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• Ident-paramdecl-map

Ident-paramdecl-map-equiv

Basic equivalence relation for ident-paramdecl-map structures.

Definitions and Theorems

Function: ident-paramdecl-map-equiv$inline

(defun ident-paramdecl-map-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (ident-paramdecl-mapp acl2::x)
                              (ident-paramdecl-mapp acl2::y))))
  (equal (ident-paramdecl-map-fix acl2::x)
         (ident-paramdecl-map-fix acl2::y)))

Theorem: ident-paramdecl-map-equiv-is-an-equivalence

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

Theorem: ident-paramdecl-map-equiv-implies-equal-ident-paramdecl-map-fix-1

(defthm
  ident-paramdecl-map-equiv-implies-equal-ident-paramdecl-map-fix-1
  (implies (ident-paramdecl-map-equiv acl2::x x-equiv)
           (equal (ident-paramdecl-map-fix acl2::x)
                  (ident-paramdecl-map-fix x-equiv)))
  :rule-classes (:congruence))

Theorem: ident-paramdecl-map-fix-under-ident-paramdecl-map-equiv

(defthm ident-paramdecl-map-fix-under-ident-paramdecl-map-equiv
  (ident-paramdecl-map-equiv (ident-paramdecl-map-fix acl2::x)
                             acl2::x)
  :rule-classes (:rewrite :rewrite-quoted-constant))

Theorem: equal-of-ident-paramdecl-map-fix-1-forward-to-ident-paramdecl-map-equiv

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

Theorem: equal-of-ident-paramdecl-map-fix-2-forward-to-ident-paramdecl-map-equiv

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

Theorem: ident-paramdecl-map-equiv-of-ident-paramdecl-map-fix-1-forward

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

Theorem: ident-paramdecl-map-equiv-of-ident-paramdecl-map-fix-2-forward

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