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• Lhs.lisp

Lhs-vars-normorderedp

Signature

(lhs-vars-normorderedp bound offset x) → *

Arguments

bound — Guard (integerp bound).

offset — Guard (natp offset).

x — Guard (lhs-p x).

Definitions and Theorems

Function: lhs-vars-normorderedp

(defun lhs-vars-normorderedp (bound offset x)
 (declare (xargs :guard (and (integerp bound)
                             (natp offset)
                             (lhs-p x))))
 (let ((__function__ 'lhs-vars-normorderedp))
  (declare (ignorable __function__))
  (if (atom x)
      t
   (and
      (lhatom-normorderedp bound offset (lhrange->atom (car x)))
      (lhs-vars-normorderedp bound
                             (+ (lhrange->w (car x)) (lnfix offset))
                             (cdr x))))))

Theorem: lhs-vars-normorderedp-of-ifix-bound

(defthm lhs-vars-normorderedp-of-ifix-bound
  (equal (lhs-vars-normorderedp (ifix bound)
                                offset x)
         (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-int-equiv-congruence-on-bound

(defthm lhs-vars-normorderedp-int-equiv-congruence-on-bound
  (implies (int-equiv bound bound-equiv)
           (equal (lhs-vars-normorderedp bound offset x)
                  (lhs-vars-normorderedp bound-equiv offset x)))
  :rule-classes :congruence)

Theorem: lhs-vars-normorderedp-of-nfix-offset

(defthm lhs-vars-normorderedp-of-nfix-offset
  (equal (lhs-vars-normorderedp bound (nfix offset)
                                x)
         (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-nat-equiv-congruence-on-offset

(defthm lhs-vars-normorderedp-nat-equiv-congruence-on-offset
  (implies (nat-equiv offset offset-equiv)
           (equal (lhs-vars-normorderedp bound offset x)
                  (lhs-vars-normorderedp bound offset-equiv x)))
  :rule-classes :congruence)

Theorem: lhs-vars-normorderedp-of-lhs-fix-x

(defthm lhs-vars-normorderedp-of-lhs-fix-x
  (equal (lhs-vars-normorderedp bound offset (lhs-fix x))
         (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-lhs-equiv-congruence-on-x

(defthm lhs-vars-normorderedp-lhs-equiv-congruence-on-x
  (implies (lhs-equiv x x-equiv)
           (equal (lhs-vars-normorderedp bound offset x)
                  (lhs-vars-normorderedp bound offset x-equiv)))
  :rule-classes :congruence)

Theorem: lhs-vars-normorderedp-of-lhs-cons

(defthm lhs-vars-normorderedp-of-lhs-cons
  (equal (lhs-vars-normorderedp bound offset (lhs-cons x y))
         (lhs-vars-normorderedp bound offset (cons x y))))

Theorem: lhs-vars-normorderedp-of-lhs-norm-x

(defthm lhs-vars-normorderedp-of-lhs-norm-x
  (equal (lhs-vars-normorderedp bound offset (lhs-norm x))
         (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-lhs-norm-equiv-congruence-on-x

(defthm lhs-vars-normorderedp-lhs-norm-equiv-congruence-on-x
  (implies (lhs-norm-equiv x x-equiv)
           (equal (lhs-vars-normorderedp bound offset x)
                  (lhs-vars-normorderedp bound offset x-equiv)))
  :rule-classes :congruence)

Theorem: lhs-vars-normorderedp-implies-rest

(defthm lhs-vars-normorderedp-implies-rest
  (implies (and (lhs-vars-normorderedp bound offset x)
                (equal (nfix offset1)
                       (+ (nfix offset) (lhrange->w (car x)))))
           (lhs-vars-normorderedp bound offset1 (cdr x))))

Theorem: lhs-vars-normorderedp-implies-rest-strict

(defthm lhs-vars-normorderedp-implies-rest-strict
 (implies
  (and (<= (lhrange->w (car x)) (nfix offset))
       (lhs-vars-normorderedp bound
                              (- (nfix offset) (lhrange->w (car x)))
                              x))
  (lhs-vars-normorderedp bound offset (cdr x))))

Theorem: lhs-vars-normorderedp-implies-atom-normorderedp-of-car

(defthm lhs-vars-normorderedp-implies-atom-normorderedp-of-car
  (implies
       (and (lhs-vars-normorderedp bound offset x)
            (consp x))
       (lhatom-normorderedp bound offset (lhrange->atom (car x)))))

Theorem: lhs-vars-normorderedp-implies-index-bound

(defthm lhs-vars-normorderedp-implies-index-bound
  (implies
       (and (lhs-vars-normorderedp bound offset x)
            (consp x)
            (eq (lhatom-kind (lhrange->atom (car x)))
                :var))
       (<= (svar-index (lhatom-var->name (lhrange->atom (car x))))
           (ifix bound)))
  :rule-classes :linear)

Theorem: lhs-vars-normorderedp-implies-rsh-when-index-equal

(defthm lhs-vars-normorderedp-implies-rsh-when-index-equal
 (implies
  (and
      (lhs-vars-normorderedp bound offset x)
      (consp x)
      (eq (lhatom-kind (lhrange->atom (car x)))
          :var)
      (equal (svar-index (lhatom-var->name (lhrange->atom (car x))))
             (ifix bound)))
  (<= (lhatom-var->rsh (lhrange->atom (car x)))
      (nfix offset)))
 :rule-classes :linear)

Theorem: lhs-vars-normorderedp-implies-lhs-bitproj-index

(defthm lhs-vars-normorderedp-implies-lhs-bitproj-index
  (implies (and (lhs-vars-normorderedp bound offset x)
                (eq (lhbit-kind (lhs-bitproj idx x))
                    :var))
           (svar-index (lhbit-var->name (lhs-bitproj idx x)))))

Theorem: lhs-vars-normorderedp-implies-lhs-bitproj-index-bound

(defthm lhs-vars-normorderedp-implies-lhs-bitproj-index-bound
  (implies (and (lhs-vars-normorderedp bound offset x)
                (eq (lhbit-kind (lhs-bitproj idx x))
                    :var))
           (<= (svar-index (lhbit-var->name (lhs-bitproj idx x)))
               (ifix bound)))
  :rule-classes :linear)

Theorem: lhs-vars-normorderedp-implies-lhs-bitproj-idx-when-index-bound

(defthm
     lhs-vars-normorderedp-implies-lhs-bitproj-idx-when-index-bound
 (implies
      (and (lhs-vars-normorderedp bound offset x)
           (eq (lhbit-kind (lhs-bitproj idx x))
               :var)
           (equal (svar-index (lhbit-var->name (lhs-bitproj idx x)))
                  (ifix bound)))
      (<= (lhbit-var->idx (lhs-bitproj idx x))
          (+ (nfix idx) (nfix offset))))
 :rule-classes :linear)

Theorem: lhs-vars-normorderedp-of-greater

(defthm lhs-vars-normorderedp-of-greater
  (implies (and (lhs-vars-normorderedp bound1 offset1 x)
                (<= (ifix bound1) (ifix bound))
                (<= (nfix offset1) (nfix offset)))
           (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-of-greater-bound

(defthm lhs-vars-normorderedp-of-greater-bound
  (implies (and (lhs-vars-normorderedp bound1 offset1 x)
                (< (ifix bound1) (ifix bound)))
           (lhs-vars-normorderedp bound offset x)))

Theorem: lhs-vars-normorderedp-of-rsh

(defthm lhs-vars-normorderedp-of-rsh
  (implies (and (lhs-vars-normorderedp bound offset x)
                (<= (+ (nfix offset) (nfix sh))
                    (nfix offset1)))
           (lhs-vars-normorderedp bound offset1 (lhs-rsh sh x))))

Theorem: lhs-vars-normorderedp-of-rsh-strict

(defthm lhs-vars-normorderedp-of-rsh-strict
  (implies (and (<= (nfix sh) (nfix offset1))
                (equal offset (- (nfix offset1) (nfix sh)))
                (lhs-vars-normorderedp bound offset x))
           (lhs-vars-normorderedp bound offset1 (lhs-rsh sh x))))

Theorem: lhs-vars-normorderedp-of-rsh-greater-bound

(defthm lhs-vars-normorderedp-of-rsh-greater-bound
  (implies (and (lhs-vars-normorderedp bound offset x)
                (< (ifix bound) (ifix bound1)))
           (lhs-vars-normorderedp bound1 offset1 (lhs-rsh sh x))))

Theorem: lhs-vars-normorderedp-of-concat-strict

(defthm lhs-vars-normorderedp-of-concat-strict
  (implies
       (and (lhs-vars-normorderedp bound offset x1)
            (lhs-vars-normorderedp bound (+ (nfix offset) (nfix w))
                                   x2))
       (lhs-vars-normorderedp bound offset (lhs-concat w x1 x2))))

Theorem: lhs-vars-normorderedp-of-concat

(defthm lhs-vars-normorderedp-of-concat
  (implies
       (and (lhs-vars-normorderedp bound offset x1)
            (lhs-vars-normorderedp bound offset2 x2)
            (<= (nfix offset2)
                (+ (nfix offset) (nfix w))))
       (lhs-vars-normorderedp bound offset (lhs-concat w x1 x2))))

Theorem: lhs-vars-normorderedp-implies-svarlist-boundedp

(defthm lhs-vars-normorderedp-implies-svarlist-boundedp
  (implies (lhs-vars-normorderedp bound offset x)
           (svarlist-boundedp (lhs-vars x)
                              (+ 1 (ifix bound)))))