10.64/4.40	YES
12.48/4.91	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.48/4.91	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.48/4.91	
12.48/4.91	
12.48/4.91	H-Termination with start terms of the given HASKELL could be proven:
12.48/4.91	
12.48/4.91	(0) HASKELL
12.48/4.91	(1) BR [EQUIVALENT, 0 ms]
12.48/4.91	(2) HASKELL
12.48/4.91	(3) COR [EQUIVALENT, 0 ms]
12.48/4.91	(4) HASKELL
12.48/4.91	(5) Narrow [SOUND, 0 ms]
12.48/4.91	(6) QDP
12.48/4.91	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.48/4.91	(8) YES
12.48/4.91	
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(0)
12.48/4.91	Obligation:
12.48/4.91	mainModule Main 
12.48/4.91	module Maybe where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	module List where {
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.48/4.91	isPrefixOf [] _ = True;
12.48/4.91	isPrefixOf _ [] = False;
12.48/4.91	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.48/4.91	
12.48/4.91	}
12.48/4.91	module Main where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(1) BR (EQUIVALENT)
12.48/4.91	Replaced joker patterns by fresh variables and removed binding patterns.
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(2)
12.48/4.91	Obligation:
12.48/4.91	mainModule Main 
12.48/4.91	module Maybe where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	module List where {
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.48/4.91	isPrefixOf [] vy = True;
12.48/4.91	isPrefixOf vz [] = False;
12.48/4.91	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.48/4.91	
12.48/4.91	}
12.48/4.91	module Main where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(3) COR (EQUIVALENT)
12.48/4.91	Cond Reductions:
12.48/4.91	The following Function with conditions
12.48/4.91	"undefined |Falseundefined;
12.48/4.91	"
12.48/4.91	is transformed to
12.48/4.91	"undefined  = undefined1;
12.48/4.91	"
12.48/4.91	"undefined0 True = undefined;
12.48/4.91	"
12.48/4.91	"undefined1  = undefined0 False;
12.48/4.91	"
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(4)
12.48/4.91	Obligation:
12.48/4.91	mainModule Main 
12.48/4.91	module Maybe where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	module List where {
12.48/4.91	import qualified Main;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.48/4.91	isPrefixOf [] vy = True;
12.48/4.91	isPrefixOf vz [] = False;
12.48/4.91	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.48/4.91	
12.48/4.91	}
12.48/4.91	module Main where {
12.48/4.91	import qualified List;
12.48/4.91	import qualified Maybe;
12.48/4.91	import qualified Prelude;
12.48/4.91	}
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(5) Narrow (SOUND)
12.48/4.91	Haskell To QDPs
12.48/4.91	
12.48/4.91	digraph dp_graph {
12.48/4.91	node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.48/4.91	3[label="List.isPrefixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.48/4.91	4[label="List.isPrefixOf wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];39[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	39 -> 5[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	40[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	40 -> 6[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	5[label="List.isPrefixOf (wu30 : wu31) wu4",fontsize=16,color="burlywood",shape="box"];41[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 41[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	41 -> 7[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	42[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 42[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	42 -> 8[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	6[label="List.isPrefixOf [] wu4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
12.48/4.91	7[label="List.isPrefixOf (wu30 : wu31) (wu40 : wu41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
12.48/4.91	8[label="List.isPrefixOf (wu30 : wu31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
12.48/4.91	9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
12.48/4.91	10[label="wu30 == wu40 && List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
12.48/4.91	11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
12.48/4.91	13[label="List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
12.48/4.91	13 -> 15[label="",style="dashed", color="magenta", weight=3];
12.48/4.91	12[label="wu30 == wu40 && wu5",fontsize=16,color="burlywood",shape="triangle"];43[label="wu30/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 43[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	43 -> 16[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	44[label="wu30/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 44[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	44 -> 17[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	45[label="wu30/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 45[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	45 -> 18[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	14[label="wu41",fontsize=16,color="green",shape="box"];15[label="wu31",fontsize=16,color="green",shape="box"];16[label="LT == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];46[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 46[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	46 -> 19[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	47[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 47[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	47 -> 20[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	48[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 48[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	48 -> 21[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	17[label="EQ == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];49[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 49[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	49 -> 22[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	50[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 50[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	50 -> 23[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	51[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 51[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	51 -> 24[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	18[label="GT == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];52[label="wu40/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 52[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	52 -> 25[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	53[label="wu40/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 53[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	53 -> 26[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	54[label="wu40/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 54[label="",style="solid", color="burlywood", weight=9];
12.48/4.91	54 -> 27[label="",style="solid", color="burlywood", weight=3];
12.48/4.91	19[label="LT == LT && wu5",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
12.48/4.91	20[label="LT == EQ && wu5",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
12.48/4.91	21[label="LT == GT && wu5",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
12.48/4.91	22[label="EQ == LT && wu5",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
12.48/4.91	23[label="EQ == EQ && wu5",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
12.48/4.91	24[label="EQ == GT && wu5",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
12.48/4.91	25[label="GT == LT && wu5",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
12.48/4.91	26[label="GT == EQ && wu5",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
12.48/4.91	27[label="GT == GT && wu5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
12.48/4.91	28[label="True && wu5",fontsize=16,color="black",shape="triangle"];28 -> 37[label="",style="solid", color="black", weight=3];
12.48/4.91	29[label="False && wu5",fontsize=16,color="black",shape="triangle"];29 -> 38[label="",style="solid", color="black", weight=3];
12.48/4.91	30 -> 29[label="",style="dashed", color="red", weight=0];
12.48/4.91	30[label="False && wu5",fontsize=16,color="magenta"];31 -> 29[label="",style="dashed", color="red", weight=0];
12.48/4.91	31[label="False && wu5",fontsize=16,color="magenta"];32 -> 28[label="",style="dashed", color="red", weight=0];
12.48/4.91	32[label="True && wu5",fontsize=16,color="magenta"];33 -> 29[label="",style="dashed", color="red", weight=0];
12.48/4.91	33[label="False && wu5",fontsize=16,color="magenta"];34 -> 29[label="",style="dashed", color="red", weight=0];
12.48/4.91	34[label="False && wu5",fontsize=16,color="magenta"];35 -> 29[label="",style="dashed", color="red", weight=0];
12.48/4.91	35[label="False && wu5",fontsize=16,color="magenta"];36 -> 28[label="",style="dashed", color="red", weight=0];
12.48/4.91	36[label="True && wu5",fontsize=16,color="magenta"];37[label="wu5",fontsize=16,color="green",shape="box"];38[label="False",fontsize=16,color="green",shape="box"];}
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(6)
12.48/4.91	Obligation:
12.48/4.91	Q DP problem:
12.48/4.91	The TRS P consists of the following rules:
12.48/4.91	
12.48/4.91	   new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
12.48/4.91	
12.48/4.91	R is empty.
12.48/4.91	Q is empty.
12.48/4.91	We have to consider all minimal (P,Q,R)-chains.
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(7) QDPSizeChangeProof (EQUIVALENT)
12.48/4.91	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. 
12.48/4.91	
12.48/4.91	From the DPs we obtained the following set of size-change graphs:
12.48/4.91	*new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
12.48/4.91	The graph contains the following edges 1 > 1, 2 > 2
12.48/4.91	
12.48/4.91	
12.48/4.91	----------------------------------------
12.48/4.91	
12.48/4.91	(8)
12.48/4.91	YES
12.64/4.98	EOF