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• Module

Module-equiv

Basic equivalence relation for module structures.

Definitions and Theorems

Function: module-equiv$inline

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

Theorem: module-equiv-is-an-equivalence

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

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

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

Theorem: module-fix-under-module-equiv

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

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

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

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

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

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

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

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

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