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• Elab-modlist

Elab-modlist-equiv

Basic equivalence relation for elab-modlist structures.

Definitions and Theorems

Function: elab-modlist-equiv$inline

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

Theorem: elab-modlist-equiv-is-an-equivalence

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

Theorem: elab-modlist-equiv-implies-equal-elab-modlist-fix-1

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

Theorem: elab-modlist-fix-under-elab-modlist-equiv

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

Theorem: equal-of-elab-modlist-fix-1-forward-to-elab-modlist-equiv

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

Theorem: equal-of-elab-modlist-fix-2-forward-to-elab-modlist-equiv

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

Theorem: elab-modlist-equiv-of-elab-modlist-fix-1-forward

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

Theorem: elab-modlist-equiv-of-elab-modlist-fix-2-forward

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