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Atc-premise

Atc-premise-equiv

Basic equivalence relation for atc-premise structures.

Definitions and Theorems

Function: atc-premise-equiv$inline

(defun atc-premise-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (atc-premisep acl2::x)
                              (atc-premisep acl2::y))))
  (equal (atc-premise-fix acl2::x)
         (atc-premise-fix acl2::y)))

Theorem: atc-premise-equiv-is-an-equivalence

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

Theorem: atc-premise-equiv-implies-equal-atc-premise-fix-1

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

Theorem: atc-premise-fix-under-atc-premise-equiv

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

Theorem: equal-of-atc-premise-fix-1-forward-to-atc-premise-equiv

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

Theorem: equal-of-atc-premise-fix-2-forward-to-atc-premise-equiv

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

Theorem: atc-premise-equiv-of-atc-premise-fix-1-forward

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

Theorem: atc-premise-equiv-of-atc-premise-fix-2-forward

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