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• Truth

Swap-vars-aux

Signature

(swap-vars-aux 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-aux

(defun swap-vars-aux (n m truth numvars)
  (declare (xargs :guard (and (natp n)
                              (natp m)
                              (integerp truth)
                              (natp numvars))))
  (declare (xargs :guard (and (< n numvars) (< m n))))
  (let ((__function__ 'swap-vars-aux))
    (declare (ignorable __function__))
    (b* ((truth (loghead (ash 1 (lnfix numvars)) truth))
         (varn (var n numvars))
         (varm (var m numvars))
         (same (logeqv varn varm))
         (n&~m (logand varn (lognot varm)))
         (m&~n (logand varm (lognot varn)))
         (shift (- (ash 1 (lnfix n))
                   (ash 1 (lnfix m)))))
      (logior (logand same truth)
              (ash (logand n&~m truth) (- shift))
              (ash (logand m&~n truth) shift)))))

Theorem: integerp-of-swap-vars-aux

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

Theorem: eval-of-swap-vars-aux

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

Theorem: size-of-swap-vars-aux

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

Theorem: truth-norm-of-swap-vars-aux

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

Theorem: swap-vars-aux-of-truth-norm

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

Theorem: swap-vars-aux-of-nfix-n

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

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

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

Theorem: swap-vars-aux-of-nfix-m

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

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

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

Theorem: swap-vars-aux-of-ifix-truth

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

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

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

Theorem: swap-vars-aux-of-nfix-numvars

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

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

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