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Truthmap

Truthmap-equiv

Basic equivalence relation for truthmap structures.

Definitions and Theorems

Function: truthmap-equiv$inline

(defun truthmap-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (truthmap-p x)
                              (truthmap-p acl2::y))))
  (equal (truthmap-fix x)
         (truthmap-fix acl2::y)))

Theorem: truthmap-equiv-is-an-equivalence

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

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

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

Theorem: truthmap-fix-under-truthmap-equiv

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

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

(defthm equal-of-truthmap-fix-1-forward-to-truthmap-equiv
  (implies (equal (truthmap-fix x) acl2::y)
           (truthmap-equiv x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm equal-of-truthmap-fix-2-forward-to-truthmap-equiv
  (implies (equal x (truthmap-fix acl2::y))
           (truthmap-equiv x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm truthmap-equiv-of-truthmap-fix-1-forward
  (implies (truthmap-equiv (truthmap-fix x)
                           acl2::y)
           (truthmap-equiv x acl2::y))
  :rule-classes :forward-chaining)

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

(defthm truthmap-equiv-of-truthmap-fix-2-forward
  (implies (truthmap-equiv x (truthmap-fix acl2::y))
           (truthmap-equiv x acl2::y))
  :rule-classes :forward-chaining)