\DOC ASM_REAL_ARITH_TAC

\TYPE {ASM_REAL_ARITH_TAC : tactic}

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

\DESCRIBE
The tactic {ASM_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. Unlike plain
{REAL_ARITH_TAC}, {ASM_REAL_ARITH_TAC} uses any assumptions that are not
universally quantified as additional hypotheses.

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

\EXAMPLE
This example illustrates how {ASM_REAL_ARITH_TAC} uses assumptions while
{REAL_ARITH_TAC} does not. Of course, this is for illustration only: plain
{REAL_ARITH_TAC} would solve the entire goal before application of {STRIP_TAC}.
{
  # g `!x y z:real. abs(x) <= y ==> abs(x - z) <= abs(y + abs(z))`;;
  val it : goalstack = 1 subgoal (1 total)

  `!x y z. abs x <= y ==> abs (x - z) <= abs (y + abs z)`

  # e(REPEAT STRIP_TAC);;
  val it : goalstack = 1 subgoal (1 total)

    0 [`abs x <= y`]

  `abs (x - z) <= abs (y + abs z)`

  # e REAL_ARITH_TAC;;
  Exception: Failure "linear_ineqs: no contradiction".
  # e ASM_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.

\SEEALSO
ARITH_TAC, INT_ARITH_TAC, REAL_ARITH, REAL_ARITH_TAC, REAL_FIELD, REAL_RING.

\ENDDOC