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• Lhrange

Lhrange-fix

Fixing function for lhrange structures.

Signature

(lhrange-fix x) → new-x

Arguments

x — Guard (lhrange-p x).

Returns

new-x — Type (lhrange-p new-x).

Definitions and Theorems

Function: lhrange-fix$inline

(defun lhrange-fix$inline (x)
 (declare (xargs :guard (lhrange-p x)))
 (let ((__function__ 'lhrange-fix))
  (declare (ignorable __function__))
  (mbe :logic
       (b* ((w (pos-fix (if (and (consp x) (integerp (car x)))
                            (car x)
                          1)))
            (atom (lhatom-fix (if (and (consp x) (integerp (car x)))
                                  (cdr x)
                                x))))
         (if (eql w 1) atom (cons w atom)))
       :exec x)))

Theorem: lhrange-p-of-lhrange-fix

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

Theorem: lhrange-fix-when-lhrange-p

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

Function: lhrange-equiv$inline

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

Theorem: lhrange-equiv-is-an-equivalence

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

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

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

Theorem: lhrange-fix-under-lhrange-equiv

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

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

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

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

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

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

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

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

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