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• Fnsym

Fnsym-equiv

Equivalence relation for fnsyms.

Definitions and Theorems

Function: fnsym-equiv$inline

(defun fnsym-equiv$inline (x y)
  (declare (xargs :guard (and (fnsym-p x) (fnsym-p y))))
  (eq (fnsym-fix x) (fnsym-fix y)))

Theorem: fnsym-equiv-is-an-equivalence

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

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

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

Theorem: fnsym-fix-under-fnsym-equiv

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

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

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

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

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

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

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

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

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