RES_TAC

Tactic.RES_TAC : tactic

Enriches assumptions by repeatedly resolving them against each other.

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 REPEATed. 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.

See also

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