn=un)]
         t[x1,...,xn]
         (A |- t[t1,...,tn])

returns the theorem A1 u ... An |- t[u1,...,un]. The term argument t[x1,...,xn] is a template which should match the conclusion of the theorem being substituted into, with the variables x1, ... , xn marking those places where occurrences of t1, ... , tn are to be replaced by the terms u1, ... , un, respectively. The occurrence of ti at the places marked by xi must be free (i.e. ti must not contain any bound variables). SUBST automatically renames bound variables to prevent free variables in ui becoming bound after substitution.

SUBST is a complex primitive because it performs both parallel simultaneous substitution and renaming of variables. This is for efficiency reasons, but it would be logically cleaner if SUBST were simpler.

FAILURE

If the template does not match the conclusion of the hypothesis, or the terms in the conclusion marked by the variables x1, ... , xn in the template are not identical to the left hand sides of the supplied equations (i.e. the terms t1, ... , tn).

EXAMPLE

  - val x = --`x:num`--
    and y = --`y:num`--
    and th0 = SPEC (--`0`--) arithmeticTheory.ADD1 
    and th1 = SPEC (--`1`--) arithmeticTheory.ADD1;

(*    x = (--`x`--)
      y = (--`y`--)
    th0 = |- SUC 0 = 0 + 1
    th1 = |- SUC 1 = 1 + 1     *)

  - SUBST [x |-> th0, y |-> th1] (--`(x+y) > SUC 0`--)
        (ASSUME (--`(SUC 0 + SUC 1) > SUC 0`--));

  val it = [.] |- (0 + 1) + 1 + 1 > SUC 0 : thm


  - SUBST [x |-> th0, y |-> th1] (--`(SUC 0 + y) > SUC 0`--)
         (ASSUME (--`(SUC 0 + SUC 1) > SUC 0`--));

  val it = [.] |- SUC 0 + 1 + 1 > SUC 0 : thm


  - SUBST [x |-> th0, y |-> th1] (--`(x+y) > x`--)
         (ASSUME (--`(SUC 0 + SUC 1) > SUC 0`--));

  val it = [.] |- (0 + 1) + 1 + 1 > 0 + 1 : thm

USES

For substituting at selected occurrences. Often useful for writing special purpose derived inference rules.

SEEALSO  SUBS