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Mappingdecl

Mappingdecl-equiv

Basic equivalence relation for mappingdecl structures.

Definitions and Theorems

Function: mappingdecl-equiv$inline

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

Theorem: mappingdecl-equiv-is-an-equivalence

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

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

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

Theorem: mappingdecl-fix-under-mappingdecl-equiv

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

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

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

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

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

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

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

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

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