• 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

Index-perm

Signature

(index-perm n perm x numvars) → perm-idx

Arguments

n — current position in the list.
    Guard (natp n).

perm — indices to permute.
    Guard (nat-listp perm).

x — index to permute.
    Guard (natp x).

numvars — Guard (natp numvars).

Returns

perm-idx — Type (natp perm-idx).

Definitions and Theorems

Function: index-perm

(defun index-perm (n perm x numvars)
       (declare (xargs :guard (and (natp n)
                                   (nat-listp perm)
                                   (natp x)
                                   (natp numvars))))
       (declare (xargs :guard (and (<= n numvars)
                                   (eql (len perm) numvars))))
       (let ((__function__ 'index-perm))
            (declare (ignorable __function__))
            (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                             :exec (eql n numvars)))
                  (lnfix x))
                 (x (index-swap n (nth n perm) x)))
                (index-perm (1+ (lnfix n))
                            perm x numvars))))

Theorem: natp-of-index-perm

(defthm natp-of-index-perm
        (b* ((perm-idx (index-perm n perm x numvars)))
            (natp perm-idx))
        :rule-classes :type-prescription)

Theorem: bound-of-index-perm

(defthm bound-of-index-perm
        (b* ((?perm-idx (index-perm n perm x numvars)))
            (implies (and (< (nfix x) (nfix numvars))
                          (index-listp perm numvars))
                     (< perm-idx (nfix numvars))))
        :rule-classes :linear)

Theorem: index-perm-unique

(defthm index-perm-unique
        (iff (equal (index-perm n perm x numvars)
                    (index-perm n perm y numvars))
             (equal (nfix x) (nfix y))))

Theorem: index-perm-of-nfix-n

(defthm index-perm-of-nfix-n
        (equal (index-perm (nfix n) perm x numvars)
               (index-perm n perm x numvars)))

Theorem: index-perm-nat-equiv-congruence-on-n

(defthm index-perm-nat-equiv-congruence-on-n
        (implies (nat-equiv n n-equiv)
                 (equal (index-perm n perm x numvars)
                        (index-perm n-equiv perm x numvars)))
        :rule-classes :congruence)

Theorem: index-perm-of-nfix-x

(defthm index-perm-of-nfix-x
        (equal (index-perm n perm (nfix x) numvars)
               (index-perm n perm x numvars)))

Theorem: index-perm-nat-equiv-congruence-on-x

(defthm index-perm-nat-equiv-congruence-on-x
        (implies (nat-equiv x x-equiv)
                 (equal (index-perm n perm x numvars)
                        (index-perm n perm x-equiv numvars)))
        :rule-classes :congruence)

Theorem: index-perm-of-nfix-numvars

(defthm index-perm-of-nfix-numvars
        (equal (index-perm n perm x (nfix numvars))
               (index-perm n perm x numvars)))

Theorem: index-perm-nat-equiv-congruence-on-numvars

(defthm index-perm-nat-equiv-congruence-on-numvars
        (implies (nat-equiv numvars numvars-equiv)
                 (equal (index-perm n perm x numvars)
                        (index-perm n perm x numvars-equiv)))
        :rule-classes :congruence)