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Term-bvars

Term-bvars-equiv

Basic equivalence relation for term-bvars structures.

Definitions and Theorems

Function: term-bvars-equiv$inline

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

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

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

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

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

Theorem: term-bvars-fix-under-term-bvars-equiv

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

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

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

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

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

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

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

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

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