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• Truth

Env-move-var-down

Signature

(env-move-var-down 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-down

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

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

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

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

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

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

(defthm
   env-move-var-down-of-env-move-var-up
   (implies (<= (nfix m) (nfix n))
            (equal (env-move-var-down m n (env-move-var-up m n env))
                   (nfix env))))

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

(defthm
   env-move-var-up-of-env-move-var-down
   (implies (<= (nfix m) (nfix n))
            (equal (env-move-var-up m n (env-move-var-down m n env))
                   (nfix env))))

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

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

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

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

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

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

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

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

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

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

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

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