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• Std/lists/abstract

Element-list-equiv

Generic typed list equivalence relation

Definitions and Theorems

Function: element-list-equiv

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

Theorem: element-list-equiv-is-an-equivalence

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

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

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

Theorem: element-list-fix-under-element-list-equiv

(defthm element-list-fix-under-element-list-equiv
        (element-list-equiv (element-list-fix x)
                            x)
        :rule-classes :rewrite)

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

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

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

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

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

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

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

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

Theorem: element-list-equiv-implies-element-equiv-car-1

(defthm element-list-equiv-implies-element-equiv-car-1
        (implies (element-list-equiv x x-equiv)
                 (element-equiv (car x) (car x-equiv)))
        :rule-classes (:congruence))

Theorem: element-list-equiv-implies-element-list-equiv-cdr-1

(defthm element-list-equiv-implies-element-list-equiv-cdr-1
        (implies (element-list-equiv x x-equiv)
                 (element-list-equiv (cdr x)
                                     (cdr x-equiv)))
        :rule-classes (:congruence))

Theorem: element-equiv-implies-element-list-equiv-cons-1

(defthm element-equiv-implies-element-list-equiv-cons-1
        (implies (element-equiv x x-equiv)
                 (element-list-equiv (cons x y)
                                     (cons x-equiv y)))
        :rule-classes (:congruence))

Theorem: element-list-equiv-implies-element-list-equiv-cons-2

(defthm element-list-equiv-implies-element-list-equiv-cons-2
        (implies (element-list-equiv y y-equiv)
                 (element-list-equiv (cons x y)
                                     (cons x y-equiv)))
        :rule-classes (:congruence))