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• Filepath-plexeme-list-alist

Filepath-plexeme-list-alist-equiv

Basic equivalence relation for filepath-plexeme-list-alist structures.

Definitions and Theorems

Function: filepath-plexeme-list-alist-equiv$inline

(defun filepath-plexeme-list-alist-equiv$inline (acl2::x acl2::y)
  (declare
       (xargs :guard (and (filepath-plexeme-list-alistp acl2::x)
                          (filepath-plexeme-list-alistp acl2::y))))
  (equal (filepath-plexeme-list-alist-fix acl2::x)
         (filepath-plexeme-list-alist-fix acl2::y)))

Theorem: filepath-plexeme-list-alist-equiv-is-an-equivalence

(defthm filepath-plexeme-list-alist-equiv-is-an-equivalence
  (and (booleanp (filepath-plexeme-list-alist-equiv x y))
       (filepath-plexeme-list-alist-equiv x x)
       (implies (filepath-plexeme-list-alist-equiv x y)
                (filepath-plexeme-list-alist-equiv y x))
       (implies (and (filepath-plexeme-list-alist-equiv x y)
                     (filepath-plexeme-list-alist-equiv y z))
                (filepath-plexeme-list-alist-equiv x z)))
  :rule-classes (:equivalence))

Theorem: filepath-plexeme-list-alist-equiv-implies-equal-filepath-plexeme-list-alist-fix-1

(defthm
 filepath-plexeme-list-alist-equiv-implies-equal-filepath-plexeme-list-alist-fix-1
 (implies (filepath-plexeme-list-alist-equiv acl2::x x-equiv)
          (equal (filepath-plexeme-list-alist-fix acl2::x)
                 (filepath-plexeme-list-alist-fix x-equiv)))
 :rule-classes (:congruence))

Theorem: filepath-plexeme-list-alist-fix-under-filepath-plexeme-list-alist-equiv

(defthm
 filepath-plexeme-list-alist-fix-under-filepath-plexeme-list-alist-equiv
 (filepath-plexeme-list-alist-equiv
      (filepath-plexeme-list-alist-fix acl2::x)
      acl2::x)
 :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-filepath-plexeme-list-alist-fix-1-forward-to-filepath-plexeme-list-alist-equiv

(defthm
 equal-of-filepath-plexeme-list-alist-fix-1-forward-to-filepath-plexeme-list-alist-equiv
 (implies (equal (filepath-plexeme-list-alist-fix acl2::x)
                 acl2::y)
          (filepath-plexeme-list-alist-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: equal-of-filepath-plexeme-list-alist-fix-2-forward-to-filepath-plexeme-list-alist-equiv

(defthm
 equal-of-filepath-plexeme-list-alist-fix-2-forward-to-filepath-plexeme-list-alist-equiv
 (implies (equal acl2::x
                 (filepath-plexeme-list-alist-fix acl2::y))
          (filepath-plexeme-list-alist-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: filepath-plexeme-list-alist-equiv-of-filepath-plexeme-list-alist-fix-1-forward

(defthm
 filepath-plexeme-list-alist-equiv-of-filepath-plexeme-list-alist-fix-1-forward
 (implies (filepath-plexeme-list-alist-equiv
               (filepath-plexeme-list-alist-fix acl2::x)
               acl2::y)
          (filepath-plexeme-list-alist-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)

Theorem: filepath-plexeme-list-alist-equiv-of-filepath-plexeme-list-alist-fix-2-forward

(defthm
 filepath-plexeme-list-alist-equiv-of-filepath-plexeme-list-alist-fix-2-forward
 (implies (filepath-plexeme-list-alist-equiv
               acl2::x
               (filepath-plexeme-list-alist-fix acl2::y))
          (filepath-plexeme-list-alist-equiv acl2::x acl2::y))
 :rule-classes :forward-chaining)