REAL_ARITH_TAC : tactic

SYNOPSIS

Attempt to prove goal using basic algebra and linear arithmetic over the reals.

DESCRIPTION

The tactic REAL_ARITH_TAC is the tactic form of REAL_ARITH. Roughly speaking, it will automatically prove any formulas over the reals that are effectively universally quantified and can be proved valid by algebraic normalization and linear equational and inequality reasoning. See REAL_ARITH for more information about the algorithm used and its scope.

FAILURE CONDITIONS

Fails if the goal is not in the subset solvable by these means, or is not valid.

EXAMPLE

Here is a goal that holds by virtue of pure algebraic normalization:

  # g `(x1 pow 2 + x2 pow 2 + x3 pow 2 + x4 pow 2) pow 2 =
       ((x1 + x2) pow 4 + (x1 + x3) pow 4 + (x1 + x4) pow 4 +
        (x2 + x3) pow 4 + (x2 + x4) pow 4 + (x3 + x4) pow 4 +
        (x1 - x2) pow 4 + (x1 - x3) pow 4 + (x1 - x4) pow 4 +
        (x2 - x3) pow 4 + (x2 - x4) pow 4 + (x3 - x4) pow 4) / &6`;;

and here is one that holds by linear inequality reasoning:

  # g `&26 < x / &2 ==> abs(x / &4 + &1) < abs(x / &3)`;;

so either goal is solved simply by:

  # e REAL_ARITH_TAC;;
  val it : goalstack = No subgoals

COMMENTS

For nonlinear equational reasoning, use CONV_TAC REAL_RING or CONV_TAC REAL_FIELD. For nonlinear inequality reasoning, there are no powerful rules built into HOL Light, but the additional derived rules defined in Examples/sos.ml and Rqe/make.ml may be useful.