7.91/3.54	YES
9.49/3.96	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.49/3.96	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.49/3.96	
9.49/3.96	
9.49/3.96	H-Termination with start terms of the given HASKELL could be proven:
9.49/3.96	
9.49/3.96	(0) HASKELL
9.49/3.96	(1) BR [EQUIVALENT, 0 ms]
9.49/3.96	(2) HASKELL
9.49/3.96	(3) COR [EQUIVALENT, 0 ms]
9.49/3.96	(4) HASKELL
9.49/3.96	(5) Narrow [SOUND, 0 ms]
9.49/3.96	(6) QDP
9.49/3.96	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
9.49/3.96	(8) AND
9.49/3.96	    (9) QDP
9.49/3.96	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.49/3.96	        (11) YES
9.49/3.96	    (12) QDP
9.49/3.96	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.49/3.96	        (14) YES
9.49/3.96	
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(0)
9.49/3.96	Obligation:
9.49/3.96	mainModule Main 
9.49/3.96	module Main where {
9.49/3.96	import qualified Prelude;
9.49/3.96	}
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(1) BR (EQUIVALENT)
9.49/3.96	Replaced joker patterns by fresh variables and removed binding patterns.
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(2)
9.49/3.96	Obligation:
9.49/3.96	mainModule Main 
9.49/3.96	module Main where {
9.49/3.96	import qualified Prelude;
9.49/3.96	}
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(3) COR (EQUIVALENT)
9.49/3.96	Cond Reductions:
9.49/3.96	The following Function with conditions
9.49/3.96	"lookup k [] = Nothing;
9.49/3.96	lookup k ((x,y) : xys)|k == xJust y|otherwiselookup k xys;
9.49/3.96	"
9.49/3.96	is transformed to
9.49/3.96	"lookup k [] = lookup3 k [];
9.49/3.96	lookup k ((x,y) : xys) = lookup2 k ((x,y) : xys);
9.49/3.96	"
9.49/3.96	"lookup0 k x y xys True = lookup k xys;
9.49/3.96	"
9.49/3.96	"lookup1 k x y xys True = Just y;
9.49/3.96	lookup1 k x y xys False = lookup0 k x y xys otherwise;
9.49/3.96	"
9.49/3.96	"lookup2 k ((x,y) : xys) = lookup1 k x y xys (k == x);
9.49/3.96	"
9.49/3.96	"lookup3 k [] = Nothing;
9.49/3.96	lookup3 wu wv = lookup2 wu wv;
9.49/3.96	"
9.49/3.96	The following Function with conditions
9.49/3.96	"undefined |Falseundefined;
9.49/3.96	"
9.49/3.96	is transformed to
9.49/3.96	"undefined  = undefined1;
9.49/3.96	"
9.49/3.96	"undefined0 True = undefined;
9.49/3.96	"
9.49/3.96	"undefined1  = undefined0 False;
9.49/3.96	"
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(4)
9.49/3.96	Obligation:
9.49/3.96	mainModule Main 
9.49/3.96	module Main where {
9.49/3.96	import qualified Prelude;
9.49/3.96	}
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(5) Narrow (SOUND)
9.49/3.96	Haskell To QDPs
9.49/3.96	
9.49/3.96	digraph dp_graph {
9.49/3.96	node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.49/3.96	3[label="lookup ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.49/3.96	4[label="lookup ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];34[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	34 -> 5[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	35[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 35[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	35 -> 6[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	5[label="lookup ww3 (ww40 : ww41)",fontsize=16,color="burlywood",shape="box"];36[label="ww40/(ww400,ww401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 36[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	36 -> 7[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	6[label="lookup ww3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.49/3.96	7[label="lookup ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.49/3.96	8[label="lookup3 ww3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.49/3.96	9[label="lookup2 ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.49/3.96	10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 ww3 ww400 ww401 ww41 (ww3 == ww400)",fontsize=16,color="burlywood",shape="box"];37[label="ww3/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 37[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	37 -> 12[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	38[label="ww3/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 38[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	38 -> 13[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	12[label="lookup1 False ww400 ww401 ww41 (False == ww400)",fontsize=16,color="burlywood",shape="box"];39[label="ww400/False",fontsize=10,color="white",style="solid",shape="box"];12 -> 39[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	39 -> 14[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	40[label="ww400/True",fontsize=10,color="white",style="solid",shape="box"];12 -> 40[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	40 -> 15[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	13[label="lookup1 True ww400 ww401 ww41 (True == ww400)",fontsize=16,color="burlywood",shape="box"];41[label="ww400/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	41 -> 16[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	42[label="ww400/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 42[label="",style="solid", color="burlywood", weight=9];
9.49/3.96	42 -> 17[label="",style="solid", color="burlywood", weight=3];
9.49/3.96	14[label="lookup1 False False ww401 ww41 (False == False)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
9.49/3.96	15[label="lookup1 False True ww401 ww41 (False == True)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
9.49/3.96	16[label="lookup1 True False ww401 ww41 (True == False)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
9.49/3.96	17[label="lookup1 True True ww401 ww41 (True == True)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
9.49/3.96	18[label="lookup1 False False ww401 ww41 True",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
9.49/3.96	19[label="lookup1 False True ww401 ww41 False",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
9.49/3.96	20[label="lookup1 True False ww401 ww41 False",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
9.49/3.96	21[label="lookup1 True True ww401 ww41 True",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.49/3.96	22[label="Just ww401",fontsize=16,color="green",shape="box"];23[label="lookup0 False True ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.49/3.96	24[label="lookup0 True False ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.49/3.96	25[label="Just ww401",fontsize=16,color="green",shape="box"];26[label="lookup0 False True ww401 ww41 True",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
9.49/3.96	27[label="lookup0 True False ww401 ww41 True",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
9.49/3.96	28 -> 4[label="",style="dashed", color="red", weight=0];
9.49/3.96	28[label="lookup False ww41",fontsize=16,color="magenta"];28 -> 30[label="",style="dashed", color="magenta", weight=3];
9.49/3.96	28 -> 31[label="",style="dashed", color="magenta", weight=3];
9.49/3.96	29 -> 4[label="",style="dashed", color="red", weight=0];
9.49/3.96	29[label="lookup True ww41",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
9.49/3.96	29 -> 33[label="",style="dashed", color="magenta", weight=3];
9.49/3.96	30[label="False",fontsize=16,color="green",shape="box"];31[label="ww41",fontsize=16,color="green",shape="box"];32[label="True",fontsize=16,color="green",shape="box"];33[label="ww41",fontsize=16,color="green",shape="box"];}
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(6)
9.49/3.96	Obligation:
9.49/3.96	Q DP problem:
9.49/3.96	The TRS P consists of the following rules:
9.49/3.96	
9.49/3.96	   new_lookup(True, :(@2(False, ww401), ww41), h) -> new_lookup(True, ww41, h)
9.49/3.96	   new_lookup(False, :(@2(True, ww401), ww41), h) -> new_lookup(False, ww41, h)
9.49/3.96	
9.49/3.96	R is empty.
9.49/3.96	Q is empty.
9.49/3.96	We have to consider all minimal (P,Q,R)-chains.
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(7) DependencyGraphProof (EQUIVALENT)
9.49/3.96	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs.
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(8)
9.49/3.96	Complex Obligation (AND)
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(9)
9.49/3.96	Obligation:
9.49/3.96	Q DP problem:
9.49/3.96	The TRS P consists of the following rules:
9.49/3.96	
9.49/3.96	   new_lookup(False, :(@2(True, ww401), ww41), h) -> new_lookup(False, ww41, h)
9.49/3.96	
9.49/3.96	R is empty.
9.49/3.96	Q is empty.
9.49/3.96	We have to consider all minimal (P,Q,R)-chains.
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(10) QDPSizeChangeProof (EQUIVALENT)
9.49/3.96	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. 
9.49/3.96	
9.49/3.96	From the DPs we obtained the following set of size-change graphs:
9.49/3.96	*new_lookup(False, :(@2(True, ww401), ww41), h) -> new_lookup(False, ww41, h)
9.49/3.96	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.49/3.96	
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(11)
9.49/3.96	YES
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(12)
9.49/3.96	Obligation:
9.49/3.96	Q DP problem:
9.49/3.96	The TRS P consists of the following rules:
9.49/3.96	
9.49/3.96	   new_lookup(True, :(@2(False, ww401), ww41), h) -> new_lookup(True, ww41, h)
9.49/3.96	
9.49/3.96	R is empty.
9.49/3.96	Q is empty.
9.49/3.96	We have to consider all minimal (P,Q,R)-chains.
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(13) QDPSizeChangeProof (EQUIVALENT)
9.49/3.96	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. 
9.49/3.96	
9.49/3.96	From the DPs we obtained the following set of size-change graphs:
9.49/3.96	*new_lookup(True, :(@2(False, ww401), ww41), h) -> new_lookup(True, ww41, h)
9.49/3.96	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
9.49/3.96	
9.49/3.96	
9.49/3.96	----------------------------------------
9.49/3.96	
9.49/3.96	(14)
9.49/3.96	YES
9.59/4.03	EOF