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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)