formal/hol/Help/ARITH_RULE.doc

\DOC ARITH_RULE

\TYPE {ARITH_RULE : term -> thm}

\SYNOPSIS
Automatically proves natural number arithmetic theorems needing basic
rearrangement and linear inequality reasoning only.

\DESCRIBE
The function {ARITH_RULE} can automatically prove natural number theorems using
basic algebraic normalization and inequality reasoning. For nonlinear
equational reasoning use {NUM_RING}.

\FAILURE
Fails if the term is not boolean or if it cannot be proved using the basic
methods employed, e.g. requiring nonlinear inequality reasoning.

\EXAMPLE
{
  # ARITH_RULE `x = 1 ==> y <= 1 \/ x < y`;;
  val it : thm = |- x = 1 ==> y <= 1 \/ x < y

  # ARITH_RULE `x <= 127 ==> ((86 * x) DIV 256 = x DIV 3)`;;
  val it : thm = |- x <= 127 ==> (86 * x) DIV 256 = x DIV 3

  # ARITH_RULE
     `2 * a * b EXP 2 <= b * a * b ==> (SUC c - SUC(a * b * b) <= c)`;;
  val it : thm =
    |- 2 * a * b EXP 2 <= b * a * b ==> SUC c - SUC (a * b * b) <= c
}

\USES
Disposing of elementary arithmetic goals.

\SEEALSO
ARITH_TAC, INT_ARITH, NUM_RING, REAL_ARITH, REAL_FIELD, REAL_RING.

\ENDDOC