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• Abc-comb-simp-config

Abc-comb-simp-config-equiv

Basic equivalence relation for abc-comb-simp-config structures.

Definitions and Theorems

Function: abc-comb-simp-config-equiv$inline

(defun abc-comb-simp-config-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (abc-comb-simp-config-p x)
                              (abc-comb-simp-config-p acl2::y))))
  (equal (abc-comb-simp-config-fix x)
         (abc-comb-simp-config-fix acl2::y)))

Theorem: abc-comb-simp-config-equiv-is-an-equivalence

(defthm abc-comb-simp-config-equiv-is-an-equivalence
  (and (booleanp (abc-comb-simp-config-equiv x y))
       (abc-comb-simp-config-equiv x x)
       (implies (abc-comb-simp-config-equiv x y)
                (abc-comb-simp-config-equiv y x))
       (implies (and (abc-comb-simp-config-equiv x y)
                     (abc-comb-simp-config-equiv y z))
                (abc-comb-simp-config-equiv x z)))
  :rule-classes (:equivalence))

Theorem: abc-comb-simp-config-equiv-implies-equal-abc-comb-simp-config-fix-1

(defthm
 abc-comb-simp-config-equiv-implies-equal-abc-comb-simp-config-fix-1
 (implies (abc-comb-simp-config-equiv x x-equiv)
          (equal (abc-comb-simp-config-fix x)
                 (abc-comb-simp-config-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: abc-comb-simp-config-fix-under-abc-comb-simp-config-equiv

(defthm abc-comb-simp-config-fix-under-abc-comb-simp-config-equiv
  (abc-comb-simp-config-equiv (abc-comb-simp-config-fix x)
                              x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-abc-comb-simp-config-fix-1-forward-to-abc-comb-simp-config-equiv

(defthm
 equal-of-abc-comb-simp-config-fix-1-forward-to-abc-comb-simp-config-equiv
 (implies (equal (abc-comb-simp-config-fix x)
                 acl2::y)
          (abc-comb-simp-config-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-abc-comb-simp-config-fix-2-forward-to-abc-comb-simp-config-equiv

(defthm
 equal-of-abc-comb-simp-config-fix-2-forward-to-abc-comb-simp-config-equiv
 (implies (equal x (abc-comb-simp-config-fix acl2::y))
          (abc-comb-simp-config-equiv x acl2::y))
 :rule-classes :forward-chaining)

Theorem: abc-comb-simp-config-equiv-of-abc-comb-simp-config-fix-1-forward

(defthm
   abc-comb-simp-config-equiv-of-abc-comb-simp-config-fix-1-forward
  (implies (abc-comb-simp-config-equiv (abc-comb-simp-config-fix x)
                                       acl2::y)
           (abc-comb-simp-config-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: abc-comb-simp-config-equiv-of-abc-comb-simp-config-fix-2-forward

(defthm
   abc-comb-simp-config-equiv-of-abc-comb-simp-config-fix-2-forward
 (implies
   (abc-comb-simp-config-equiv x (abc-comb-simp-config-fix acl2::y))
   (abc-comb-simp-config-equiv x acl2::y))
 :rule-classes :forward-chaining)