8.31/3.64	YES
9.82/4.10	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.82/4.10	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.82/4.10	
9.82/4.10	
9.82/4.10	H-Termination with start terms of the given HASKELL could be proven:
9.82/4.10	
9.82/4.10	(0) HASKELL
9.82/4.10	(1) BR [EQUIVALENT, 0 ms]
9.82/4.10	(2) HASKELL
9.82/4.10	(3) COR [EQUIVALENT, 0 ms]
9.82/4.10	(4) HASKELL
9.82/4.10	(5) Narrow [SOUND, 0 ms]
9.82/4.10	(6) QDP
9.82/4.10	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.82/4.10	(8) YES
9.82/4.10	
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(0)
9.82/4.10	Obligation:
9.82/4.10	mainModule Main 
9.82/4.10	module Main where {
9.82/4.10	import qualified Prelude;
9.82/4.10	}
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(1) BR (EQUIVALENT)
9.82/4.10	Replaced joker patterns by fresh variables and removed binding patterns.
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(2)
9.82/4.10	Obligation:
9.82/4.10	mainModule Main 
9.82/4.10	module Main where {
9.82/4.10	import qualified Prelude;
9.82/4.10	}
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(3) COR (EQUIVALENT)
9.82/4.10	Cond Reductions:
9.82/4.10	The following Function with conditions
9.82/4.10	"min x y|x <= yx|otherwisey;
9.82/4.10	"
9.82/4.10	is transformed to
9.82/4.10	"min x y = min2 x y;
9.82/4.10	"
9.82/4.10	"min0 x y True = y;
9.82/4.10	"
9.82/4.10	"min1 x y True = x;
9.82/4.10	min1 x y False = min0 x y otherwise;
9.82/4.10	"
9.82/4.10	"min2 x y = min1 x y (x <= y);
9.82/4.10	"
9.82/4.10	The following Function with conditions
9.82/4.10	"undefined |Falseundefined;
9.82/4.10	"
9.82/4.10	is transformed to
9.82/4.10	"undefined  = undefined1;
9.82/4.10	"
9.82/4.10	"undefined0 True = undefined;
9.82/4.10	"
9.82/4.10	"undefined1  = undefined0 False;
9.82/4.10	"
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(4)
9.82/4.10	Obligation:
9.82/4.10	mainModule Main 
9.82/4.10	module Main where {
9.82/4.10	import qualified Prelude;
9.82/4.10	}
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(5) Narrow (SOUND)
9.82/4.10	Haskell To QDPs
9.82/4.10	
9.82/4.10	digraph dp_graph {
9.82/4.10	node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.82/4.10	3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.82/4.10	4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];53[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 53[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	53 -> 5[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	54[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	54 -> 6[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.82/4.10	6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.82/4.10	7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];55[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 55[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	55 -> 9[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	56[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 56[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	56 -> 10[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl min vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.82/4.10	10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.82/4.10	11 -> 7[label="",style="dashed", color="red", weight=0];
9.82/4.10	11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
9.82/4.10	11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.82/4.10	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="min vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
9.82/4.10	15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.82/4.10	16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="burlywood",shape="box"];57[label="vx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 57[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	57 -> 17[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	58[label="vx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 58[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	58 -> 18[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	59[label="vx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	59 -> 19[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	17[label="min1 LT vx310 (LT <= vx310)",fontsize=16,color="burlywood",shape="box"];60[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 60[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	60 -> 20[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	61[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	61 -> 21[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	62[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	62 -> 22[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	18[label="min1 EQ vx310 (EQ <= vx310)",fontsize=16,color="burlywood",shape="box"];63[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 63[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	63 -> 23[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	64[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 64[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	64 -> 24[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	65[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 65[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	65 -> 25[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	19[label="min1 GT vx310 (GT <= vx310)",fontsize=16,color="burlywood",shape="box"];66[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 66[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	66 -> 26[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	67[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 67[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	67 -> 27[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	68[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 68[label="",style="solid", color="burlywood", weight=9];
9.82/4.10	68 -> 28[label="",style="solid", color="burlywood", weight=3];
9.82/4.10	20[label="min1 LT LT (LT <= LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
9.82/4.10	21[label="min1 LT EQ (LT <= EQ)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
9.82/4.10	22[label="min1 LT GT (LT <= GT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
9.82/4.10	23[label="min1 EQ LT (EQ <= LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
9.82/4.10	24[label="min1 EQ EQ (EQ <= EQ)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
9.82/4.10	25[label="min1 EQ GT (EQ <= GT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
9.82/4.10	26[label="min1 GT LT (GT <= LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
9.82/4.10	27[label="min1 GT EQ (GT <= EQ)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
9.82/4.10	28[label="min1 GT GT (GT <= GT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
9.82/4.10	29[label="min1 LT LT True",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
9.82/4.10	30[label="min1 LT EQ True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
9.82/4.10	31[label="min1 LT GT True",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
9.82/4.10	32[label="min1 EQ LT False",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
9.82/4.10	33[label="min1 EQ EQ True",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
9.82/4.10	34[label="min1 EQ GT True",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3];
9.82/4.10	35[label="min1 GT LT False",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3];
9.82/4.10	36[label="min1 GT EQ False",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3];
9.82/4.10	37[label="min1 GT GT True",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3];
9.82/4.10	38[label="LT",fontsize=16,color="green",shape="box"];39[label="LT",fontsize=16,color="green",shape="box"];40[label="LT",fontsize=16,color="green",shape="box"];41[label="min0 EQ LT otherwise",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3];
9.82/4.10	42[label="EQ",fontsize=16,color="green",shape="box"];43[label="EQ",fontsize=16,color="green",shape="box"];44[label="min0 GT LT otherwise",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
9.82/4.10	45[label="min0 GT EQ otherwise",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
9.82/4.10	46[label="GT",fontsize=16,color="green",shape="box"];47[label="min0 EQ LT True",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3];
9.82/4.10	48[label="min0 GT LT True",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3];
9.82/4.10	49[label="min0 GT EQ True",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
9.82/4.10	50[label="LT",fontsize=16,color="green",shape="box"];51[label="LT",fontsize=16,color="green",shape="box"];52[label="EQ",fontsize=16,color="green",shape="box"];}
9.82/4.10	
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(6)
9.82/4.10	Obligation:
9.82/4.10	Q DP problem:
9.82/4.10	The TRS P consists of the following rules:
9.82/4.10	
9.82/4.10	   new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.82/4.10	
9.82/4.10	The TRS R consists of the following rules:
9.82/4.10	
9.82/4.10	   new_min1(LT, GT) -> LT
9.82/4.10	   new_min1(GT, LT) -> LT
9.82/4.10	   new_min1(GT, GT) -> GT
9.82/4.10	   new_min1(EQ, EQ) -> EQ
9.82/4.10	   new_min1(LT, EQ) -> LT
9.82/4.10	   new_min1(EQ, LT) -> LT
9.82/4.10	   new_min1(EQ, GT) -> EQ
9.82/4.10	   new_min1(GT, EQ) -> EQ
9.82/4.10	   new_min1(LT, LT) -> LT
9.82/4.10	
9.82/4.10	The set Q consists of the following terms:
9.82/4.10	
9.82/4.10	   new_min1(LT, EQ)
9.82/4.10	   new_min1(EQ, LT)
9.82/4.10	   new_min1(EQ, GT)
9.82/4.10	   new_min1(GT, EQ)
9.82/4.10	   new_min1(LT, GT)
9.82/4.10	   new_min1(GT, LT)
9.82/4.10	   new_min1(EQ, EQ)
9.82/4.10	   new_min1(GT, GT)
9.82/4.10	   new_min1(LT, LT)
9.82/4.10	
9.82/4.10	We have to consider all minimal (P,Q,R)-chains.
9.82/4.10	----------------------------------------
9.82/4.10	
9.82/4.10	(7) QDPSizeChangeProof (EQUIVALENT)
9.82/4.10	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. 
9.82/4.10	
9.82/4.10	From the DPs we obtained the following set of size-change graphs:
9.82/4.10	*new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.82/4.11	The graph contains the following edges 2 > 2
9.82/4.11	
9.82/4.11	
9.82/4.11	----------------------------------------
9.82/4.11	
9.82/4.11	(8)
9.82/4.11	YES
10.04/4.16	EOF