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Programid

Programid-equiv

Basic equivalence relation for programid structures.

Definitions and Theorems

Function: programid-equiv$inline

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

Theorem: programid-equiv-is-an-equivalence

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

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

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

Theorem: programid-fix-under-programid-equiv

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

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

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

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

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

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

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

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

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