irule : thm_tactic
Reduces the goal using a supplied implication, with matching.
When applied to a theorem of the form
   A' |- !x1...xn. s ==> !y1...ym. t ==> !z1...zk. u
irule produces a tactic that reduces a goal whose conclusion u' is a substitution and/or type instance of u to the corresponding instances of s and of t. Any variables free in s or t but not in u will be existentially quantified in the resulting subgoal, and a variable free in both s and t will result in a subgoal which is s /\ t, existentially quantified

The following diagram is simplified: more implications and quantified variables than shown are allowed.

     A ?- u'
  =========================  irule (A' |- !x. s ==> !y. t ==> u)
   A ?- (?z. s') /\ ?w. t'
where z and w are (type instances of) variables among x, y that do not occur free in s, t. The assumptions A', instantiated, are added as further new subgoals.
Fails unless the theorem, when stripped of universal quantification and antecedents of implications, can be instantiated to match the goal.
The supplied theorem is pre-processed using IRULE_CANON, which removes the universal quantifiers and antecedents of implications, and existentially quantifies variables which were instantiated but appeared only in the antecedents of implications.

Then MATCH_MP_TAC or MATCH_ACCEPT_TAC is applied (depending on whether or not the result of the preprocessing is an implication). To avoid preprocessing entirely, one can use prim_irule.

With goal R a (f b) and theorem thm being
   |- !x u. P u x ==> !w y. Q w x y ==> R x (f y)
the proof step e (irule thm) gives new goal (?u. P u a) /\ ?w. Q w a b.

With goal a = b and theorem trans

   |- !x y. (x = y) ==> !z. (y = z) ==> (x = z)
the proof step e (irule trans) gives new goal
   ?y. (a = y) /\ (y = b)
HOL  Kananaskis-14