Basic equivalence relation for op/en-p structures.
Function:
(defun op/en-p-equiv$inline (x y) (declare (xargs :guard (and (op/en-p-p x) (op/en-p-p y)))) (equal (op/en-p-fix x) (op/en-p-fix y)))
Theorem:
(defthm op/en-p-equiv-is-an-equivalence (and (booleanp (op/en-p-equiv x y)) (op/en-p-equiv x x) (implies (op/en-p-equiv x y) (op/en-p-equiv y x)) (implies (and (op/en-p-equiv x y) (op/en-p-equiv y z)) (op/en-p-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm op/en-p-equiv-implies-equal-op/en-p-fix-1 (implies (op/en-p-equiv x x-equiv) (equal (op/en-p-fix x) (op/en-p-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm op/en-p-fix-under-op/en-p-equiv (op/en-p-equiv (op/en-p-fix x) x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-op/en-p-fix-1-forward-to-op/en-p-equiv (implies (equal (op/en-p-fix x) y) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-op/en-p-fix-2-forward-to-op/en-p-equiv (implies (equal x (op/en-p-fix y)) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm op/en-p-equiv-of-op/en-p-fix-1-forward (implies (op/en-p-equiv (op/en-p-fix x) y) (op/en-p-equiv x y)) :rule-classes :forward-chaining)
Theorem:
(defthm op/en-p-equiv-of-op/en-p-fix-2-forward (implies (op/en-p-equiv x (op/en-p-fix y)) (op/en-p-equiv x y)) :rule-classes :forward-chaining)