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• Maybe-fgl-generic-rule

Maybe-fgl-generic-rule-equiv

Basic equivalence relation for maybe-fgl-generic-rule structures.

Definitions and Theorems

Function: maybe-fgl-generic-rule-equiv$inline

(defun maybe-fgl-generic-rule-equiv$inline (x y)
  (declare (xargs :guard (and (maybe-fgl-generic-rule-p x)
                              (maybe-fgl-generic-rule-p y))))
  (equal (maybe-fgl-generic-rule-fix x)
         (maybe-fgl-generic-rule-fix y)))

Theorem: maybe-fgl-generic-rule-equiv-is-an-equivalence

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

Theorem: maybe-fgl-generic-rule-equiv-implies-equal-maybe-fgl-generic-rule-fix-1

(defthm
 maybe-fgl-generic-rule-equiv-implies-equal-maybe-fgl-generic-rule-fix-1
 (implies (maybe-fgl-generic-rule-equiv x x-equiv)
          (equal (maybe-fgl-generic-rule-fix x)
                 (maybe-fgl-generic-rule-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: maybe-fgl-generic-rule-fix-under-maybe-fgl-generic-rule-equiv

(defthm
      maybe-fgl-generic-rule-fix-under-maybe-fgl-generic-rule-equiv
  (maybe-fgl-generic-rule-equiv (maybe-fgl-generic-rule-fix x)
                                x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-maybe-fgl-generic-rule-fix-1-forward-to-maybe-fgl-generic-rule-equiv

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

Theorem: equal-of-maybe-fgl-generic-rule-fix-2-forward-to-maybe-fgl-generic-rule-equiv

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

Theorem: maybe-fgl-generic-rule-equiv-of-maybe-fgl-generic-rule-fix-1-forward

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

Theorem: maybe-fgl-generic-rule-equiv-of-maybe-fgl-generic-rule-fix-2-forward

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