• 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
  • Projects
  • Debugging
  • Std
  • Proof-automation
  • Macro-libraries
  • ACL2
  • Interfacing-tools
  • Hardware-verification
  • Software-verification
  • Testing-utilities
  • Math

• 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)