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String-option

String-option-equiv

Basic equivalence relation for string-option structures.

Definitions and Theorems

Function: string-option-equiv$inline

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

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

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

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

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

Theorem: string-option-fix-under-string-option-equiv

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

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

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

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

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

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

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

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

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