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