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