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