8.02/3.59	YES
9.60/4.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.60/4.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.60/4.05	
9.60/4.05	
9.60/4.05	H-Termination with start terms of the given HASKELL could be proven:
9.60/4.05	
9.60/4.05	(0) HASKELL
9.60/4.05	(1) BR [EQUIVALENT, 0 ms]
9.60/4.05	(2) HASKELL
9.60/4.05	(3) COR [EQUIVALENT, 0 ms]
9.60/4.05	(4) HASKELL
9.60/4.05	(5) Narrow [SOUND, 0 ms]
9.60/4.05	(6) QDP
9.60/4.05	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.60/4.05	(8) YES
9.60/4.05	
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(0)
9.60/4.05	Obligation:
9.60/4.05	mainModule Main 
9.60/4.05	module Main where {
9.60/4.05	import qualified Prelude;
9.60/4.05	}
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(1) BR (EQUIVALENT)
9.60/4.05	Replaced joker patterns by fresh variables and removed binding patterns.
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(2)
9.60/4.05	Obligation:
9.60/4.05	mainModule Main 
9.60/4.05	module Main where {
9.60/4.05	import qualified Prelude;
9.60/4.05	}
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(3) COR (EQUIVALENT)
9.60/4.05	Cond Reductions:
9.60/4.05	The following Function with conditions
9.60/4.05	"min x y|x <= yx|otherwisey;
9.60/4.05	"
9.60/4.05	is transformed to
9.60/4.05	"min x y = min2 x y;
9.60/4.05	"
9.60/4.05	"min1 x y True = x;
9.60/4.05	min1 x y False = min0 x y otherwise;
9.60/4.05	"
9.60/4.05	"min0 x y True = y;
9.60/4.05	"
9.60/4.05	"min2 x y = min1 x y (x <= y);
9.60/4.05	"
9.60/4.05	The following Function with conditions
9.60/4.05	"undefined |Falseundefined;
9.60/4.05	"
9.60/4.05	is transformed to
9.60/4.05	"undefined  = undefined1;
9.60/4.05	"
9.60/4.05	"undefined0 True = undefined;
9.60/4.05	"
9.60/4.05	"undefined1  = undefined0 False;
9.60/4.05	"
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(4)
9.60/4.05	Obligation:
9.60/4.05	mainModule Main 
9.60/4.05	module Main where {
9.60/4.05	import qualified Prelude;
9.60/4.05	}
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(5) Narrow (SOUND)
9.60/4.05	Haskell To QDPs
9.60/4.05	
9.60/4.05	digraph dp_graph {
9.60/4.05	node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.60/4.05	3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.60/4.05	4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];33[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 33[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	33 -> 5[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	34[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 34[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	34 -> 6[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.60/4.05	6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.60/4.05	7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];35[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 35[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	35 -> 9[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	36[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 36[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	36 -> 10[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	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];
9.60/4.05	10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.60/4.05	11 -> 7[label="",style="dashed", color="red", weight=0];
9.60/4.05	11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
9.60/4.05	11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.60/4.05	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];
9.60/4.05	15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.60/4.05	16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="burlywood",shape="box"];37[label="vx30/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 37[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	37 -> 17[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	38[label="vx30/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 38[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	38 -> 18[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	17[label="min1 False vx310 (False <= vx310)",fontsize=16,color="burlywood",shape="box"];39[label="vx310/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 39[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	39 -> 19[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	40[label="vx310/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 40[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	40 -> 20[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	18[label="min1 True vx310 (True <= vx310)",fontsize=16,color="burlywood",shape="box"];41[label="vx310/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 41[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	41 -> 21[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	42[label="vx310/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 42[label="",style="solid", color="burlywood", weight=9];
9.60/4.05	42 -> 22[label="",style="solid", color="burlywood", weight=3];
9.60/4.05	19[label="min1 False False (False <= False)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
9.60/4.05	20[label="min1 False True (False <= True)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
9.60/4.05	21[label="min1 True False (True <= False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
9.60/4.05	22[label="min1 True True (True <= True)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
9.60/4.05	23[label="min1 False False True",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
9.60/4.05	24[label="min1 False True True",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
9.60/4.05	25[label="min1 True False False",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
9.60/4.05	26[label="min1 True True True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
9.60/4.05	27[label="False",fontsize=16,color="green",shape="box"];28[label="False",fontsize=16,color="green",shape="box"];29[label="min0 True False otherwise",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
9.60/4.05	30[label="True",fontsize=16,color="green",shape="box"];31[label="min0 True False True",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3];
9.60/4.05	32[label="False",fontsize=16,color="green",shape="box"];}
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(6)
9.60/4.05	Obligation:
9.60/4.05	Q DP problem:
9.60/4.05	The TRS P consists of the following rules:
9.60/4.05	
9.60/4.05	   new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.60/4.05	
9.60/4.05	The TRS R consists of the following rules:
9.60/4.05	
9.60/4.05	   new_min1(True, True) -> True
9.60/4.05	   new_min1(False, True) -> False
9.60/4.05	   new_min1(True, False) -> False
9.60/4.05	   new_min1(False, False) -> False
9.60/4.05	
9.60/4.05	The set Q consists of the following terms:
9.60/4.05	
9.60/4.05	   new_min1(False, False)
9.60/4.05	   new_min1(True, True)
9.60/4.05	   new_min1(False, True)
9.60/4.05	   new_min1(True, False)
9.60/4.05	
9.60/4.05	We have to consider all minimal (P,Q,R)-chains.
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(7) QDPSizeChangeProof (EQUIVALENT)
9.60/4.05	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. 
9.60/4.05	
9.60/4.05	From the DPs we obtained the following set of size-change graphs:
9.60/4.05	*new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.60/4.05	The graph contains the following edges 2 > 2
9.60/4.05	
9.60/4.05	
9.60/4.05	----------------------------------------
9.60/4.05	
9.60/4.05	(8)
9.60/4.05	YES
9.70/4.12	EOF