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