IMP_RES_THEN

Thm_cont.IMP_RES_THEN : thm_tactical

Resolves an implication with the assumptions of a goal.

The function IMP_RES_THEN is the basic building block for resolution in HOL. This is not full higher-order, or even first-order, resolution with unification, but simply one way simultaneous pattern-matching (resulting in term and type instantiation) of the antecedent of an implicative theorem to the conclusion of another theorem (the candidate antecedent).

Given a theorem-tactic ttac and a theorem th, the theorem-tactical IMP_RES_THEN uses RES_CANON to derive a canonical list of implications from th, each of which has the form:

   Ai |- !x1...xn. ui ==> vi

IMP_RES_THEN then produces a tactic that, when applied to a goal A ?- g attempts to match each antecedent ui to each assumption aj |- aj in the assumptions A. If the antecedent ui of any implication matches the conclusion aj of any assumption, then an instance of the theorem Ai u {aj} |- vi, called a ‘resolvent’, is obtained by specialization of the variables x1, …, xn and type instantiation, followed by an application of modus ponens. There may be more than one canonical implication and each implication is tried against every assumption of the goal, so there may be several resolvents (or, indeed, none).

Tactics are produced using the theorem-tactic ttac from all these resolvents (failures of ttac at this stage are filtered out) and these tactics are then applied in an unspecified sequence to the goal. That is,

   IMP_RES_THEN ttac th  (A ?- g)

has the effect of:

   MAP_EVERY (mapfilter ttac [... , (Ai u {aj} |- vi) , ...]) (A ?- g)

where the theorems Ai u {aj} |- vi are all the consequences that can be drawn by a (single) matching modus-ponens inference from the assumptions of the goal A ?- g and the implications derived from the supplied theorem th. The sequence in which the theorems Ai u {aj} |- vi are generated and the corresponding tactics applied is unspecified.

Failure

Evaluating IMP_RES_THEN ttac th fails if the supplied theorem th is not an implication, or if no implications can be derived from th by the transformation process described under the entry for RES_CANON. Evaluating IMP_RES_THEN ttac th (A ?- g) fails if no assumption of the goal A ?- g can be resolved with the implication or implications derived from th. Evaluation also fails if there are resolvents, but for every resolvent Ai u {aj} |- vi evaluating the application ttac (Ai u {aj} |- vi) fails—that is, if for every resolvent ttac fails to produce a tactic. Finally, failure is propagated if any of the tactics that are produced from the resolvents by ttac fails when applied in sequence to the goal.

Example

The following example shows a straightforward use of IMP_RES_THEN to infer an equational consequence of the assumptions of a goal, use it once as a substitution in the conclusion of goal, and then ‘throw it away’. Suppose the goal is:

   a + n = a  ?- !k. k - n = k

By the built-in theorem:

   ADD_INV_0 = |- !m n. (m + n = m) ==> (n = 0)

the assumption of this goal implies that n equals 0. A single-step resolution with this theorem followed by substitution:

   IMP_RES_THEN SUBST1_TAC ADD_INV_0

can therefore be used to reduce the goal to:

   a + n = a  ?- !k. k - 0 = m

Here, a single resolvent a + n = a |- n = 0 is obtained by matching the antecedent of ADD_INV_0 to the assumption of the goal. This is then used to substitute 0 for n in the conclusion of the goal.

See also

Tactic.IMP_RES_TAC, Drule.MATCH_MP, Drule.RES_CANON, Tactic.RES_TAC, Thm_cont.RES_THEN