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