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• Node

Node-fix

Fixing function for node structures.

Signature

(node-fix x) → new-x

Arguments

x — Guard (node-p x).

Returns

new-x — Type (node-p new-x).

Definitions and Theorems

Function: node-fix$inline

(defun node-fix$inline (x)
  (declare (xargs :guard (node-p x)))
  (let ((__function__ 'node-fix))
    (declare (ignorable __function__))
    (mbe :logic
         (case (stype x)
           (:const nil)
           (:pi '(:pi))
           (:reg '(:reg))
           (:and (b* ((fanin0 (lit-fix (cadr x)))
                      (fanin1 (lit-fix (caddr x))))
                   (list :and fanin0 fanin1)))
           (:xor (b* ((fanin0 (lit-fix (cadr x)))
                      (fanin1 (lit-fix (caddr x))))
                   (list :xor fanin0 fanin1)))
           (:po (b* ((fanin (lit-fix (cadr x))))
                  (list :po fanin)))
           (:nxst (b* ((fanin (lit-fix (cadr x)))
                       (reg (nfix (caddr x))))
                    (list :nxst fanin reg))))
         :exec x)))

Theorem: node-p-of-node-fix

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

Theorem: node-fix-when-node-p

(defthm node-fix-when-node-p
  (implies (node-p x)
           (equal (node-fix x) x)))

Function: node-equiv$inline

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

Theorem: node-equiv-is-an-equivalence

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

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

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

Theorem: node-fix-under-node-equiv

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

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

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

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

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

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

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

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

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

Theorem: stype$inline-of-node-fix-x

(defthm stype$inline-of-node-fix-x
  (equal (stype$inline (node-fix x))
         (stype$inline x)))

Theorem: stype$inline-node-equiv-congruence-on-x

(defthm stype$inline-node-equiv-congruence-on-x
  (implies (node-equiv x x-equiv)
           (equal (stype$inline x)
                  (stype$inline x-equiv)))
  :rule-classes :congruence)