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• Truth

Swap-vars

Signature

(swap-vars n m truth numvars) → swapped-truth

Arguments

n — Guard (natp n).

m — Guard (natp m).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

swapped-truth — Type (integerp swapped-truth).

Definitions and Theorems

Function: swap-vars

(defun swap-vars (n m truth numvars)
  (declare (xargs :guard (and (natp n)
                              (natp m)
                              (integerp truth)
                              (natp numvars))))
  (declare (xargs :guard (and (< n numvars) (< m numvars))))
  (let ((__function__ 'swap-vars))
    (declare (ignorable __function__))
    (cond ((< (lnfix n) (lnfix m))
           (swap-vars-aux m n truth numvars))
          ((< (lnfix m) (lnfix n))
           (swap-vars-aux n m truth numvars))
          (t (truth-norm truth numvars)))))

Theorem: integerp-of-swap-vars

(defthm integerp-of-swap-vars
  (b* ((swapped-truth (swap-vars n m truth numvars)))
    (integerp swapped-truth))
  :rule-classes :type-prescription)

Theorem: swap-vars-commute

(defthm swap-vars-commute
  (b* nil
    (equal (swap-vars m n truth numvars)
           (swap-vars n m truth numvars)))
  :rule-classes ((:rewrite :loop-stopper ((n m)))))

Theorem: eval-of-swap-vars

(defthm eval-of-swap-vars
  (b* ((?swapped-truth (swap-vars n m truth numvars)))
    (implies (and (< (nfix n) (nfix numvars))
                  (< (nfix m) (nfix numvars)))
             (equal (truth-eval swapped-truth env numvars)
                    (truth-eval truth (env-swap-vars n m env)
                                numvars)))))

Theorem: size-of-swap-vars

(defthm size-of-swap-vars
  (b* ((?swapped-truth (swap-vars n m truth numvars)))
    (implies (and (< (nfix n) (nfix numvars))
                  (< (nfix m) (nfix numvars))
                  (natp size)
                  (<= (ash 1 numvars) size))
             (unsigned-byte-p size swapped-truth))))

Theorem: swap-vars-self

(defthm swap-vars-self
  (b* nil
    (equal (swap-vars n n truth numvars)
           (truth-norm truth numvars))))

Theorem: truth-norm-of-swap-vars

(defthm truth-norm-of-swap-vars
  (b* ((?swapped-truth (swap-vars n m truth numvars)))
    (implies (and (< (nfix n) (nfix numvars))
                  (< (nfix m) (nfix numvars)))
             (equal (truth-norm swapped-truth numvars)
                    swapped-truth))))

Theorem: swap-vars-of-truth-norm

(defthm swap-vars-of-truth-norm
  (equal (swap-vars n m (truth-norm truth numvars)
                    numvars)
         (swap-vars n m truth numvars)))

Theorem: swap-vars-of-nfix-n

(defthm swap-vars-of-nfix-n
  (equal (swap-vars (nfix n) m truth numvars)
         (swap-vars n m truth numvars)))

Theorem: swap-vars-nat-equiv-congruence-on-n

(defthm swap-vars-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (swap-vars n m truth numvars)
                  (swap-vars n-equiv m truth numvars)))
  :rule-classes :congruence)

Theorem: swap-vars-of-nfix-m

(defthm swap-vars-of-nfix-m
  (equal (swap-vars n (nfix m) truth numvars)
         (swap-vars n m truth numvars)))

Theorem: swap-vars-nat-equiv-congruence-on-m

(defthm swap-vars-nat-equiv-congruence-on-m
  (implies (nat-equiv m m-equiv)
           (equal (swap-vars n m truth numvars)
                  (swap-vars n m-equiv truth numvars)))
  :rule-classes :congruence)

Theorem: swap-vars-of-ifix-truth

(defthm swap-vars-of-ifix-truth
  (equal (swap-vars n m (ifix truth) numvars)
         (swap-vars n m truth numvars)))

Theorem: swap-vars-int-equiv-congruence-on-truth

(defthm swap-vars-int-equiv-congruence-on-truth
  (implies (int-equiv truth truth-equiv)
           (equal (swap-vars n m truth numvars)
                  (swap-vars n m truth-equiv numvars)))
  :rule-classes :congruence)

Theorem: swap-vars-of-nfix-numvars

(defthm swap-vars-of-nfix-numvars
  (equal (swap-vars n m truth (nfix numvars))
         (swap-vars n m truth numvars)))

Theorem: swap-vars-nat-equiv-congruence-on-numvars

(defthm swap-vars-nat-equiv-congruence-on-numvars
  (implies (nat-equiv numvars numvars-equiv)
           (equal (swap-vars n m truth numvars)
                  (swap-vars n m truth numvars-equiv)))
  :rule-classes :congruence)