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Envmap

Envmap-equiv

Basic equivalence relation for envmap structures.

Definitions and Theorems

Function: envmap-equiv$inline

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

Theorem: envmap-equiv-is-an-equivalence

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

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

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

Theorem: envmap-fix-under-envmap-equiv

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

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

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

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

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

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

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

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

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