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• Truth

Env-move-var-up

Signature

(env-move-var-up m n env) → perm-env

Arguments

m — Guard (natp m).

n — Guard (natp n).

env — Guard (natp env).

Returns

perm-env — Type (natp perm-env).

Definitions and Theorems

Function: env-move-var-up

(defun env-move-var-up (m n env)
  (declare (xargs :guard (and (natp m) (natp n) (natp env))))
  (declare (xargs :guard (<= m n)))
  (let ((__function__ 'env-move-var-up))
    (declare (ignorable __function__))
    (b* (((when (mbe :logic (zp (- (nfix n) (nfix m)))
                     :exec (eql n m)))
          (lnfix env))
         (next (1+ (lnfix m)))
         (env (env-update next (env-lookup m env)
                          (env-update m (env-lookup next env)
                                      env))))
      (env-move-var-up next n env))))

Theorem: natp-of-env-move-var-up

(defthm natp-of-env-move-var-up
  (b* ((perm-env (env-move-var-up m n env)))
    (natp perm-env))
  :rule-classes :type-prescription)

Theorem: lookup-in-env-move-var-up

(defthm lookup-in-env-move-var-up
  (b* ((?perm-env (env-move-var-up m n env)))
    (equal (env-lookup k perm-env)
           (env-lookup (index-move-down m n k)
                       env))))

Theorem: env-move-var-up-of-nfix-m

(defthm env-move-var-up-of-nfix-m
  (equal (env-move-var-up (nfix m) n env)
         (env-move-var-up m n env)))

Theorem: env-move-var-up-nat-equiv-congruence-on-m

(defthm env-move-var-up-nat-equiv-congruence-on-m
  (implies (nat-equiv m m-equiv)
           (equal (env-move-var-up m n env)
                  (env-move-var-up m-equiv n env)))
  :rule-classes :congruence)

Theorem: env-move-var-up-of-nfix-n

(defthm env-move-var-up-of-nfix-n
  (equal (env-move-var-up m (nfix n) env)
         (env-move-var-up m n env)))

Theorem: env-move-var-up-nat-equiv-congruence-on-n

(defthm env-move-var-up-nat-equiv-congruence-on-n
  (implies (nat-equiv n n-equiv)
           (equal (env-move-var-up m n env)
                  (env-move-var-up m n-equiv env)))
  :rule-classes :congruence)

Theorem: env-move-var-up-of-nfix-env

(defthm env-move-var-up-of-nfix-env
  (equal (env-move-var-up m n (nfix env))
         (env-move-var-up m n env)))

Theorem: env-move-var-up-nat-equiv-congruence-on-env

(defthm env-move-var-up-nat-equiv-congruence-on-env
  (implies (nat-equiv env env-equiv)
           (equal (env-move-var-up m n env)
                  (env-move-var-up m n env-equiv)))
  :rule-classes :congruence)