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• Std/lists/abstract

Element-equiv

Generic equivalence relation among typed list elements.

Definitions and Theorems

Function: element-equiv

(defun element-equiv (x y)
  (equal (element-fix x) (element-fix y)))

Theorem: element-equiv-is-an-equivalence

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

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

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

Theorem: element-fix-under-element-equiv

(defthm element-fix-under-element-equiv
  (element-equiv (element-fix x) x))

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

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

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

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

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

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

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

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