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• Svdecomp-symenv

Svdecomp-symenv-equiv

Basic equivalence relation for svdecomp-symenv structures.

Definitions and Theorems

Function: svdecomp-symenv-equiv$inline

(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: svdecomp-symenv-equiv-is-an-equivalence

(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: svdecomp-symenv-equiv-implies-equal-svdecomp-symenv-fix-1

(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: svdecomp-symenv-fix-under-svdecomp-symenv-equiv

(defthm svdecomp-symenv-fix-under-svdecomp-symenv-equiv
  (svdecomp-symenv-equiv (svdecomp-symenv-fix x)
                         x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-svdecomp-symenv-fix-1-forward-to-svdecomp-symenv-equiv

(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: equal-of-svdecomp-symenv-fix-2-forward-to-svdecomp-symenv-equiv

(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: svdecomp-symenv-equiv-of-svdecomp-symenv-fix-1-forward

(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: svdecomp-symenv-equiv-of-svdecomp-symenv-fix-2-forward

(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)