• Top
  • Documentation
  • Books
  • Recursion-and-induction
  • Boolean-reasoning
    • Ipasir
    • Aignet
    • Aig
    • Satlink
    • Truth
      • Index-permute-shrink
      • Permute-stretch
      • Env-mismatch-aux
      • Permute-shrink
      • Permute-polarity
      • Env-permute-polarity
      • Env-permute-shrink
      • Permute-var-down
      • Swap-vars-aux
      • Env-permute-stretch
      • Swap-vars
      • Permute-var-up
      • Negative-cofactor
      • Truth-perm-rev
      • Index-permute-stretch
      • Env-mismatch
      • Truth-perm
      • Swap-polarity
      • Positive-cofactor
      • Index-perm-rev
      • Env-perm-rev
      • Nth-set-bit-pos
      • Index-swap
      • Env-move-var-down
      • Is-xor-with-var
      • Index-perm
      • Env-swap-vars
      • Var
      • Truth-eval
      • Index-move-down
      • Env-update
      • Env-perm
      • Depends-on-witness
      • Env-swap-polarity
      • Var-repetitions
      • Env-move-var-up
      • Index-move-up
      • Depends-on
      • Truth-norm
      • Index-listp
      • Env-diff-index
      • Env-lookup
      • True
      • False
    • Ubdds
    • Bdd
    • Faig
    • Bed
    • 4v
  • Debugging
  • Projects
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Testing-utilities
  • Math

• Truth

Swap-polarity

Signature

(swap-polarity n truth numvars) → new-truth

Arguments

n — Guard (natp n).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

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

Definitions and Theorems

Function: swap-polarity

(defun
 swap-polarity (n truth numvars)
 (declare (xargs :guard (and (natp n)
                             (integerp truth)
                             (natp numvars))))
 (declare (xargs :guard (< n numvars)))
 (let
   ((__function__ 'swap-polarity))
   (declare (ignorable __function__))
   (b* ((truth (truth-norm truth numvars))
        (var (var n numvars))
        (shift (ash 1 (lnfix n))))
       (logior (ash (logand var truth) (- shift))
               (ash (logand (lognot var)
                            (loghead (ash 1 (lnfix numvars)) truth))
                    shift)))))

Theorem: integerp-of-swap-polarity

(defthm integerp-of-swap-polarity
        (b* ((new-truth (swap-polarity n truth numvars)))
            (integerp new-truth))
        :rule-classes :type-prescription)

Theorem: swap-polarity-correct

(defthm
     swap-polarity-correct
     (b* ((?new-truth (swap-polarity n truth numvars)))
         (implies (< (nfix n) (nfix numvars))
                  (equal (truth-eval new-truth env numvars)
                         (truth-eval truth (env-swap-polarity n env)
                                     numvars)))))

Theorem: size-of-logand-by-size-of-loghead-2

(defthm size-of-logand-by-size-of-loghead-2
        (implies (and (unsigned-byte-p m a)
                      (unsigned-byte-p n (loghead m b)))
                 (unsigned-byte-p n (logand b a))))

Theorem: size-of-logand-with-loghead

(defthm size-of-logand-with-loghead
        (implies (unsigned-byte-p n (loghead m b))
                 (unsigned-byte-p n (logand b (loghead m a)))))

Theorem: swap-polarity-size-basic

(defthm swap-polarity-size-basic
        (b* ((?new-truth (swap-polarity n truth numvars)))
            (implies (< (nfix n) (nfix numvars))
                     (unsigned-byte-p (ash 1 numvars)
                                      new-truth))))

Theorem: swap-polarity-size

(defthm swap-polarity-size
        (b* ((?new-truth (swap-polarity n truth numvars)))
            (implies (and (natp size)
                          (<= (ash 1 (nfix numvars)) size)
                          (< (nfix n) (nfix numvars)))
                     (unsigned-byte-p size new-truth))))

Theorem: swap-polarity-of-truth-norm

(defthm swap-polarity-of-truth-norm
        (equal (swap-polarity n (truth-norm truth numvars)
                              numvars)
               (swap-polarity n truth numvars)))

Theorem: swap-polarity-of-nfix-n

(defthm swap-polarity-of-nfix-n
        (equal (swap-polarity (nfix n) truth numvars)
               (swap-polarity n truth numvars)))

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

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

Theorem: swap-polarity-of-ifix-truth

(defthm swap-polarity-of-ifix-truth
        (equal (swap-polarity n (ifix truth) numvars)
               (swap-polarity n truth numvars)))

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

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

Theorem: swap-polarity-of-nfix-numvars

(defthm swap-polarity-of-nfix-numvars
        (equal (swap-polarity n truth (nfix numvars))
               (swap-polarity n truth numvars)))

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

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