H-Termination proof of ../tpdb/qualif/Queue_addToQueue

YES H-Termination proof of ../tpdb/qualif/Queue_addToQueue_1.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Main

((Queue.addToQueue :: Queue.Queue a -> a -> Queue.Queue a) :: Queue.Queue a -> a -> Queue.Queue a)

module Main where

import qualified Prelude
import qualified Queue

module Queue where

import qualified Main
import qualified Prelude

data Queue a = Q [a] [a] [a]

addToQueue :: Queue a -> a -> Queue a

addToQueue

(Q xs ys xs') y

makeQ xs (y : ys) xs'

listToQueue :: [a] -> Queue a

listToQueue

Q xs [] xs

makeQ :: [a] -> [a] -> [a] -> Queue a

makeQ	xs ys []	=	listToQueue (rotate xs ys [])
makeQ	xs ys (_ : xs')	=	Q xs ys xs'

rotate :: [a] -> [a] -> [a] -> [a]

rotate	[] (y : _) zs	=	y : zs
rotate	(x : xs) (y : ys) zs	=	x : rotate xs ys (y : zs)

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

mainModule Main

((Queue.addToQueue :: Queue.Queue a -> a -> Queue.Queue a) :: Queue.Queue a -> a -> Queue.Queue a)

module Main where

import qualified Prelude
import qualified Queue

module Queue where

import qualified Main
import qualified Prelude

data Queue a = Q [a] [a] [a]

addToQueue :: Queue a -> a -> Queue a

addToQueue

(Q xs ys xs') y

makeQ xs (y : ys) xs'

listToQueue :: [a] -> Queue a

listToQueue

Q xs [] xs

makeQ :: [a] -> [a] -> [a] -> Queue a

makeQ	xs ys []	=	listToQueue (rotate xs ys [])
makeQ	xs ys (vv : xs')	=	Q xs ys xs'

rotate :: [a] -> [a] -> [a] -> [a]

rotate	[] (y : vw) zs	=	y : zs
rotate	(x : xs) (y : ys) zs	=	x : rotate xs ys (y : zs)

Num Reduction:All numbers are transformed to thier corresponding representation with Succ, Pred and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(Queue.addToQueue :: Queue.Queue a -> a -> Queue.Queue a)

module Main where

import qualified Prelude
import qualified Queue

module Queue where

import qualified Main
import qualified Prelude

data Queue a = Q [a] [a] [a]

addToQueue :: Queue a -> a -> Queue a

addToQueue

(Q xs ys xs') y

makeQ xs (y : ys) xs'

listToQueue :: [a] -> Queue a

listToQueue

Q xs [] xs

makeQ :: [a] -> [a] -> [a] -> Queue a

makeQ	xs ys []	=	listToQueue (rotate xs ys [])
makeQ	xs ys (vv : xs')	=	Q xs ys xs'

rotate :: [a] -> [a] -> [a] -> [a]

rotate	[] (y : vw) zs	=	y : zs
rotate	(x : xs) (y : ys) zs	=	x : rotate xs ys (y : zs)

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="Queue.addToQueue\n",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="Queue.addToQueue vz3\n",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="Queue.addToQueue vz3 vz4\n",fontsize=16,color="burlywood",shape="triangle"];159[label="vz3/Queue.Q vz30 vz31 vz32",fontsize=10,color="white",style="solid",shape="box"];4 -> 159[label="",style="solid", color="burlywood", weight=9];
159 -> 5[label="",style="solid", color="burlywood", weight=3];
5[label="Queue.addToQueue (Queue.Q vz30 vz31 vz32) vz4\n",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="Queue.makeQ vz30 (vz4 : vz31) vz32\n",fontsize=16,color="burlywood",shape="box"];160[label="vz32/vz320 : vz321",fontsize=10,color="white",style="solid",shape="box"];6 -> 160[label="",style="solid", color="burlywood", weight=9];
160 -> 7[label="",style="solid", color="burlywood", weight=3];
161[label="vz32/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 161[label="",style="solid", color="burlywood", weight=9];
161 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="Queue.makeQ vz30 (vz4 : vz31) (vz320 : vz321)\n",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Queue.makeQ vz30 (vz4 : vz31) []\n",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Queue.Q vz30 (vz4 : vz31) vz321\n",fontsize=16,color="green",shape="box"];10[label="Queue.listToQueue (Queue.rotate vz30 (vz4 : vz31) [])\n",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="Queue.Q (Queue.rotate vz30 (vz4 : vz31) []) [] (Queue.rotate vz30 (vz4 : vz31) [])\n",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
11 -> 13[label="",style="dashed", color="green", weight=3];
12[label="Queue.rotate vz30 (vz4 : vz31) []\n",fontsize=16,color="burlywood",shape="triangle"];162[label="vz30/vz300 : vz301",fontsize=10,color="white",style="solid",shape="box"];12 -> 162[label="",style="solid", color="burlywood", weight=9];
162 -> 14[label="",style="solid", color="burlywood", weight=3];
163[label="vz30/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 163[label="",style="solid", color="burlywood", weight=9];
163 -> 15[label="",style="solid", color="burlywood", weight=3];
13 -> 12[label="",style="dashed", color="red", weight=0];
13[label="Queue.rotate vz30 (vz4 : vz31) []\n",fontsize=16,color="magenta"];14[label="Queue.rotate (vz300 : vz301) (vz4 : vz31) []\n",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="Queue.rotate [] (vz4 : vz31) []\n",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vz300 : Queue.rotate vz301 vz31 (vz4 : [])\n",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vz4 : []\n",fontsize=16,color="green",shape="box"];18 -> 107[label="",style="dashed", color="red", weight=0];
18[label="Queue.rotate vz301 vz31 (vz4 : [])\n",fontsize=16,color="magenta"];18 -> 108[label="",style="dashed", color="magenta", weight=3];
18 -> 109[label="",style="dashed", color="magenta", weight=3];
18 -> 110[label="",style="dashed", color="magenta", weight=3];
18 -> 111[label="",style="dashed", color="magenta", weight=3];
108[label="[]\n",fontsize=16,color="green",shape="box"];109[label="vz301\n",fontsize=16,color="green",shape="box"];110[label="vz31\n",fontsize=16,color="green",shape="box"];111[label="vz4\n",fontsize=16,color="green",shape="box"];107[label="Queue.rotate vz6 vz7 (vz8 : vz9)\n",fontsize=16,color="burlywood",shape="triangle"];166[label="vz6/vz60 : vz61",fontsize=10,color="white",style="solid",shape="box"];107 -> 166[label="",style="solid", color="burlywood", weight=9];
166 -> 144[label="",style="solid", color="burlywood", weight=3];
167[label="vz6/[]",fontsize=10,color="white",style="solid",shape="box"];107 -> 167[label="",style="solid", color="burlywood", weight=9];
167 -> 145[label="",style="solid", color="burlywood", weight=3];
144[label="Queue.rotate (vz60 : vz61) vz7 (vz8 : vz9)\n",fontsize=16,color="burlywood",shape="box"];168[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];144 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 146[label="",style="solid", color="burlywood", weight=3];
169[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];144 -> 169[label="",style="solid", color="burlywood", weight=9];
169 -> 147[label="",style="solid", color="burlywood", weight=3];
145[label="Queue.rotate [] vz7 (vz8 : vz9)\n",fontsize=16,color="burlywood",shape="box"];170[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];145 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 148[label="",style="solid", color="burlywood", weight=3];
171[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];145 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 149[label="",style="solid", color="burlywood", weight=3];
146[label="Queue.rotate (vz60 : vz61) (vz70 : vz71) (vz8 : vz9)\n",fontsize=16,color="black",shape="box"];146 -> 150[label="",style="solid", color="black", weight=3];
147[label="Queue.rotate (vz60 : vz61) [] (vz8 : vz9)\n",fontsize=16,color="black",shape="box"];147 -> 151[label="",style="solid", color="black", weight=3];
148[label="Queue.rotate [] (vz70 : vz71) (vz8 : vz9)\n",fontsize=16,color="black",shape="box"];148 -> 152[label="",style="solid", color="black", weight=3];
149[label="Queue.rotate [] [] (vz8 : vz9)\n",fontsize=16,color="black",shape="box"];149 -> 153[label="",style="solid", color="black", weight=3];
150[label="vz60 : Queue.rotate vz61 vz71 (vz70 : vz8 : vz9)\n",fontsize=16,color="green",shape="box"];150 -> 154[label="",style="dashed", color="green", weight=3];
151[label="error []\n",fontsize=16,color="red",shape="box"];152[label="vz70 : vz8 : vz9\n",fontsize=16,color="green",shape="box"];153[label="error []\n",fontsize=16,color="red",shape="box"];154 -> 107[label="",style="dashed", color="red", weight=0];
154[label="Queue.rotate vz61 vz71 (vz70 : vz8 : vz9)\n",fontsize=16,color="magenta"];154 -> 155[label="",style="dashed", color="magenta", weight=3];
154 -> 156[label="",style="dashed", color="magenta", weight=3];
154 -> 157[label="",style="dashed", color="magenta", weight=3];
154 -> 158[label="",style="dashed", color="magenta", weight=3];
155[label="vz8 : vz9\n",fontsize=16,color="green",shape="box"];156[label="vz61\n",fontsize=16,color="green",shape="box"];157[label="vz71\n",fontsize=16,color="green",shape="box"];158[label="vz70\n",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ NumRed

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_rotate(:(vz60, vz61), :(vz70, vz71), vz8, vz9, h) → new_rotate(vz61, vz71, vz70, :(vz8, vz9), h)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

new_rotate(:(vz60, vz61), :(vz70, vz71), vz8, vz9, h) → new_rotate(vz61, vz71, vz70, :(vz8, vz9), h)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 5 >= 5