H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.6950000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/Queue.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule Queue

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

module Queue where

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

      ↳ COR

mainModule Queue

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

module Queue where

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)

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Queue

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

module Queue where

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="addToQueue",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="addToQueue vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="addToQueue vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];159[label="vz3/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="addToQueue (Q vz30 vz31 vz32) vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="makeQ vz30 (vz4 : vz31) vz32",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="makeQ vz30 (vz4 : vz31) (vz320 : vz321)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="makeQ vz30 (vz4 : vz31) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="Q vz30 (vz4 : vz31) vz321",fontsize=16,color="green",shape="box"];10[label="listToQueue (rotate vz30 (vz4 : vz31) [])",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
11[label="Q (rotate vz30 (vz4 : vz31) []) [] (rotate vz30 (vz4 : vz31) [])",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="rotate vz30 (vz4 : vz31) []",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="rotate vz30 (vz4 : vz31) []",fontsize=16,color="magenta"];14[label="rotate (vz300 : vz301) (vz4 : vz31) []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="rotate [] (vz4 : vz31) []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="vz300 : rotate vz301 vz31 (vz4 : [])",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="vz4 : []",fontsize=16,color="green",shape="box"];18 -> 107[label="",style="dashed", color="red", weight=0];
18[label="rotate vz301 vz31 (vz4 : [])",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="[]",fontsize=16,color="green",shape="box"];109[label="vz4",fontsize=16,color="green",shape="box"];110[label="vz31",fontsize=16,color="green",shape="box"];111[label="vz301",fontsize=16,color="green",shape="box"];107[label="rotate vz6 vz7 (vz8 : vz9)",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="rotate (vz60 : vz61) vz7 (vz8 : vz9)",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="rotate [] vz7 (vz8 : vz9)",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="rotate (vz60 : vz61) (vz70 : vz71) (vz8 : vz9)",fontsize=16,color="black",shape="box"];146 -> 150[label="",style="solid", color="black", weight=3];
147[label="rotate (vz60 : vz61) [] (vz8 : vz9)",fontsize=16,color="black",shape="box"];147 -> 151[label="",style="solid", color="black", weight=3];
148[label="rotate [] (vz70 : vz71) (vz8 : vz9)",fontsize=16,color="black",shape="box"];148 -> 152[label="",style="solid", color="black", weight=3];
149[label="rotate [] [] (vz8 : vz9)",fontsize=16,color="black",shape="box"];149 -> 153[label="",style="solid", color="black", weight=3];
150[label="vz60 : rotate vz61 vz71 (vz70 : vz8 : vz9)",fontsize=16,color="green",shape="box"];150 -> 154[label="",style="dashed", color="green", weight=3];
151[label="error []",fontsize=16,color="red",shape="box"];152[label="vz70 : vz8 : vz9",fontsize=16,color="green",shape="box"];153[label="error []",fontsize=16,color="red",shape="box"];154 -> 107[label="",style="dashed", color="red", weight=0];
154[label="rotate vz61 vz71 (vz70 : vz8 : vz9)",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",fontsize=16,color="green",shape="box"];156[label="vz70",fontsize=16,color="green",shape="box"];157[label="vz71",fontsize=16,color="green",shape="box"];158[label="vz61",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ 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