8.22/3.64	YES
9.78/4.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.78/4.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.78/4.12	
9.78/4.12	
9.78/4.12	H-Termination with start terms of the given HASKELL could be proven:
9.78/4.12	
9.78/4.12	(0) HASKELL
9.78/4.12	(1) BR [EQUIVALENT, 0 ms]
9.78/4.12	(2) HASKELL
9.78/4.12	(3) COR [EQUIVALENT, 0 ms]
9.78/4.12	(4) HASKELL
9.78/4.12	(5) Narrow [SOUND, 0 ms]
9.78/4.12	(6) QDP
9.78/4.12	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.78/4.12	(8) YES
9.78/4.12	
9.78/4.12	
9.78/4.12	----------------------------------------
9.78/4.12	
9.78/4.12	(0)
9.78/4.12	Obligation:
9.78/4.12	mainModule Main 
9.78/4.12	module Main where {
9.78/4.12	import qualified Prelude;
9.78/4.12	}
9.78/4.12	
9.78/4.12	----------------------------------------
9.78/4.12	
9.78/4.12	(1) BR (EQUIVALENT)
9.78/4.12	Replaced joker patterns by fresh variables and removed binding patterns.
9.78/4.12	----------------------------------------
9.78/4.12	
9.78/4.12	(2)
9.78/4.12	Obligation:
9.78/4.12	mainModule Main 
9.78/4.12	module Main where {
9.78/4.12	import qualified Prelude;
9.78/4.12	}
9.78/4.12	
9.78/4.12	----------------------------------------
9.78/4.12	
9.78/4.12	(3) COR (EQUIVALENT)
9.78/4.12	Cond Reductions:
9.78/4.12	The following Function with conditions
9.78/4.12	"max x y|x <= yy|otherwisex;
9.78/4.12	"
9.78/4.12	is transformed to
9.78/4.12	"max x y = max2 x y;
9.78/4.12	"
9.78/4.12	"max1 x y True = y;
9.78/4.12	max1 x y False = max0 x y otherwise;
9.78/4.13	"
9.78/4.13	"max0 x y True = x;
9.78/4.13	"
9.78/4.13	"max2 x y = max1 x y (x <= y);
9.78/4.13	"
9.78/4.13	The following Function with conditions
9.78/4.13	"undefined |Falseundefined;
9.78/4.13	"
9.78/4.13	is transformed to
9.78/4.13	"undefined  = undefined1;
9.78/4.13	"
9.78/4.13	"undefined0 True = undefined;
9.78/4.13	"
9.78/4.13	"undefined1  = undefined0 False;
9.78/4.13	"
9.78/4.13	
9.78/4.13	----------------------------------------
9.78/4.13	
9.78/4.13	(4)
9.78/4.13	Obligation:
9.78/4.13	mainModule Main 
9.78/4.13	module Main where {
9.78/4.13	import qualified Prelude;
9.78/4.13	}
9.78/4.13	
9.78/4.13	----------------------------------------
9.78/4.13	
9.78/4.13	(5) Narrow (SOUND)
9.78/4.13	Haskell To QDPs
9.78/4.13	
9.78/4.13	digraph dp_graph {
9.78/4.13	node [outthreshold=100, inthreshold=100];1[label="maximum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.78/4.13	3[label="maximum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.78/4.13	4[label="foldl1 max 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];
9.78/4.13	25 -> 5[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	26[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 26[label="",style="solid", color="burlywood", weight=9];
9.78/4.13	26 -> 6[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	5[label="foldl1 max (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.78/4.13	6[label="foldl1 max []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.78/4.13	7[label="foldl max 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];
9.78/4.13	27 -> 9[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	28[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
9.78/4.13	28 -> 10[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl max vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.78/4.13	10[label="foldl max vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.78/4.13	11 -> 7[label="",style="dashed", color="red", weight=0];
9.78/4.13	11[label="foldl max (max vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3];
9.78/4.13	11 -> 14[label="",style="dashed", color="magenta", weight=3];
9.78/4.13	12[label="vx30",fontsize=16,color="green",shape="box"];13[label="max vx30 vx310",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.78/4.13	14[label="vx311",fontsize=16,color="green",shape="box"];15[label="max2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
9.78/4.13	16[label="max1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
9.78/4.13	17[label="max1 vx30 vx310 (compare vx30 vx310 /= GT)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
9.78/4.13	18[label="max1 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];
9.78/4.13	29 -> 19[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	19[label="max1 () 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];
9.78/4.13	30 -> 20[label="",style="solid", color="burlywood", weight=3];
9.78/4.13	20[label="max1 () () (not (compare () () == GT))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3];
9.78/4.13	21[label="max1 () () (not (EQ == GT))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
9.78/4.13	22[label="max1 () () (not False)",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
9.78/4.13	23[label="max1 () () True",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3];
9.78/4.13	24[label="()",fontsize=16,color="green",shape="box"];}
9.78/4.13	
9.78/4.13	----------------------------------------
9.78/4.13	
9.78/4.13	(6)
9.78/4.13	Obligation:
9.78/4.13	Q DP problem:
9.78/4.13	The TRS P consists of the following rules:
9.78/4.13	
9.78/4.13	   new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311)
9.78/4.13	
9.78/4.13	The TRS R consists of the following rules:
9.78/4.13	
9.78/4.13	   new_max1(@0, @0) -> @0
9.78/4.13	
9.78/4.13	The set Q consists of the following terms:
9.78/4.13	
9.78/4.13	   new_max1(@0, @0)
9.78/4.13	
9.78/4.13	We have to consider all minimal (P,Q,R)-chains.
9.78/4.13	----------------------------------------
9.78/4.13	
9.78/4.13	(7) QDPSizeChangeProof (EQUIVALENT)
9.78/4.13	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. 
9.78/4.13	
9.78/4.13	From the DPs we obtained the following set of size-change graphs:
9.78/4.13	*new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311)
9.78/4.13	The graph contains the following edges 2 > 2
9.78/4.13	
9.78/4.13	
9.78/4.13	----------------------------------------
9.78/4.13	
9.78/4.13	(8)
9.78/4.13	YES
10.07/4.27	EOF