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RES_TAC                                                       (Tactic)
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RES_TAC : tactic

SYNOPSIS
Enriches assumptions by repeatedly resolving them against each other.

KEYWORDS
tactic, resolution, implication.

DESCRIBE
{RES_TAC} searches for pairs of assumed assumptions of a goal (that is, for a
candidate implication and a candidate antecedent, respectively) which can be
‘resolved’ to yield new results. The conclusions of all the new results are
returned as additional assumptions of the subgoal(s).  The effect of {RES_TAC}
on a goal is to enrich the assumptions set with some of its collective
consequences.

When applied to a goal {A ?- g}, the tactic {RES_TAC} uses {RES_CANON} to
obtain a set of implicative theorems in canonical form from the assumptions {A}
of the goal. Each of the resulting theorems (if there are any) will have the
form:

   A |- u1 ==> u2 ==> ... ==> un ==> v

{RES_TAC} then tries to repeatedly ‘resolve’ these theorems
against the assumptions of a goal by attempting to match the antecedents {u1},
{u2}, ..., {un} (in that order) to some assumption of the goal (i.e. to some
candidate antecedents among the assumptions).  If all the antecedents can be
matched to assumptions of the goal, then an instance of the theorem

   A u {a1,...,an} |- v

called a ‘final resolvent’ is obtained by repeated specialization of
the variables in the implicative theorem, type instantiation, and applications
of modus ponens.  If only the first {i} antecedents {u1}, ..., {ui} can be
matched to assumptions and then no further matching is possible, then the final
resolvent is an instance of the theorem:

   A u {a1,...,ai} |- u(i+1) ==> ... ==> v

All the final resolvents obtained in this way (there may be several,
since an antecedent {ui} may match several assumptions) are added to the
assumptions of the goal, in the stripped form produced by using
{STRIP_ASSUME_TAC}.  If the conclusion of any final resolvent is a
contradiction ‘{F}’ or is alpha-equivalent to the conclusion of the goal, then
{RES_TAC} solves the goal.

FAILURE
{RES_TAC} cannot fail and so should not be unconditionally {REPEAT}ed. However,
since the final resolvents added to the original assumptions are never used as
‘candidate antecedents’ it is sometimes necessary to apply {RES_TAC} more than
once to derive the desired result.

SEEALSO
Tactic.IMP_RES_TAC, Thm_cont.IMP_RES_THEN, Drule.RES_CANON,
Thm_cont.RES_THEN.

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