\DOC AP_THM

\TYPE {AP_THM : thm -> term -> thm}

\SYNOPSIS
Proves equality of equal functions applied to a term.

\KEYWORDS
rule.

\DESCRIBE
When applied to a theorem {A |- f = g} and a term {x}, the inference
rule {AP_THM} returns the theorem {A |- f x = g x}.
{
      A |- f = g
   ----------------  AP_THM (A |- f = g) `x`
    A |- f x = g x
}

\FAILURE
Fails unless the conclusion of the theorem is an equation, both sides
of which are functions whose domain type is the same as that of the
supplied term.

\EXAMPLE
{
  # REWRITE_RULE[GSYM FUN_EQ_THM] ADD1;;
  val it : thm = |- SUC = (\m. m + 1)

  # AP_THM it `11`;;
  val it : thm = |- SUC 11 = (\m. m + 1) 11
}

\SEEALSO
AP_TERM, ETA_CONV, MK_COMB.

\ENDDOC