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Concatenation

Concatenation-equiv

Basic equivalence relation for concatenation structures.

Definitions and Theorems

Function: concatenation-equiv$inline

(defun concatenation-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (concatenationp acl2::x)
                              (concatenationp acl2::y))))
  (equal (concatenation-fix acl2::x)
         (concatenation-fix acl2::y)))

Theorem: concatenation-equiv-is-an-equivalence

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

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

(defthm concatenation-equiv-implies-equal-concatenation-fix-1
  (implies (concatenation-equiv acl2::x x-equiv)
           (equal (concatenation-fix acl2::x)
                  (concatenation-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: concatenation-fix-under-concatenation-equiv

(defthm concatenation-fix-under-concatenation-equiv
  (concatenation-equiv (concatenation-fix acl2::x)
                       acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

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

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

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

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

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

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

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

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