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• 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)