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• Var-decl-map

Var-decl-map-equiv

Basic equivalence relation for var-decl-map structures.

Definitions and Theorems

Function: var-decl-map-equiv$inline

(defun var-decl-map-equiv$inline (x y)
  (declare (xargs :guard (and (var-decl-map-p x)
                              (var-decl-map-p y))))
  (equal (var-decl-map-fix x)
         (var-decl-map-fix y)))

Theorem: var-decl-map-equiv-is-an-equivalence

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

Theorem: var-decl-map-equiv-implies-equal-var-decl-map-fix-1

(defthm var-decl-map-equiv-implies-equal-var-decl-map-fix-1
  (implies (var-decl-map-equiv x x-equiv)
           (equal (var-decl-map-fix x)
                  (var-decl-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: var-decl-map-fix-under-var-decl-map-equiv

(defthm var-decl-map-fix-under-var-decl-map-equiv
  (var-decl-map-equiv (var-decl-map-fix x)
                      x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-var-decl-map-fix-1-forward-to-var-decl-map-equiv

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

Theorem: equal-of-var-decl-map-fix-2-forward-to-var-decl-map-equiv

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

Theorem: var-decl-map-equiv-of-var-decl-map-fix-1-forward

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

Theorem: var-decl-map-equiv-of-var-decl-map-fix-2-forward

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