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• Fgl-env

Fgl-env-equiv

Basic equivalence relation for fgl-env structures.

Definitions and Theorems

Function: fgl-env-equiv$inline

(defun fgl-env-equiv$inline (x y)
  (declare (xargs :guard (and (fgl-env-p x) (fgl-env-p y))))
  (equal (fgl-env-fix x) (fgl-env-fix y)))

Theorem: fgl-env-equiv-is-an-equivalence

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

Theorem: fgl-env-equiv-implies-equal-fgl-env-fix-1

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

Theorem: fgl-env-fix-under-fgl-env-equiv

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

Theorem: equal-of-fgl-env-fix-1-forward-to-fgl-env-equiv

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

Theorem: equal-of-fgl-env-fix-2-forward-to-fgl-env-equiv

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

Theorem: fgl-env-equiv-of-fgl-env-fix-1-forward

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

Theorem: fgl-env-equiv-of-fgl-env-fix-2-forward

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