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Axi-term

Axi-term-equiv

Basic equivalence relation for axi-term structures.

Definitions and Theorems

Function: axi-term-equiv$inline

(defun axi-term-equiv$inline (x acl2::y)
  (declare (xargs :guard (and (axi-term-p x)
                              (axi-term-p acl2::y))))
  (equal (axi-term-fix x)
         (axi-term-fix acl2::y)))

Theorem: axi-term-equiv-is-an-equivalence

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

Theorem: axi-term-equiv-implies-equal-axi-term-fix-1

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

Theorem: axi-term-fix-under-axi-term-equiv

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

Theorem: equal-of-axi-term-fix-1-forward-to-axi-term-equiv

(defthm equal-of-axi-term-fix-1-forward-to-axi-term-equiv
  (implies (equal (axi-term-fix x) acl2::y)
           (axi-term-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-axi-term-fix-2-forward-to-axi-term-equiv

(defthm equal-of-axi-term-fix-2-forward-to-axi-term-equiv
  (implies (equal x (axi-term-fix acl2::y))
           (axi-term-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: axi-term-equiv-of-axi-term-fix-1-forward

(defthm axi-term-equiv-of-axi-term-fix-1-forward
  (implies (axi-term-equiv (axi-term-fix x)
                           acl2::y)
           (axi-term-equiv x acl2::y))
  :rule-classes :forward-chaining)

Theorem: axi-term-equiv-of-axi-term-fix-2-forward

(defthm axi-term-equiv-of-axi-term-fix-2-forward
  (implies (axi-term-equiv x (axi-term-fix acl2::y))
           (axi-term-equiv x acl2::y))
  :rule-classes :forward-chaining)