formal/hol/Help/ASM_ARITH_TAC.doc

\DOC ASM_ARITH_TAC

\TYPE {ASM_ARITH_TAC : tactic}

\SYNOPSIS
Tactic for proving arithmetic goals needing basic rearrangement and linear
inequality reasoning only, using assumptions

\DESCRIBE
{ASM_ARITH_TAC} will automatically prove goals that require basic algebraic
normalization and inequality reasoning over the natural numbers. For nonlinear
equational reasoning use {NUM_RING} and derivatives. Unlike plain {ARITH_TAC},
{ASM_ARITH_TAC} uses any assumptions that are not universally quantified as
additional hypotheses.

\FAILURE
Fails if the automated methods do not suffice.

\EXAMPLE
This example illustrates how {ASM_ARITH_TAC} uses assumptions while {ARITH_TAC}
does not. Of course, this is for illustration only: plain {ARITH_TAC} would
solve the entire goal before application of {STRIP_TAC}.
{
  # g `1 <= 6 * x /\ 2 * x <= 3 ==> x = 1`;;
  Warning: Free variables in goal: x
  val it : goalstack = 1 subgoal (1 total)

  `1 <= 6 * x /\ 2 * x <= 3 ==> x = 1`

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

    0 [`1 <= 6 * x`]
    1 [`2 * x <= 3`]

  `x = 1`

  # e ARITH_TAC;;
  Exception: Failure "linear_ineqs: no contradiction".
  # e ASM_ARITH_TAC;;
  val it : goalstack = No subgoals
}

\USES
Solving basic arithmetic goals.

\SEEALSO
ARITH_RULE, ARITH_TAC, INT_ARITH_TAC, NUM_RING, REAL_ARITH_TAC.

\ENDDOC