• Top
  • Documentation
  • Books
  • Boolean-reasoning
    • Ipasir
    • Aignet
    • Aig
    • Satlink
    • Truth
      • Index-permute-shrink
      • Permute-stretch
      • Permute-shrink
      • Env-mismatch-aux
      • Env-permute-shrink
      • Permute-polarity
      • Env-permute-polarity
      • Permute-var-down
      • Env-permute-stretch
      • Swap-vars-aux
      • Swap-vars
      • Permute-var-up
      • Truth-perm-rev
      • Negative-cofactor
      • Index-permute-stretch
      • Env-mismatch
      • Swap-polarity
      • Positive-cofactor
      • Truth-perm
      • Index-perm-rev
      • Nth-set-bit-pos
      • Env-perm-rev
      • Is-xor-with-var
      • Index-swap
      • Index-perm
      • Env-move-var-down
      • Truth-eval
      • Env-swap-vars
      • Env-perm
      • Depends-on-witness
      • Var
      • Index-move-down
      • Env-update
      • Env-swap-polarity
      • Var-repetitions
      • Env-move-var-up
      • Depends-on
      • Index-move-up
      • 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
  • Math
  • Testing-utilities

• Truth

Permute-var-down

Signature

(permute-var-down m n truth numvars) → perm-truth

Arguments

m — Guard (natp m).

n — Guard (natp n).

truth — Guard (integerp truth).

numvars — Guard (natp numvars).

Returns

perm-truth — Type (natp perm-truth).

Definitions and Theorems

Function: permute-var-down

(defun permute-var-down (m n truth numvars)
  (declare (xargs :guard (and (natp m)
                              (natp n)
                              (integerp truth)
                              (natp numvars))))
  (declare (xargs :guard (and (<= m n) (< n numvars))))
  (let ((__function__ 'permute-var-down))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix n) (nfix m)))
                     :exec (eql n m)))
          (truth-norm truth numvars))
         (next (1+ (lnfix m)))
         (truth (permute-var-down next n truth numvars)))
      (swap-vars next m truth numvars))))

Theorem: natp-of-permute-var-down

(defthm natp-of-permute-var-down
  (b* ((perm-truth (permute-var-down m n truth numvars)))
    (natp perm-truth))
  :rule-classes :type-prescription)

Theorem: eval-of-permute-var-down-with-env-move-var-down

(defthm eval-of-permute-var-down-with-env-move-var-down
 (b* ((?perm-truth (permute-var-down m n truth numvars)))
  (implies (and (<= (nfix m) (nfix n))
                (< (nfix n) (nfix numvars)))
           (equal (truth-eval perm-truth (env-move-var-down m n env)
                              numvars)
                  (truth-eval truth env numvars)))))

Theorem: eval-of-permute-var-down

(defthm eval-of-permute-var-down
  (b* ((?perm-truth (permute-var-down m n truth numvars)))
    (implies (and (<= (nfix m) (nfix n))
                  (< (nfix n) (nfix numvars)))
             (equal (truth-eval perm-truth env numvars)
                    (truth-eval truth (env-move-var-up m n env)
                                numvars)))))

Theorem: size-of-permute-var-down

(defthm size-of-permute-var-down
  (b* ((?perm-truth (permute-var-down m n truth numvars)))
    (implies (and (< (nfix n) (nfix numvars))
                  (natp size)
                  (<= (ash 1 numvars) size))
             (unsigned-byte-p size perm-truth))))

Theorem: permute-var-down-of-truth-norm

(defthm permute-var-down-of-truth-norm
  (equal (permute-var-down m n (truth-norm truth numvars)
                           numvars)
         (permute-var-down m n truth numvars)))

Theorem: permute-var-down-of-nfix-m

(defthm permute-var-down-of-nfix-m
  (equal (permute-var-down (nfix m)
                           n truth numvars)
         (permute-var-down m n truth numvars)))

Theorem: permute-var-down-nat-equiv-congruence-on-m

(defthm permute-var-down-nat-equiv-congruence-on-m
  (implies (nat-equiv m m-equiv)
           (equal (permute-var-down m n truth numvars)
                  (permute-var-down m-equiv n truth numvars)))
  :rule-classes :congruence)

Theorem: permute-var-down-of-nfix-n

(defthm permute-var-down-of-nfix-n
  (equal (permute-var-down m (nfix n)
                           truth numvars)
         (permute-var-down m n truth numvars)))

Theorem: permute-var-down-nat-equiv-congruence-on-n

(defthm permute-var-down-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (permute-var-down m n truth numvars)
                  (permute-var-down m n-equiv truth numvars)))
  :rule-classes :congruence)

Theorem: permute-var-down-of-ifix-truth

(defthm permute-var-down-of-ifix-truth
  (equal (permute-var-down m n (ifix truth)
                           numvars)
         (permute-var-down m n truth numvars)))

Theorem: permute-var-down-int-equiv-congruence-on-truth

(defthm permute-var-down-int-equiv-congruence-on-truth
  (implies (int-equiv truth truth-equiv)
           (equal (permute-var-down m n truth numvars)
                  (permute-var-down m n truth-equiv numvars)))
  :rule-classes :congruence)

Theorem: permute-var-down-of-nfix-numvars

(defthm permute-var-down-of-nfix-numvars
  (equal (permute-var-down m n truth (nfix numvars))
         (permute-var-down m n truth numvars)))

Theorem: permute-var-down-nat-equiv-congruence-on-numvars

(defthm permute-var-down-nat-equiv-congruence-on-numvars
  (implies (nat-equiv numvars numvars-equiv)
           (equal (permute-var-down m n truth numvars)
                  (permute-var-down m n truth numvars-equiv)))
  :rule-classes :congruence)