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• Nat-option

Nat-option-equiv

Basic equivalence relation for nat-option structures.

Definitions and Theorems

Function: nat-option-equiv$inline

(defun nat-option-equiv$inline (x y)
  (declare (xargs :guard (and (nat-optionp x) (nat-optionp y))))
  (equal (nat-option-fix x)
         (nat-option-fix y)))

Theorem: nat-option-equiv-is-an-equivalence

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

Theorem: nat-option-equiv-implies-equal-nat-option-fix-1

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

Theorem: nat-option-fix-under-nat-option-equiv

(defthm nat-option-fix-under-nat-option-equiv
  (nat-option-equiv (nat-option-fix x) x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-nat-option-fix-1-forward-to-nat-option-equiv

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

Theorem: equal-of-nat-option-fix-2-forward-to-nat-option-equiv

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

Theorem: nat-option-equiv-of-nat-option-fix-1-forward

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

Theorem: nat-option-equiv-of-nat-option-fix-2-forward

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