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Pseudo-term-subst

Pseudo-term-subst-equiv

Basic equivalence relation for pseudo-term-subst structures.

Definitions and Theorems

Function: pseudo-term-subst-equiv$inline

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

Theorem: pseudo-term-subst-equiv-is-an-equivalence

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

Theorem: pseudo-term-subst-equiv-implies-equal-pseudo-term-subst-fix-1

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

Theorem: pseudo-term-subst-fix-under-pseudo-term-subst-equiv

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

Theorem: equal-of-pseudo-term-subst-fix-1-forward-to-pseudo-term-subst-equiv

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

Theorem: equal-of-pseudo-term-subst-fix-2-forward-to-pseudo-term-subst-equiv

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

Theorem: pseudo-term-subst-equiv-of-pseudo-term-subst-fix-1-forward

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

Theorem: pseudo-term-subst-equiv-of-pseudo-term-subst-fix-2-forward

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