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• Bin-expo-prefix

Bin-expo-prefix-equiv

Basic equivalence relation for bin-expo-prefix structures.

Definitions and Theorems

Function: bin-expo-prefix-equiv$inline

(defun bin-expo-prefix-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (bin-expo-prefixp acl2::x)
                              (bin-expo-prefixp acl2::y))))
  (equal (bin-expo-prefix-fix acl2::x)
         (bin-expo-prefix-fix acl2::y)))

Theorem: bin-expo-prefix-equiv-is-an-equivalence

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

Theorem: bin-expo-prefix-equiv-implies-equal-bin-expo-prefix-fix-1

(defthm bin-expo-prefix-equiv-implies-equal-bin-expo-prefix-fix-1
  (implies (bin-expo-prefix-equiv acl2::x x-equiv)
           (equal (bin-expo-prefix-fix acl2::x)
                  (bin-expo-prefix-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: bin-expo-prefix-fix-under-bin-expo-prefix-equiv

(defthm bin-expo-prefix-fix-under-bin-expo-prefix-equiv
  (bin-expo-prefix-equiv (bin-expo-prefix-fix acl2::x)
                         acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-bin-expo-prefix-fix-1-forward-to-bin-expo-prefix-equiv

(defthm
    equal-of-bin-expo-prefix-fix-1-forward-to-bin-expo-prefix-equiv
  (implies (equal (bin-expo-prefix-fix acl2::x)
                  acl2::y)
           (bin-expo-prefix-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: equal-of-bin-expo-prefix-fix-2-forward-to-bin-expo-prefix-equiv

(defthm
    equal-of-bin-expo-prefix-fix-2-forward-to-bin-expo-prefix-equiv
  (implies (equal acl2::x (bin-expo-prefix-fix acl2::y))
           (bin-expo-prefix-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bin-expo-prefix-equiv-of-bin-expo-prefix-fix-1-forward

(defthm bin-expo-prefix-equiv-of-bin-expo-prefix-fix-1-forward
  (implies (bin-expo-prefix-equiv (bin-expo-prefix-fix acl2::x)
                                  acl2::y)
           (bin-expo-prefix-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)

Theorem: bin-expo-prefix-equiv-of-bin-expo-prefix-fix-2-forward

(defthm bin-expo-prefix-equiv-of-bin-expo-prefix-fix-2-forward
  (implies
       (bin-expo-prefix-equiv acl2::x (bin-expo-prefix-fix acl2::y))
       (bin-expo-prefix-equiv acl2::x acl2::y))
  :rule-classes :forward-chaining)