10.82/4.47	YES
12.51/4.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.51/4.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.51/4.96	
12.51/4.96	
12.51/4.96	H-Termination with start terms of the given HASKELL could be proven:
12.51/4.96	
12.51/4.96	(0) HASKELL
12.51/4.96	(1) BR [EQUIVALENT, 0 ms]
12.51/4.96	(2) HASKELL
12.51/4.96	(3) COR [EQUIVALENT, 26 ms]
12.51/4.96	(4) HASKELL
12.51/4.96	(5) Narrow [SOUND, 0 ms]
12.51/4.96	(6) QDP
12.51/4.96	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.51/4.96	(8) YES
12.51/4.96	
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(0)
12.51/4.96	Obligation:
12.51/4.96	mainModule Main 
12.51/4.96	module Maybe where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	module List where {
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.51/4.96	isPrefixOf [] _ = True;
12.51/4.96	isPrefixOf _ [] = False;
12.51/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.51/4.96	
12.51/4.96	}
12.51/4.96	module Main where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(1) BR (EQUIVALENT)
12.51/4.96	Replaced joker patterns by fresh variables and removed binding patterns.
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(2)
12.51/4.96	Obligation:
12.51/4.96	mainModule Main 
12.51/4.96	module Maybe where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	module List where {
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.51/4.96	isPrefixOf [] vy = True;
12.51/4.96	isPrefixOf vz [] = False;
12.51/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.51/4.96	
12.51/4.96	}
12.51/4.96	module Main where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(3) COR (EQUIVALENT)
12.51/4.96	Cond Reductions:
12.51/4.96	The following Function with conditions
12.51/4.96	"undefined |Falseundefined;
12.51/4.96	"
12.51/4.96	is transformed to
12.51/4.96	"undefined  = undefined1;
12.51/4.96	"
12.51/4.96	"undefined0 True = undefined;
12.51/4.96	"
12.51/4.96	"undefined1  = undefined0 False;
12.51/4.96	"
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(4)
12.51/4.96	Obligation:
12.51/4.96	mainModule Main 
12.51/4.96	module Maybe where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	module List where {
12.51/4.96	import qualified Main;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	isPrefixOf :: Eq a => [a]  ->  [a]  ->  Bool;
12.51/4.96	isPrefixOf [] vy = True;
12.51/4.96	isPrefixOf vz [] = False;
12.51/4.96	isPrefixOf (x : xs) (y : ys) = x == y && isPrefixOf xs ys;
12.51/4.96	
12.51/4.96	}
12.51/4.96	module Main where {
12.51/4.96	import qualified List;
12.51/4.96	import qualified Maybe;
12.51/4.96	import qualified Prelude;
12.51/4.96	}
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(5) Narrow (SOUND)
12.51/4.96	Haskell To QDPs
12.51/4.96	
12.51/4.96	digraph dp_graph {
12.51/4.96	node [outthreshold=100, inthreshold=100];1[label="List.isPrefixOf",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.51/4.96	3[label="List.isPrefixOf wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.51/4.96	4[label="List.isPrefixOf wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];20[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	20 -> 5[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	21[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	21 -> 6[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	5[label="List.isPrefixOf (wu30 : wu31) wu4",fontsize=16,color="burlywood",shape="box"];22[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 22[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	22 -> 7[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	23[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 23[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	23 -> 8[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	6[label="List.isPrefixOf [] wu4",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
12.51/4.96	7[label="List.isPrefixOf (wu30 : wu31) (wu40 : wu41)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
12.51/4.96	8[label="List.isPrefixOf (wu30 : wu31) []",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
12.51/4.96	9[label="True",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="red", weight=0];
12.51/4.96	10[label="wu30 == wu40 && List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
12.51/4.96	11[label="False",fontsize=16,color="green",shape="box"];13 -> 4[label="",style="dashed", color="red", weight=0];
12.51/4.96	13[label="List.isPrefixOf wu31 wu41",fontsize=16,color="magenta"];13 -> 14[label="",style="dashed", color="magenta", weight=3];
12.51/4.96	13 -> 15[label="",style="dashed", color="magenta", weight=3];
12.51/4.96	12[label="wu30 == wu40 && wu5",fontsize=16,color="burlywood",shape="triangle"];24[label="wu30/()",fontsize=10,color="white",style="solid",shape="box"];12 -> 24[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	24 -> 16[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	14[label="wu41",fontsize=16,color="green",shape="box"];15[label="wu31",fontsize=16,color="green",shape="box"];16[label="() == wu40 && wu5",fontsize=16,color="burlywood",shape="box"];25[label="wu40/()",fontsize=10,color="white",style="solid",shape="box"];16 -> 25[label="",style="solid", color="burlywood", weight=9];
12.51/4.96	25 -> 17[label="",style="solid", color="burlywood", weight=3];
12.51/4.96	17[label="() == () && wu5",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
12.51/4.96	18[label="True && wu5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
12.51/4.96	19[label="wu5",fontsize=16,color="green",shape="box"];}
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(6)
12.51/4.96	Obligation:
12.51/4.96	Q DP problem:
12.51/4.96	The TRS P consists of the following rules:
12.51/4.96	
12.51/4.96	   new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
12.51/4.96	
12.51/4.96	R is empty.
12.51/4.96	Q is empty.
12.51/4.96	We have to consider all minimal (P,Q,R)-chains.
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(7) QDPSizeChangeProof (EQUIVALENT)
12.51/4.96	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. 
12.51/4.96	
12.51/4.96	From the DPs we obtained the following set of size-change graphs:
12.51/4.96	*new_isPrefixOf(:(wu30, wu31), :(wu40, wu41)) -> new_isPrefixOf(wu31, wu41)
12.51/4.96	The graph contains the following edges 1 > 1, 2 > 2
12.51/4.96	
12.51/4.96	
12.51/4.96	----------------------------------------
12.51/4.96	
12.51/4.96	(8)
12.51/4.96	YES
12.71/5.01	EOF