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Equiv-contexts

Equiv-contexts-equiv

Basic equivalence relation for equiv-contexts structures.

Definitions and Theorems

Function: equiv-contexts-equiv$inline

(defun equiv-contexts-equiv$inline (x y)
  (declare (xargs :guard (and (equiv-contextsp x)
                              (equiv-contextsp y))))
  (equal (equiv-contexts-fix x)
         (equiv-contexts-fix y)))

Theorem: equiv-contexts-equiv-is-an-equivalence

(defthm equiv-contexts-equiv-is-an-equivalence
  (and (booleanp (equiv-contexts-equiv x y))
       (equiv-contexts-equiv x x)
       (implies (equiv-contexts-equiv x y)
                (equiv-contexts-equiv y x))
       (implies (and (equiv-contexts-equiv x y)
                     (equiv-contexts-equiv y z))
                (equiv-contexts-equiv x z)))
  :rule-classes (:equivalence))

Theorem: equiv-contexts-equiv-implies-equal-equiv-contexts-fix-1

(defthm equiv-contexts-equiv-implies-equal-equiv-contexts-fix-1
  (implies (equiv-contexts-equiv x x-equiv)
           (equal (equiv-contexts-fix x)
                  (equiv-contexts-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: equiv-contexts-fix-under-equiv-contexts-equiv

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

Theorem: equal-of-equiv-contexts-fix-1-forward-to-equiv-contexts-equiv

(defthm
      equal-of-equiv-contexts-fix-1-forward-to-equiv-contexts-equiv
  (implies (equal (equiv-contexts-fix x) y)
           (equiv-contexts-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equal-of-equiv-contexts-fix-2-forward-to-equiv-contexts-equiv

(defthm
      equal-of-equiv-contexts-fix-2-forward-to-equiv-contexts-equiv
  (implies (equal x (equiv-contexts-fix y))
           (equiv-contexts-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equiv-contexts-equiv-of-equiv-contexts-fix-1-forward

(defthm equiv-contexts-equiv-of-equiv-contexts-fix-1-forward
  (implies (equiv-contexts-equiv (equiv-contexts-fix x)
                                 y)
           (equiv-contexts-equiv x y))
  :rule-classes :forward-chaining)

Theorem: equiv-contexts-equiv-of-equiv-contexts-fix-2-forward

(defthm equiv-contexts-equiv-of-equiv-contexts-fix-2-forward
  (implies (equiv-contexts-equiv x (equiv-contexts-fix y))
           (equiv-contexts-equiv x y))
  :rule-classes :forward-chaining)