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Op-funct

Op-funct-equiv

Basic equivalence relation for op-funct structures.

Definitions and Theorems

Function: op-funct-equiv$inline

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

Theorem: op-funct-equiv-is-an-equivalence

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

Theorem: op-funct-equiv-implies-equal-op-funct-fix-1

(defthm op-funct-equiv-implies-equal-op-funct-fix-1
  (implies (op-funct-equiv acl2::x x-equiv)
           (equal (op-funct-fix acl2::x)
                  (op-funct-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: op-funct-fix-under-op-funct-equiv

(defthm op-funct-fix-under-op-funct-equiv
  (op-funct-equiv (op-funct-fix acl2::x)
                  acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-op-funct-fix-1-forward-to-op-funct-equiv

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

Theorem: equal-of-op-funct-fix-2-forward-to-op-funct-equiv

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

Theorem: op-funct-equiv-of-op-funct-fix-1-forward

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

Theorem: op-funct-equiv-of-op-funct-fix-2-forward

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