7.98/3.58	YES
9.40/4.01	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.40/4.01	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.40/4.01	
9.40/4.01	
9.40/4.01	H-Termination with start terms of the given HASKELL could be proven:
9.40/4.01	
9.40/4.01	(0) HASKELL
9.40/4.01	(1) BR [EQUIVALENT, 0 ms]
9.40/4.01	(2) HASKELL
9.40/4.01	(3) COR [EQUIVALENT, 0 ms]
9.40/4.01	(4) HASKELL
9.40/4.01	(5) Narrow [SOUND, 0 ms]
9.40/4.01	(6) QDP
9.40/4.01	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.40/4.01	(8) YES
9.40/4.01	
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(0)
9.40/4.01	Obligation:
9.40/4.01	mainModule Main 
9.40/4.01	module Main where {
9.40/4.01	import qualified Prelude;
9.40/4.01	}
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(1) BR (EQUIVALENT)
9.40/4.01	Replaced joker patterns by fresh variables and removed binding patterns.
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(2)
9.40/4.01	Obligation:
9.40/4.01	mainModule Main 
9.40/4.01	module Main where {
9.40/4.01	import qualified Prelude;
9.40/4.01	}
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(3) COR (EQUIVALENT)
9.40/4.01	Cond Reductions:
9.40/4.01	The following Function with conditions
9.40/4.01	"min x y|x <= yx|otherwisey;
9.40/4.01	"
9.40/4.01	is transformed to
9.40/4.01	"min x y = min2 x y;
9.40/4.01	"
9.40/4.01	"min1 x y True = x;
9.40/4.01	min1 x y False = min0 x y otherwise;
9.40/4.01	"
9.40/4.01	"min0 x y True = y;
9.40/4.01	"
9.40/4.01	"min2 x y = min1 x y (x <= y);
9.40/4.01	"
9.40/4.01	The following Function with conditions
9.40/4.01	"undefined |Falseundefined;
9.40/4.01	"
9.40/4.01	is transformed to
9.40/4.01	"undefined  = undefined1;
9.40/4.01	"
9.40/4.01	"undefined0 True = undefined;
9.40/4.01	"
9.40/4.01	"undefined1  = undefined0 False;
9.40/4.01	"
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(4)
9.40/4.01	Obligation:
9.40/4.01	mainModule Main 
9.40/4.01	module Main where {
9.40/4.01	import qualified Prelude;
9.40/4.01	}
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(5) Narrow (SOUND)
9.40/4.01	Haskell To QDPs
9.40/4.01	
9.40/4.01	digraph dp_graph {
9.40/4.01	node [outthreshold=100, inthreshold=100];1[label="minimum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.40/4.01	3[label="minimum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.40/4.01	4[label="foldl1 min vx3",fontsize=16,color="burlywood",shape="box"];25[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 25[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	25 -> 5[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	26[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	26 -> 6[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	5[label="foldl1 min (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.40/4.01	6[label="foldl1 min []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.40/4.01	7[label="foldl min vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];27[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 27[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	27 -> 9[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	28[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	28 -> 10[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	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];
9.40/4.01	10[label="foldl min vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.40/4.01	11 -> 7[label="",style="dashed", color="red", weight=0];
9.40/4.01	11[label="foldl min (min vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
9.40/4.01	11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.40/4.01	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="min vx30 vx310",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.40/4.01	14[label="vx311",fontsize=16,color="green",shape="box"];15[label="min2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.40/4.01	16[label="min1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
9.40/4.01	17[label="min1 vx30 vx310 (compare vx30 vx310 /= GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
9.40/4.01	18[label="min1 vx30 vx310 (not (compare vx30 vx310 == GT))",fontsize=16,color="burlywood",shape="box"];29[label="vx30/()",fontsize=10,color="white",style="solid",shape="box"];18 -> 29[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	29 -> 19[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	19[label="min1 () vx310 (not (compare () vx310 == GT))",fontsize=16,color="burlywood",shape="box"];30[label="vx310/()",fontsize=10,color="white",style="solid",shape="box"];19 -> 30[label="",style="solid", color="burlywood", weight=9];
9.40/4.01	30 -> 20[label="",style="solid", color="burlywood", weight=3];
9.40/4.01	20[label="min1 () () (not (compare () () == GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.40/4.01	21[label="min1 () () (not (EQ == GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.40/4.01	22[label="min1 () () (not False)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
9.40/4.01	23[label="min1 () () True",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
9.40/4.01	24[label="()",fontsize=16,color="green",shape="box"];}
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(6)
9.40/4.01	Obligation:
9.40/4.01	Q DP problem:
9.40/4.01	The TRS P consists of the following rules:
9.40/4.01	
9.40/4.01	   new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.40/4.01	
9.40/4.01	The TRS R consists of the following rules:
9.40/4.01	
9.40/4.01	   new_min1(@0, @0) -> @0
9.40/4.01	
9.40/4.01	The set Q consists of the following terms:
9.40/4.01	
9.40/4.01	   new_min1(@0, @0)
9.40/4.01	
9.40/4.01	We have to consider all minimal (P,Q,R)-chains.
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(7) QDPSizeChangeProof (EQUIVALENT)
9.40/4.01	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. 
9.40/4.01	
9.40/4.01	From the DPs we obtained the following set of size-change graphs:
9.40/4.01	*new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_min1(vx30, vx310), vx311)
9.40/4.01	The graph contains the following edges 2 > 2
9.40/4.01	
9.40/4.01	
9.40/4.01	----------------------------------------
9.40/4.01	
9.40/4.01	(8)
9.40/4.01	YES
9.65/4.05	EOF