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\DOC RES_THEN
\TYPE {RES_THEN : (thm_tactic -> tactic)}
\SYNOPSIS
Resolves all implicative assumptions against the rest.
\KEYWORDS
theorem-tactic, resolution, assumptions.
\DESCRIBE
Like the basic resolution function {IMP_RES_THEN}, the resolution tactic
{RES_THEN} performs a single-step resolution of an implication and the
assumptions of a goal. {RES_THEN} differs from {IMP_RES_THEN} only in that the
implications used for resolution are taken from the assumptions of the goal
itself, rather than supplied as an argument.
When applied to a goal {A ?- g}, the tactic {RES_THEN ttac} 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:
{
ai |- !x1...xn. ui ==> vi
}
\noindent where {ai} is one of the assumptions of the goal. Having obtained
these implications, {RES_THEN} then 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, 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 derivable from the assumptions of the goal and each
such implication is tried against every assumption, 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,
{
RES_THEN ttac (A ?- g)
}
\noindent has the effect of:
{
MAP_EVERY (mapfilter ttac [... ; (ai,aj |- vi) ; ...]) (A ?- g)
}
\noindent where the theorems {ai,aj |- vi} are all the consequences that can be
drawn by a (single) matching modus-ponens inference from the assumptions {A}
and the implications derived using {RES_CANON} from the assumptions. The
sequence in which the theorems {ai,aj |- vi} are generated and the
corresponding tactics applied is unspecified.
\FAILURE
Evaluating {RES_THEN ttac th} fails with `{no implication}' if no
implication(s) can be derived from the assumptions of the goal by the
transformation process described under the entry for {RES_CANON}. Evaluating
{RES_THEN ttac (A ?- g)} fails with `{no resolvents}' if no assumption of the
goal {A ?- g} can be resolved with the derived implication or implications.
Evaluation also fails, with `{no tactics}', if there are resolvents, but for
every resolvent {ai,aj |- vi} evaluating the application {ttac (ai,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.
\SEEALSO
IMP_RES_TAC, IMP_RES_THEN, MATCH_MP, RES_CANON, RES_TAC.
\ENDDOC
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