10.28/4.40	YES
11.88/4.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.88/4.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.88/4.86	
11.88/4.86	
11.88/4.86	H-Termination with start terms of the given HASKELL could be proven:
11.88/4.86	
11.88/4.86	(0) HASKELL
11.88/4.86	(1) BR [EQUIVALENT, 0 ms]
11.88/4.86	(2) HASKELL
11.88/4.86	(3) COR [EQUIVALENT, 0 ms]
11.88/4.86	(4) HASKELL
11.88/4.86	(5) Narrow [SOUND, 0 ms]
11.88/4.86	(6) QDP
11.88/4.86	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.88/4.86	(8) YES
11.88/4.86	
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(0)
11.88/4.86	Obligation:
11.88/4.86	mainModule Main 
11.88/4.86	module Maybe where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	module List where {
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
11.88/4.86	isPrefixOf [] _ = True;
11.88/4.86	isPrefixOf _ [] = False;
11.88/4.86	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
11.88/4.86	
11.88/4.86	}
11.88/4.86	module Main where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(1) BR (EQUIVALENT)
11.88/4.86	Replaced joker patterns by fresh variables and removed binding patterns.
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(2)
11.88/4.86	Obligation:
11.88/4.86	mainModule Main 
11.88/4.86	module Maybe where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	module List where {
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
11.88/4.86	isPrefixOf [] vy = True;
11.88/4.86	isPrefixOf vz [] = False;
11.88/4.86	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
11.88/4.86	
11.88/4.86	}
11.88/4.86	module Main where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(3) COR (EQUIVALENT)
11.88/4.86	Cond Reductions:
11.88/4.86	The following Function with conditions
11.88/4.86	"undefined |Falseundefined;
11.88/4.86	"
11.88/4.86	is transformed to
11.88/4.86	"undefined  = undefined1;
11.88/4.86	"
11.88/4.86	"undefined0 True = undefined;
11.88/4.86	"
11.88/4.86	"undefined1  = undefined0 False;
11.88/4.86	"
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(4)
11.88/4.86	Obligation:
11.88/4.86	mainModule Main 
11.88/4.86	module Maybe where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	module List where {
11.88/4.86	import qualified Main;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
11.88/4.86	isPrefixOf [] vy = True;
11.88/4.86	isPrefixOf vz [] = False;
11.88/4.86	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
11.88/4.86	
11.88/4.86	}
11.88/4.86	module Main where {
11.88/4.86	import qualified List;
11.88/4.86	import qualified Maybe;
11.88/4.86	import qualified Prelude;
11.88/4.86	}
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(5) Narrow (SOUND)
11.88/4.86	Haskell To QDPs
11.88/4.86	
11.88/4.86	digraph dp_graph {
11.88/4.86	node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.88/4.86	3[label="List.isPrefixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.88/4.86	4[label="List.isPrefixOf wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];28[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];4 -> 28[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	28 -> 5[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	29[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	29 -> 6[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	5[label="List.isPrefixOf (wu30 : wu31) wu4",fontsize=16,color="burlywood",shape="box"];30[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 30[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	30 -> 7[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	31[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	31 -> 8[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	6[label="List.isPrefixOf [] wu4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
11.88/4.86	7[label="List.isPrefixOf (wu30 : wu31) (wu40 : wu41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
11.88/4.86	8[label="List.isPrefixOf (wu30 : wu31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
11.88/4.86	9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
11.88/4.86	10[label="wu30 == wu40 && List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
11.88/4.86	11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
11.88/4.86	13[label="List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
11.88/4.86	13 -> 15[label="",style="dashed", color="magenta", weight=3];
11.88/4.86	12[label="wu30 == wu40 && wu5",fontsize=16,color="burlywood",shape="triangle"];32[label="wu30/False",fontsize=10,color="white",style="solid",shape="box"];12 -> 32[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	32 -> 16[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	33[label="wu30/True",fontsize=10,color="white",style="solid",shape="box"];12 -> 33[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	33 -> 17[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	14[label="wu31",fontsize=16,color="green",shape="box"];15[label="wu41",fontsize=16,color="green",shape="box"];16[label="False == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];34[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 34[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	34 -> 18[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	35[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 35[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	35 -> 19[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	17[label="True == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];36[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 36[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	36 -> 20[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	37[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 37[label="",style="solid", color="burlywood", weight=9];
11.88/4.86	37 -> 21[label="",style="solid", color="burlywood", weight=3];
11.88/4.86	18[label="False == False && wu5",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
11.88/4.86	19[label="False == True && wu5",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
11.88/4.86	20[label="True == False && wu5",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
11.88/4.86	21[label="True == True && wu5",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
11.88/4.86	22[label="True && wu5",fontsize=16,color="black",shape="triangle"];22 -> 26[label="",style="solid", color="black", weight=3];
11.88/4.86	23[label="False && wu5",fontsize=16,color="black",shape="triangle"];23 -> 27[label="",style="solid", color="black", weight=3];
11.88/4.86	24 -> 23[label="",style="dashed", color="red", weight=0];
11.88/4.86	24[label="False && wu5",fontsize=16,color="magenta"];25 -> 22[label="",style="dashed", color="red", weight=0];
11.88/4.86	25[label="True && wu5",fontsize=16,color="magenta"];26[label="wu5",fontsize=16,color="green",shape="box"];27[label="False",fontsize=16,color="green",shape="box"];}
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(6)
11.88/4.86	Obligation:
11.88/4.86	Q DP problem:
11.88/4.86	The TRS P consists of the following rules:
11.88/4.86	
11.88/4.86	   new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
11.88/4.86	
11.88/4.86	R is empty.
11.88/4.86	Q is empty.
11.88/4.86	We have to consider all minimal (P,Q,R)-chains.
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(7) QDPSizeChangeProof (EQUIVALENT)
11.88/4.86	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. 
11.88/4.86	
11.88/4.86	From the DPs we obtained the following set of size-change graphs:
11.88/4.86	*new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
11.88/4.86	The graph contains the following edges 1 > 1, 2 > 2
11.88/4.86	
11.88/4.86	
11.88/4.86	----------------------------------------
11.88/4.86	
11.88/4.86	(8)
11.88/4.86	YES
12.11/4.90	EOF