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Atc-context

Atc-context-fix

Fixing function for atc-context structures.

Signature
(atc-context-fix x) → new-x
Arguments: x — Guard (atc-contextp x).
Returns: new-x — Type (atc-contextp new-x).

Definitions and Theorems

Function: atc-context-fix$inline

(defun atc-context-fix$inline (x)
 (declare (xargs :guard (atc-contextp x)))
 (let ((__function__ 'atc-context-fix))
  (declare (ignorable __function__))
  (mbe
     :logic
     (b* ((preamble (list-fix (cdr (std::da-nth 0 x))))
          (premises (atc-premise-list-fix (cdr (std::da-nth 1 x)))))
       (list (cons 'preamble preamble)
             (cons 'premises premises)))
     :exec x)))

Theorem: atc-contextp-of-atc-context-fix

(defthm atc-contextp-of-atc-context-fix
  (b* ((new-x (atc-context-fix$inline x)))
    (atc-contextp new-x))
  :rule-classes :rewrite)

Theorem: atc-context-fix-when-atc-contextp

(defthm atc-context-fix-when-atc-contextp
  (implies (atc-contextp x)
           (equal (atc-context-fix x) x)))

Function: atc-context-equiv$inline

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

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

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

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

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

Theorem: atc-context-fix-under-atc-context-equiv

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

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

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

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

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

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

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

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

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