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