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

YES 0.706 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

((deQueue :: Queue a -> Maybe (a,Queue a)) :: Queue a -> Maybe (a,Queue a))

module Queue where

import qualified Prelude

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

deQueue :: Queue a -> Maybe (a,Queue a)

deQueue	(Q [] _ _)	=	Nothing
deQueue	(Q (x : xs) ys xs')	=	Just (x,makeQ xs 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

((deQueue :: Queue a -> Maybe (a,Queue a)) :: Queue a -> Maybe (a,Queue a))

module Queue where

import qualified Prelude

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

deQueue :: Queue a -> Maybe (a,Queue a)

deQueue	(Q [] vv vw)	=	Nothing
deQueue	(Q (x : xs) ys xs')	=	Just (x,makeQ xs 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 (vx : xs')	=	Q xs ys xs'

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

rotate	[] (y : vy) 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

(deQueue :: Queue a -> Maybe (a,Queue a))

module Queue where

import qualified Prelude

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

deQueue :: Queue a -> Maybe (a,Queue a)

deQueue	(Q [] vv vw)	=	Nothing
deQueue	(Q (x : xs) ys xs')	=	Just (x,makeQ xs 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 (vx : xs')	=	Q xs ys xs'

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

rotate	[] (y : vy) 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="deQueue",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="deQueue wv3",fontsize=16,color="burlywood",shape="triangle"];168[label="wv3/Q wv30 wv31 wv32",fontsize=10,color="white",style="solid",shape="box"];3 -> 168[label="",style="solid", color="burlywood", weight=9];
168 -> 4[label="",style="solid", color="burlywood", weight=3];
4[label="deQueue (Q wv30 wv31 wv32)",fontsize=16,color="burlywood",shape="box"];169[label="wv30/wv300 : wv301",fontsize=10,color="white",style="solid",shape="box"];4 -> 169[label="",style="solid", color="burlywood", weight=9];
169 -> 5[label="",style="solid", color="burlywood", weight=3];
170[label="wv30/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 170[label="",style="solid", color="burlywood", weight=9];
170 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="deQueue (Q (wv300 : wv301) wv31 wv32)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="deQueue (Q [] wv31 wv32)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="Just (wv300,makeQ wv301 wv31 wv32)",fontsize=16,color="green",shape="box"];7 -> 9[label="",style="dashed", color="green", weight=3];
8[label="Nothing",fontsize=16,color="green",shape="box"];9[label="makeQ wv301 wv31 wv32",fontsize=16,color="burlywood",shape="box"];171[label="wv32/wv320 : wv321",fontsize=10,color="white",style="solid",shape="box"];9 -> 171[label="",style="solid", color="burlywood", weight=9];
171 -> 10[label="",style="solid", color="burlywood", weight=3];
172[label="wv32/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 172[label="",style="solid", color="burlywood", weight=9];
172 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="makeQ wv301 wv31 (wv320 : wv321)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="makeQ wv301 wv31 []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="Q wv301 wv31 wv321",fontsize=16,color="green",shape="box"];13[label="listToQueue (rotate wv301 wv31 [])",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="Q (rotate wv301 wv31 []) [] (rotate wv301 wv31 [])",fontsize=16,color="green",shape="box"];14 -> 15[label="",style="dashed", color="green", weight=3];
14 -> 16[label="",style="dashed", color="green", weight=3];
15[label="rotate wv301 wv31 []",fontsize=16,color="burlywood",shape="triangle"];173[label="wv301/wv3010 : wv3011",fontsize=10,color="white",style="solid",shape="box"];15 -> 173[label="",style="solid", color="burlywood", weight=9];
173 -> 17[label="",style="solid", color="burlywood", weight=3];
174[label="wv301/[]",fontsize=10,color="white",style="solid",shape="box"];15 -> 174[label="",style="solid", color="burlywood", weight=9];
174 -> 18[label="",style="solid", color="burlywood", weight=3];
16 -> 15[label="",style="dashed", color="red", weight=0];
16[label="rotate wv301 wv31 []",fontsize=16,color="magenta"];17[label="rotate (wv3010 : wv3011) wv31 []",fontsize=16,color="burlywood",shape="box"];176[label="wv31/wv310 : wv311",fontsize=10,color="white",style="solid",shape="box"];17 -> 176[label="",style="solid", color="burlywood", weight=9];
176 -> 19[label="",style="solid", color="burlywood", weight=3];
177[label="wv31/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 177[label="",style="solid", color="burlywood", weight=9];
177 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="rotate [] wv31 []",fontsize=16,color="burlywood",shape="box"];178[label="wv31/wv310 : wv311",fontsize=10,color="white",style="solid",shape="box"];18 -> 178[label="",style="solid", color="burlywood", weight=9];
178 -> 21[label="",style="solid", color="burlywood", weight=3];
179[label="wv31/[]",fontsize=10,color="white",style="solid",shape="box"];18 -> 179[label="",style="solid", color="burlywood", weight=9];
179 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="rotate (wv3010 : wv3011) (wv310 : wv311) []",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="rotate (wv3010 : wv3011) [] []",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="rotate [] (wv310 : wv311) []",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="rotate [] [] []",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="wv3010 : rotate wv3011 wv311 (wv310 : [])",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="green", weight=3];
24[label="error []",fontsize=16,color="red",shape="box"];25[label="wv310 : []",fontsize=16,color="green",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];27 -> 116[label="",style="dashed", color="red", weight=0];
27[label="rotate wv3011 wv311 (wv310 : [])",fontsize=16,color="magenta"];27 -> 117[label="",style="dashed", color="magenta", weight=3];
27 -> 118[label="",style="dashed", color="magenta", weight=3];
27 -> 119[label="",style="dashed", color="magenta", weight=3];
27 -> 120[label="",style="dashed", color="magenta", weight=3];
117[label="wv3011",fontsize=16,color="green",shape="box"];118[label="wv310",fontsize=16,color="green",shape="box"];119[label="[]",fontsize=16,color="green",shape="box"];120[label="wv311",fontsize=16,color="green",shape="box"];116[label="rotate wv5 wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="triangle"];181[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];116 -> 181[label="",style="solid", color="burlywood", weight=9];
181 -> 153[label="",style="solid", color="burlywood", weight=3];
182[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];116 -> 182[label="",style="solid", color="burlywood", weight=9];
182 -> 154[label="",style="solid", color="burlywood", weight=3];
153[label="rotate (wv50 : wv51) wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="box"];183[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];153 -> 183[label="",style="solid", color="burlywood", weight=9];
183 -> 155[label="",style="solid", color="burlywood", weight=3];
184[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];153 -> 184[label="",style="solid", color="burlywood", weight=9];
184 -> 156[label="",style="solid", color="burlywood", weight=3];
154[label="rotate [] wv6 (wv7 : wv8)",fontsize=16,color="burlywood",shape="box"];185[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];154 -> 185[label="",style="solid", color="burlywood", weight=9];
185 -> 157[label="",style="solid", color="burlywood", weight=3];
186[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];154 -> 186[label="",style="solid", color="burlywood", weight=9];
186 -> 158[label="",style="solid", color="burlywood", weight=3];
155[label="rotate (wv50 : wv51) (wv60 : wv61) (wv7 : wv8)",fontsize=16,color="black",shape="box"];155 -> 159[label="",style="solid", color="black", weight=3];
156[label="rotate (wv50 : wv51) [] (wv7 : wv8)",fontsize=16,color="black",shape="box"];156 -> 160[label="",style="solid", color="black", weight=3];
157[label="rotate [] (wv60 : wv61) (wv7 : wv8)",fontsize=16,color="black",shape="box"];157 -> 161[label="",style="solid", color="black", weight=3];
158[label="rotate [] [] (wv7 : wv8)",fontsize=16,color="black",shape="box"];158 -> 162[label="",style="solid", color="black", weight=3];
159[label="wv50 : rotate wv51 wv61 (wv60 : wv7 : wv8)",fontsize=16,color="green",shape="box"];159 -> 163[label="",style="dashed", color="green", weight=3];
160[label="error []",fontsize=16,color="red",shape="box"];161[label="wv60 : wv7 : wv8",fontsize=16,color="green",shape="box"];162[label="error []",fontsize=16,color="red",shape="box"];163 -> 116[label="",style="dashed", color="red", weight=0];
163[label="rotate wv51 wv61 (wv60 : wv7 : wv8)",fontsize=16,color="magenta"];163 -> 164[label="",style="dashed", color="magenta", weight=3];
163 -> 165[label="",style="dashed", color="magenta", weight=3];
163 -> 166[label="",style="dashed", color="magenta", weight=3];
163 -> 167[label="",style="dashed", color="magenta", weight=3];
164[label="wv51",fontsize=16,color="green",shape="box"];165[label="wv60",fontsize=16,color="green",shape="box"];166[label="wv7 : wv8",fontsize=16,color="green",shape="box"];167[label="wv61",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(:(wv50, wv51), :(wv60, wv61), wv7, wv8, h) → new_rotate(wv51, wv61, wv60, :(wv7, wv8), 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(:(wv50, wv51), :(wv60, wv61), wv7, wv8, h) → new_rotate(wv51, wv61, wv60, :(wv7, wv8), h)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 5 >= 5