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• Truth

Env-perm

Signature

(env-perm n perm x numvars) → perm-env

Arguments

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

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

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

numvars — Guard (natp numvars).

Returns

perm-env — Type (natp perm-env).

Definitions and Theorems

Function: env-perm

(defun env-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__ 'env-perm))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix numvars) (nfix n)))
                     :exec (eql n numvars)))
          (lnfix x))
         (x (env-swap-vars n (nth n perm) x)))
      (env-perm (1+ (lnfix n))
                perm x numvars))))

Theorem: natp-of-env-perm

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

Theorem: lookup-index-perm-in-env-perm

(defthm lookup-index-perm-in-env-perm
  (equal (env-lookup (index-perm n perm k numvars)
                     (env-perm n perm env numvars))
         (env-lookup k env)))

Theorem: lookup-in-env-perm

(defthm lookup-in-env-perm
  (equal (env-lookup k (env-perm n perm env numvars))
         (env-lookup (index-perm-rev n perm k numvars)
                     env)))

Theorem: env-perm-of-nfix-n

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

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

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

Theorem: env-perm-of-nfix-x

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

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

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

Theorem: env-perm-of-nfix-numvars

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

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

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