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Branch-funct

Branch-funct-fix

Fixing function for branch-funct structures.

Signature
(branch-funct-fix x) → new-x
Arguments: x — Guard (branch-funct-p x).
Returns: new-x — Type (branch-funct-p new-x).

Definitions and Theorems

Function: branch-funct-fix$inline

(defun branch-funct-fix$inline (x)
  (declare (xargs :guard (branch-funct-p x)))
  (let ((__function__ 'branch-funct-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (branch-funct-kind x)
           (:beq (cons :beq (list)))
           (:bne (cons :bne (list)))
           (:blt (cons :blt (list)))
           (:bltu (cons :bltu (list)))
           (:bge (cons :bge (list)))
           (:bgeu (cons :bgeu (list))))
         :exec x)))

Theorem: branch-funct-p-of-branch-funct-fix

(defthm branch-funct-p-of-branch-funct-fix
  (b* ((new-x (branch-funct-fix$inline x)))
    (branch-funct-p new-x))
  :rule-classes :rewrite)

Theorem: branch-funct-fix-when-branch-funct-p

(defthm branch-funct-fix-when-branch-funct-p
  (implies (branch-funct-p x)
           (equal (branch-funct-fix x) x)))

Function: branch-funct-equiv$inline

(defun branch-funct-equiv$inline (acl2::x acl2::y)
  (declare (xargs :guard (and (branch-funct-p acl2::x)
                              (branch-funct-p acl2::y))))
  (equal (branch-funct-fix acl2::x)
         (branch-funct-fix acl2::y)))

Theorem: branch-funct-equiv-is-an-equivalence

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

Theorem: branch-funct-equiv-implies-equal-branch-funct-fix-1

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

Theorem: branch-funct-fix-under-branch-funct-equiv

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

Theorem: equal-of-branch-funct-fix-1-forward-to-branch-funct-equiv

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

Theorem: equal-of-branch-funct-fix-2-forward-to-branch-funct-equiv

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

Theorem: branch-funct-equiv-of-branch-funct-fix-1-forward

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

Theorem: branch-funct-equiv-of-branch-funct-fix-2-forward

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

Theorem: branch-funct-kind$inline-of-branch-funct-fix-x

(defthm branch-funct-kind$inline-of-branch-funct-fix-x
  (equal (branch-funct-kind$inline (branch-funct-fix x))
         (branch-funct-kind$inline x)))

Theorem: branch-funct-kind$inline-branch-funct-equiv-congruence-on-x

(defthm branch-funct-kind$inline-branch-funct-equiv-congruence-on-x
  (implies (branch-funct-equiv x x-equiv)
           (equal (branch-funct-kind$inline x)
                  (branch-funct-kind$inline x-equiv)))
  :rule-classes :congruence)

Theorem: consp-of-branch-funct-fix

(defthm consp-of-branch-funct-fix
  (consp (branch-funct-fix x))
  :rule-classes :type-prescription)