DEFCONG

prove congruence rule

Major Section: EVENTS

Defcong is used to prove that one equivalence relation preserves
another in a given argument position of a given function.
Example:
(defcong set-equal iff (memb x y) 2)
is an abbreviation for
(defthm set-equal-implies-iff-memb-2
 (implies (set-equal y y-equiv)
 (iff (memb x y) (memb x y-equiv)))
 :rule-classes (:congruence))

See congruence and also see equivalence.

General Form:
(defcong equiv1 equiv2 term k
 :rule-classes rule-classes
 :instructions instructions
 :hints hints
 :otf-flg otf-flg
 :event-name event-name
 :doc doc)

where equiv1 and equiv2 are known equivalence relations, term is a call of a function fn on the correct number of distinct variable arguments (fn x1 ... xn), k is a positive integer less than or equal to the arity of fn, and other arguments are as specified in the documentation for defthm . The defcong macro expands into a call of defthm . The name of the defthm event is equiv1-implies-equiv2-fn-k unless an :event-name keyword argument is supplied for the name. The term of the theorem is

(implies (equiv1 xk yk)
 (equiv2 (fn x1... xk ...xn)
 (fn x1... yk ...xn))).

The rule-class :congruence is added to the rule-classes specified, if it is not already there. All other arguments to the generated defthm form are as specified by the keyword arguments above.