10.06/4.39	YES
12.18/4.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.18/4.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.18/4.99	
12.18/4.99	
12.18/4.99	H-Termination with start terms of the given HASKELL could be proven:
12.18/4.99	
12.18/4.99	(0) HASKELL
12.18/4.99	(1) CR [EQUIVALENT, 0 ms]
12.18/4.99	(2) HASKELL
12.18/4.99	(3) BR [EQUIVALENT, 0 ms]
12.18/4.99	(4) HASKELL
12.18/4.99	(5) COR [EQUIVALENT, 0 ms]
12.18/4.99	(6) HASKELL
12.18/4.99	(7) LetRed [EQUIVALENT, 0 ms]
12.18/4.99	(8) HASKELL
12.18/4.99	(9) Narrow [SOUND, 0 ms]
12.18/4.99	(10) QDP
12.18/4.99	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.18/4.99	(12) YES
12.18/4.99	
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(0)
12.18/4.99	Obligation:
12.18/4.99	mainModule Main 
12.18/4.99	module Maybe where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	module List where {
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	maximumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
12.18/4.99	maximumBy _ [] = error [];
12.18/4.99	maximumBy cmp xs = foldl1 max xs where { 
12.18/4.99	max x y = case cmp x y of { 
12.18/4.99	GT-> x; 
12.18/4.99	_-> y; 
12.18/4.99	 } ;
12.18/4.99	 };
12.18/4.99	
12.18/4.99	}
12.18/4.99	module Main where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(1) CR (EQUIVALENT)
12.18/4.99	Case Reductions:
12.18/4.99	The following Case expression
12.18/4.99	"case cmp x y of {
12.18/4.99	GT  -> x;
12.18/4.99	_  -> y}
12.18/4.99	"
12.18/4.99	is transformed to
12.18/4.99	"max0 x y GT = x;
12.18/4.99	max0 x y _ = y;
12.18/4.99	"
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(2)
12.18/4.99	Obligation:
12.18/4.99	mainModule Main 
12.18/4.99	module Maybe where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	module List where {
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	maximumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
12.18/4.99	maximumBy _ [] = error [];
12.18/4.99	maximumBy cmp xs = foldl1 max xs where { 
12.18/4.99	max x y = max0 x y (cmp x y);
12.18/4.99	max0 x y GT = x;
12.18/4.99	max0 x y _ = y;
12.18/4.99	 };
12.18/4.99	
12.18/4.99	}
12.18/4.99	module Main where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(3) BR (EQUIVALENT)
12.18/4.99	Replaced joker patterns by fresh variables and removed binding patterns.
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(4)
12.18/4.99	Obligation:
12.18/4.99	mainModule Main 
12.18/4.99	module Maybe where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	module List where {
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	maximumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
12.18/4.99	maximumBy vy [] = error [];
12.18/4.99	maximumBy cmp xs = foldl1 max xs where { 
12.18/4.99	max x y = max0 x y (cmp x y);
12.18/4.99	max0 x y GT = x;
12.18/4.99	max0 x y vz = y;
12.18/4.99	 };
12.18/4.99	
12.18/4.99	}
12.18/4.99	module Main where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(5) COR (EQUIVALENT)
12.18/4.99	Cond Reductions:
12.18/4.99	The following Function with conditions
12.18/4.99	"undefined |Falseundefined;
12.18/4.99	"
12.18/4.99	is transformed to
12.18/4.99	"undefined  = undefined1;
12.18/4.99	"
12.18/4.99	"undefined0 True = undefined;
12.18/4.99	"
12.18/4.99	"undefined1  = undefined0 False;
12.18/4.99	"
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(6)
12.18/4.99	Obligation:
12.18/4.99	mainModule Main 
12.18/4.99	module Maybe where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	module List where {
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	maximumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
12.18/4.99	maximumBy vy [] = error [];
12.18/4.99	maximumBy cmp xs = foldl1 max xs where { 
12.18/4.99	max x y = max0 x y (cmp x y);
12.18/4.99	max0 x y GT = x;
12.18/4.99	max0 x y vz = y;
12.18/4.99	 };
12.18/4.99	
12.18/4.99	}
12.18/4.99	module Main where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(7) LetRed (EQUIVALENT)
12.18/4.99	Let/Where Reductions:
12.18/4.99	The bindings of the following Let/Where expression
12.18/4.99	"foldl1 max xs where {
12.18/4.99	max x y = max0 x y (cmp x y);
12.18/4.99	;
12.18/4.99	max0 x y GT = x;
12.18/4.99	max0 x y vz = y;
12.18/4.99	} 
12.18/4.99	"
12.18/4.99	are unpacked to the following functions on top level
12.18/4.99	"maximumByMax0 wu x y GT = x;
12.18/4.99	maximumByMax0 wu x y vz = y;
12.18/4.99	"
12.18/4.99	"maximumByMax wu x y = maximumByMax0 wu x y (wu x y);
12.18/4.99	"
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(8)
12.18/4.99	Obligation:
12.18/4.99	mainModule Main 
12.18/4.99	module Maybe where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	module List where {
12.18/4.99	import qualified Main;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	maximumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
12.18/4.99	maximumBy vy [] = error [];
12.18/4.99	maximumBy cmp xs = foldl1 (maximumByMax cmp) xs;
12.18/4.99	
12.18/4.99	maximumByMax wu x y = maximumByMax0 wu x y (wu x y);
12.18/4.99	
12.18/4.99	maximumByMax0 wu x y GT = x;
12.18/4.99	maximumByMax0 wu x y vz = y;
12.18/4.99	
12.18/4.99	}
12.18/4.99	module Main where {
12.18/4.99	import qualified List;
12.18/4.99	import qualified Maybe;
12.18/4.99	import qualified Prelude;
12.18/4.99	}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(9) Narrow (SOUND)
12.18/4.99	Haskell To QDPs
12.18/4.99	
12.18/4.99	digraph dp_graph {
12.18/4.99	node [outthreshold=100, inthreshold=100];1[label="List.maximumBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.18/4.99	3[label="List.maximumBy wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.18/4.99	4[label="List.maximumBy wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];30[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	30 -> 5[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	31[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	31 -> 6[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	5[label="List.maximumBy wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.18/4.99	6[label="List.maximumBy wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.18/4.99	7[label="foldl1 (List.maximumByMax wv3) (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.18/4.99	8[label="error []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.18/4.99	9[label="foldl (List.maximumByMax wv3) wv40 wv41",fontsize=16,color="burlywood",shape="triangle"];32[label="wv41/wv410 : wv411",fontsize=10,color="white",style="solid",shape="box"];9 -> 32[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	32 -> 11[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	33[label="wv41/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 33[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	33 -> 12[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	10[label="error []",fontsize=16,color="red",shape="box"];11[label="foldl (List.maximumByMax wv3) wv40 (wv410 : wv411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12.18/4.99	12[label="foldl (List.maximumByMax wv3) wv40 []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
12.18/4.99	13 -> 9[label="",style="dashed", color="red", weight=0];
12.18/4.99	13[label="foldl (List.maximumByMax wv3) (List.maximumByMax wv3 wv40 wv410) wv411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
12.18/4.99	13 -> 16[label="",style="dashed", color="magenta", weight=3];
12.18/4.99	14[label="wv40",fontsize=16,color="green",shape="box"];15[label="wv411",fontsize=16,color="green",shape="box"];16[label="List.maximumByMax wv3 wv40 wv410",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
12.18/4.99	17 -> 18[label="",style="dashed", color="red", weight=0];
12.18/4.99	17[label="List.maximumByMax0 wv3 wv40 wv410 (wv3 wv40 wv410)",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
12.18/4.99	19[label="wv3 wv40 wv410",fontsize=16,color="green",shape="box"];19 -> 25[label="",style="dashed", color="green", weight=3];
12.18/4.99	19 -> 26[label="",style="dashed", color="green", weight=3];
12.18/4.99	18[label="List.maximumByMax0 wv3 wv40 wv410 wv5",fontsize=16,color="burlywood",shape="triangle"];34[label="wv5/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 34[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	34 -> 22[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	35[label="wv5/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 35[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	35 -> 23[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	36[label="wv5/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 36[label="",style="solid", color="burlywood", weight=9];
12.18/4.99	36 -> 24[label="",style="solid", color="burlywood", weight=3];
12.18/4.99	25[label="wv40",fontsize=16,color="green",shape="box"];26[label="wv410",fontsize=16,color="green",shape="box"];22[label="List.maximumByMax0 wv3 wv40 wv410 LT",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
12.18/4.99	23[label="List.maximumByMax0 wv3 wv40 wv410 EQ",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
12.18/4.99	24[label="List.maximumByMax0 wv3 wv40 wv410 GT",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
12.18/4.99	27[label="wv410",fontsize=16,color="green",shape="box"];28[label="wv410",fontsize=16,color="green",shape="box"];29[label="wv40",fontsize=16,color="green",shape="box"];}
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(10)
12.18/4.99	Obligation:
12.18/4.99	Q DP problem:
12.18/4.99	The TRS P consists of the following rules:
12.18/4.99	
12.18/4.99	   new_foldl(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)
12.18/4.99	
12.18/4.99	R is empty.
12.18/4.99	Q is empty.
12.18/4.99	We have to consider all minimal (P,Q,R)-chains.
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(11) QDPSizeChangeProof (EQUIVALENT)
12.18/4.99	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. 
12.18/4.99	
12.18/4.99	From the DPs we obtained the following set of size-change graphs:
12.18/4.99	*new_foldl(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)
12.18/4.99	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.18/4.99	
12.18/4.99	
12.18/4.99	----------------------------------------
12.18/4.99	
12.18/4.99	(12)
12.18/4.99	YES
12.33/7.96	EOF