formal/hol/Help/BETA_CONV.doc

\DOC BETA_CONV

\TYPE {BETA_CONV : term -> thm}

\SYNOPSIS
Performs a simple beta-conversion.

\KEYWORDS
conversion.

\DESCRIBE
The conversion {BETA_CONV} maps a beta-redex {`(\x.u)v`} to the theorem
{
   |- (\x.u)v = u[v/x]
}
\noindent where {u[v/x]} denotes the result of substituting {v} for all free
occurrences of {x} in {u}, after renaming sufficient bound variables to avoid
variable capture. This conversion is one of the primitive inference rules of
the HOL system.

\FAILURE
{BETA_CONV tm} fails if {tm} is not a beta-redex.

\EXAMPLE
{
  # BETA_CONV `(\x. x + 1) y`;;
  val it : thm = |- (\x. x + 1) y = y + 1

  # BETA_CONV `(\x y. x + y) y`;;
  val it : thm = |- (\x y. x + y) y = (\y'. y + y')
}

\COMMENTS
The HOL Light primitive rule {BETA} is the special case where the argument is
the same as the bound variable. If you know that you are in this case, {BETA}
is significantly more efficient. Though traditionally a primitive, {BETA_CONV}
is actually a derived rule in HOL Light.

\SEEALSO
BETA, BETA_RULE, BETA_TAC, GEN_BETA_CONV, MATCH_CONV.

\ENDDOC