7.64/3.66	YES
9.49/4.16	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.49/4.16	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.49/4.16	
9.49/4.16	
9.49/4.16	H-Termination with start terms of the given HASKELL could be proven:
9.49/4.16	
9.49/4.16	(0) HASKELL
9.49/4.16	(1) BR [EQUIVALENT, 0 ms]
9.49/4.16	(2) HASKELL
9.49/4.16	(3) COR [EQUIVALENT, 0 ms]
9.49/4.16	(4) HASKELL
9.49/4.16	(5) Narrow [SOUND, 0 ms]
9.49/4.16	(6) QDP
9.49/4.16	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.49/4.16	(8) YES
9.49/4.16	
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(0)
9.49/4.16	Obligation:
9.49/4.16	mainModule Main 
9.49/4.16	module Main where {
9.49/4.16	import qualified Prelude;
9.49/4.16	import qualified Queue;
9.49/4.16	}
9.49/4.16	module Queue where {
9.49/4.16	import qualified Main;
9.49/4.16	import qualified Prelude;
9.49/4.16	data Queue a = Q [a] [a] [a] ;
9.49/4.16	
9.49/4.16	addToQueue :: Queue a  ->  a  ->  Queue a;
9.49/4.16	addToQueue (Q xs ys xs') y = makeQ xs (y : ys) xs';
9.49/4.16	
9.49/4.16	listToQueue :: [a]  ->  Queue a;
9.49/4.16	listToQueue xs = Q xs [] xs;
9.49/4.16	
9.49/4.16	makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
9.49/4.16	makeQ xs ys [] = listToQueue (rotate xs ys []);
9.49/4.16	makeQ xs ys (_ : xs') = Q xs ys xs';
9.49/4.16	
9.49/4.16	rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
9.49/4.16	rotate [] (y : _) zs = y : zs;
9.49/4.16	rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);
9.49/4.16	
9.49/4.16	}
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(1) BR (EQUIVALENT)
9.49/4.16	Replaced joker patterns by fresh variables and removed binding patterns.
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(2)
9.49/4.16	Obligation:
9.49/4.16	mainModule Main 
9.49/4.16	module Main where {
9.49/4.16	import qualified Prelude;
9.49/4.16	import qualified Queue;
9.49/4.16	}
9.49/4.16	module Queue where {
9.49/4.16	import qualified Main;
9.49/4.16	import qualified Prelude;
9.49/4.16	data Queue a = Q [a] [a] [a] ;
9.49/4.16	
9.49/4.16	addToQueue :: Queue a  ->  a  ->  Queue a;
9.49/4.16	addToQueue (Q xs ys xs') y = makeQ xs (y : ys) xs';
9.49/4.16	
9.49/4.16	listToQueue :: [a]  ->  Queue a;
9.49/4.16	listToQueue xs = Q xs [] xs;
9.49/4.16	
9.49/4.16	makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
9.49/4.16	makeQ xs ys [] = listToQueue (rotate xs ys []);
9.49/4.16	makeQ xs ys (vy : xs') = Q xs ys xs';
9.49/4.16	
9.49/4.16	rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
9.49/4.16	rotate [] (y : vx) zs = y : zs;
9.49/4.16	rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);
9.49/4.16	
9.49/4.16	}
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(3) COR (EQUIVALENT)
9.49/4.16	Cond Reductions:
9.49/4.16	The following Function with conditions
9.49/4.16	"undefined |Falseundefined;
9.49/4.16	"
9.49/4.16	is transformed to
9.49/4.16	"undefined  = undefined1;
9.49/4.16	"
9.49/4.16	"undefined0 True = undefined;
9.49/4.16	"
9.49/4.16	"undefined1  = undefined0 False;
9.49/4.16	"
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(4)
9.49/4.16	Obligation:
9.49/4.16	mainModule Main 
9.49/4.16	module Main where {
9.49/4.16	import qualified Prelude;
9.49/4.16	import qualified Queue;
9.49/4.16	}
9.49/4.16	module Queue where {
9.49/4.16	import qualified Main;
9.49/4.16	import qualified Prelude;
9.49/4.16	data Queue a = Q [a] [a] [a] ;
9.49/4.16	
9.49/4.16	addToQueue :: Queue a  ->  a  ->  Queue a;
9.49/4.16	addToQueue (Q xs ys xs') y = makeQ xs (y : ys) xs';
9.49/4.16	
9.49/4.16	listToQueue :: [a]  ->  Queue a;
9.49/4.16	listToQueue xs = Q xs [] xs;
9.49/4.16	
9.49/4.16	makeQ :: [a]  ->  [a]  ->  [a]  ->  Queue a;
9.49/4.16	makeQ xs ys [] = listToQueue (rotate xs ys []);
9.49/4.16	makeQ xs ys (vy : xs') = Q xs ys xs';
9.49/4.16	
9.49/4.16	rotate :: [a]  ->  [a]  ->  [a]  ->  [a];
9.49/4.16	rotate [] (y : vx) zs = y : zs;
9.49/4.16	rotate (x : xs) (y : ys) zs = x : rotate xs ys (y : zs);
9.49/4.16	
9.49/4.16	}
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(5) Narrow (SOUND)
9.49/4.16	Haskell To QDPs
9.49/4.16	
9.49/4.16	digraph dp_graph {
9.49/4.16	node [outthreshold=100, inthreshold=100];1[label="Queue.addToQueue",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.49/4.16	3[label="Queue.addToQueue vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.49/4.16	4[label="Queue.addToQueue vz3 vz4",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];
9.49/4.16	159 -> 5[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	5[label="Queue.addToQueue (Queue.Q vz30 vz31 vz32) vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.49/4.16	6[label="Queue.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];
9.49/4.16	160 -> 7[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	161[label="vz32/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 161[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	161 -> 8[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	7[label="Queue.makeQ vz30 (vz4 : vz31) (vz320 : vz321)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.49/4.16	8[label="Queue.makeQ vz30 (vz4 : vz31) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.49/4.16	9[label="Queue.Q vz30 (vz4 : vz31) vz321",fontsize=16,color="green",shape="box"];10[label="Queue.listToQueue (Queue.rotate vz30 (vz4 : vz31) [])",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
9.49/4.16	11[label="Queue.Q (Queue.rotate vz30 (vz4 : vz31) []) [] (Queue.rotate vz30 (vz4 : vz31) [])",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
9.49/4.16	11 -> 13[label="",style="dashed", color="green", weight=3];
9.49/4.16	12[label="Queue.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];
9.49/4.16	162 -> 14[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	163[label="vz30/[]",fontsize=10,color="white",style="solid",shape="box"];12 -> 163[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	163 -> 15[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	13 -> 12[label="",style="dashed", color="red", weight=0];
9.49/4.16	13[label="Queue.rotate vz30 (vz4 : vz31) []",fontsize=16,color="magenta"];14[label="Queue.rotate (vz300 : vz301) (vz4 : vz31) []",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
9.49/4.16	15[label="Queue.rotate [] (vz4 : vz31) []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
9.49/4.16	16[label="vz300 : Queue.rotate vz301 vz31 (vz4 : [])",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
9.49/4.16	17[label="vz4 : []",fontsize=16,color="green",shape="box"];18 -> 107[label="",style="dashed", color="red", weight=0];
9.49/4.16	18[label="Queue.rotate vz301 vz31 (vz4 : [])",fontsize=16,color="magenta"];18 -> 108[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	18 -> 109[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	18 -> 110[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	18 -> 111[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	108[label="vz301",fontsize=16,color="green",shape="box"];109[label="vz4",fontsize=16,color="green",shape="box"];110[label="[]",fontsize=16,color="green",shape="box"];111[label="vz31",fontsize=16,color="green",shape="box"];107[label="Queue.rotate vz6 vz7 (vz8 : vz9)",fontsize=16,color="burlywood",shape="triangle"];164[label="vz6/vz60 : vz61",fontsize=10,color="white",style="solid",shape="box"];107 -> 164[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	164 -> 144[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	165[label="vz6/[]",fontsize=10,color="white",style="solid",shape="box"];107 -> 165[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	165 -> 145[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	144[label="Queue.rotate (vz60 : vz61) vz7 (vz8 : vz9)",fontsize=16,color="burlywood",shape="box"];166[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];144 -> 166[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	166 -> 146[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	167[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];144 -> 167[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	167 -> 147[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	145[label="Queue.rotate [] vz7 (vz8 : vz9)",fontsize=16,color="burlywood",shape="box"];168[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];145 -> 168[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	168 -> 148[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	169[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];145 -> 169[label="",style="solid", color="burlywood", weight=9];
9.49/4.16	169 -> 149[label="",style="solid", color="burlywood", weight=3];
9.49/4.16	146[label="Queue.rotate (vz60 : vz61) (vz70 : vz71) (vz8 : vz9)",fontsize=16,color="black",shape="box"];146 -> 150[label="",style="solid", color="black", weight=3];
9.49/4.16	147[label="Queue.rotate (vz60 : vz61) [] (vz8 : vz9)",fontsize=16,color="black",shape="box"];147 -> 151[label="",style="solid", color="black", weight=3];
9.49/4.16	148[label="Queue.rotate [] (vz70 : vz71) (vz8 : vz9)",fontsize=16,color="black",shape="box"];148 -> 152[label="",style="solid", color="black", weight=3];
9.49/4.16	149[label="Queue.rotate [] [] (vz8 : vz9)",fontsize=16,color="black",shape="box"];149 -> 153[label="",style="solid", color="black", weight=3];
9.49/4.16	150[label="vz60 : Queue.rotate vz61 vz71 (vz70 : vz8 : vz9)",fontsize=16,color="green",shape="box"];150 -> 154[label="",style="dashed", color="green", weight=3];
9.49/4.16	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];
9.49/4.16	154[label="Queue.rotate vz61 vz71 (vz70 : vz8 : vz9)",fontsize=16,color="magenta"];154 -> 155[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	154 -> 156[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	154 -> 157[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	154 -> 158[label="",style="dashed", color="magenta", weight=3];
9.49/4.16	155[label="vz61",fontsize=16,color="green",shape="box"];156[label="vz70",fontsize=16,color="green",shape="box"];157[label="vz8 : vz9",fontsize=16,color="green",shape="box"];158[label="vz71",fontsize=16,color="green",shape="box"];}
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(6)
9.49/4.16	Obligation:
9.49/4.16	Q DP problem:
9.49/4.16	The TRS P consists of the following rules:
9.49/4.16	
9.49/4.16	   new_rotate(:(vz60, vz61), :(vz70, vz71), vz8, vz9, h) -> new_rotate(vz61, vz71, vz70, :(vz8, vz9), h)
9.49/4.16	
9.49/4.16	R is empty.
9.49/4.16	Q is empty.
9.49/4.16	We have to consider all minimal (P,Q,R)-chains.
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(7) QDPSizeChangeProof (EQUIVALENT)
9.49/4.16	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
9.49/4.16	
9.49/4.16	From the DPs we obtained the following set of size-change graphs:
9.49/4.16	*new_rotate(:(vz60, vz61), :(vz70, vz71), vz8, vz9, h) -> new_rotate(vz61, vz71, vz70, :(vz8, vz9), h)
9.49/4.16	The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 5 >= 5
9.49/4.16	
9.49/4.16	
9.49/4.16	----------------------------------------
9.49/4.16	
9.49/4.16	(8)
9.49/4.16	YES
9.70/4.27	EOF