formal/hol/Help/BETA_RULE.doc

\DOC BETA_RULE

\TYPE {BETA_RULE : thm -> thm}

\SYNOPSIS
Beta-reduces all the beta-redexes in the conclusion of a theorem.

\KEYWORDS
rule.

\DESCRIBE
When applied to a theorem {A |- t}, the inference rule {BETA_RULE} beta-reduces
all beta-redexes, at any depth, in the conclusion {t}. Variables are renamed
where necessary to avoid free variable capture.
{
    A |- ....((\x. s1) s2)....
   ----------------------------  BETA_RULE
      A |- ....(s1[s2/x])....
}
\FAILURE
Never fails, but will have no effect if there are no beta-redexes.

\EXAMPLE
The following example is a simple reduction which illustrates variable
renaming:
{
  # let x = ASSUME `f = ((\x y. x + y) y)`;;
  val x : thm = f = (\x y. x + y) y |- f = (\x y. x + y) y

  # BETA_RULE x;;
  val it : thm = f = (\x y. x + y) y |- f = (\y'. y + y')
}
\SEEALSO
BETA_CONV, BETA_TAC, GEN_BETA_CONV.

\ENDDOC