9.39/4.72	YES
11.27/5.28	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.27/5.28	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.27/5.28	
11.27/5.28	
11.27/5.28	H-Termination with start terms of the given HASKELL could be proven:
11.27/5.28	
11.27/5.28	(0) HASKELL
11.27/5.28	(1) CR [EQUIVALENT, 0 ms]
11.27/5.28	(2) HASKELL
11.27/5.28	(3) BR [EQUIVALENT, 0 ms]
11.27/5.28	(4) HASKELL
11.27/5.28	(5) COR [EQUIVALENT, 26 ms]
11.27/5.28	(6) HASKELL
11.27/5.28	(7) LetRed [EQUIVALENT, 0 ms]
11.27/5.28	(8) HASKELL
11.27/5.28	(9) Narrow [SOUND, 0 ms]
11.27/5.28	(10) QDP
11.27/5.28	(11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.27/5.28	(12) YES
11.27/5.28	
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(0)
11.27/5.28	Obligation:
11.27/5.28	mainModule Main 
11.27/5.28	module Maybe where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	module List where {
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
11.27/5.28	minimumBy _ [] = error [];
11.27/5.28	minimumBy cmp xs = foldl1 min xs where { 
11.27/5.28	min x y = case cmp x y of { 
11.27/5.28	GT-> y; 
11.27/5.28	_-> x; 
11.27/5.28	 } ;
11.27/5.28	 };
11.27/5.28	
11.27/5.28	}
11.27/5.28	module Main where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(1) CR (EQUIVALENT)
11.27/5.28	Case Reductions:
11.27/5.28	The following Case expression
11.27/5.28	"case cmp x y of {
11.27/5.28	GT  -> y;
11.27/5.28	_  -> x}
11.27/5.28	"
11.27/5.28	is transformed to
11.27/5.28	"min0 y x GT = y;
11.27/5.28	min0 y x _ = x;
11.27/5.28	"
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(2)
11.27/5.28	Obligation:
11.27/5.28	mainModule Main 
11.27/5.28	module Maybe where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	module List where {
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
11.27/5.28	minimumBy _ [] = error [];
11.27/5.28	minimumBy cmp xs = foldl1 min xs where { 
11.27/5.28	min x y = min0 y x (cmp x y);
11.27/5.28	min0 y x GT = y;
11.27/5.28	min0 y x _ = x;
11.27/5.28	 };
11.27/5.28	
11.27/5.28	}
11.27/5.28	module Main where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(3) BR (EQUIVALENT)
11.27/5.28	Replaced joker patterns by fresh variables and removed binding patterns.
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(4)
11.27/5.28	Obligation:
11.27/5.28	mainModule Main 
11.27/5.28	module Maybe where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	module List where {
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
11.27/5.28	minimumBy vy [] = error [];
11.27/5.28	minimumBy cmp xs = foldl1 min xs where { 
11.27/5.28	min x y = min0 y x (cmp x y);
11.27/5.28	min0 y x GT = y;
11.27/5.28	min0 y x vz = x;
11.27/5.28	 };
11.27/5.28	
11.27/5.28	}
11.27/5.28	module Main where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(5) COR (EQUIVALENT)
11.27/5.28	Cond Reductions:
11.27/5.28	The following Function with conditions
11.27/5.28	"undefined |Falseundefined;
11.27/5.28	"
11.27/5.28	is transformed to
11.27/5.28	"undefined  = undefined1;
11.27/5.28	"
11.27/5.28	"undefined0 True = undefined;
11.27/5.28	"
11.27/5.28	"undefined1  = undefined0 False;
11.27/5.28	"
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(6)
11.27/5.28	Obligation:
11.27/5.28	mainModule Main 
11.27/5.28	module Maybe where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	module List where {
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
11.27/5.28	minimumBy vy [] = error [];
11.27/5.28	minimumBy cmp xs = foldl1 min xs where { 
11.27/5.28	min x y = min0 y x (cmp x y);
11.27/5.28	min0 y x GT = y;
11.27/5.28	min0 y x vz = x;
11.27/5.28	 };
11.27/5.28	
11.27/5.28	}
11.27/5.28	module Main where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(7) LetRed (EQUIVALENT)
11.27/5.28	Let/Where Reductions:
11.27/5.28	The bindings of the following Let/Where expression
11.27/5.28	"foldl1 min xs where {
11.27/5.28	min x y = min0 y x (cmp x y);
11.27/5.28	;
11.27/5.28	min0 y x GT = y;
11.27/5.28	min0 y x vz = x;
11.27/5.28	} 
11.27/5.28	"
11.27/5.28	are unpacked to the following functions on top level
11.27/5.28	"minimumByMin0 wu y x GT = y;
11.27/5.28	minimumByMin0 wu y x vz = x;
11.27/5.28	"
11.27/5.28	"minimumByMin wu x y = minimumByMin0 wu y x (wu x y);
11.27/5.28	"
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(8)
11.27/5.28	Obligation:
11.27/5.28	mainModule Main 
11.27/5.28	module Maybe where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	module List where {
11.27/5.28	import qualified Main;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	minimumBy :: (a  ->  a  ->  Ordering)  ->  [a]  ->  a;
11.27/5.28	minimumBy vy [] = error [];
11.27/5.28	minimumBy cmp xs = foldl1 (minimumByMin cmp) xs;
11.27/5.28	
11.27/5.28	minimumByMin wu x y = minimumByMin0 wu y x (wu x y);
11.27/5.28	
11.27/5.28	minimumByMin0 wu y x GT = y;
11.27/5.28	minimumByMin0 wu y x vz = x;
11.27/5.28	
11.27/5.28	}
11.27/5.28	module Main where {
11.27/5.28	import qualified List;
11.27/5.28	import qualified Maybe;
11.27/5.28	import qualified Prelude;
11.27/5.28	}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(9) Narrow (SOUND)
11.27/5.28	Haskell To QDPs
11.27/5.28	
11.27/5.28	digraph dp_graph {
11.27/5.28	node [outthreshold=100, inthreshold=100];1[label="List.minimumBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.27/5.28	3[label="List.minimumBy wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.27/5.28	4[label="List.minimumBy 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];
11.27/5.28	30 -> 5[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	31[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
11.27/5.28	31 -> 6[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	5[label="List.minimumBy wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.27/5.28	6[label="List.minimumBy wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.27/5.28	7[label="foldl1 (List.minimumByMin wv3) (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.27/5.28	8[label="error []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
11.27/5.28	9[label="foldl (List.minimumByMin 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];
11.27/5.28	32 -> 11[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	33[label="wv41/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 33[label="",style="solid", color="burlywood", weight=9];
11.27/5.28	33 -> 12[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	10[label="error []",fontsize=16,color="red",shape="box"];11[label="foldl (List.minimumByMin wv3) wv40 (wv410 : wv411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.27/5.28	12[label="foldl (List.minimumByMin wv3) wv40 []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.27/5.28	13 -> 9[label="",style="dashed", color="red", weight=0];
11.27/5.28	13[label="foldl (List.minimumByMin wv3) (List.minimumByMin wv3 wv40 wv410) wv411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
11.27/5.28	13 -> 16[label="",style="dashed", color="magenta", weight=3];
11.27/5.28	14[label="wv40",fontsize=16,color="green",shape="box"];15[label="wv411",fontsize=16,color="green",shape="box"];16[label="List.minimumByMin wv3 wv40 wv410",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
11.27/5.28	17 -> 18[label="",style="dashed", color="red", weight=0];
11.27/5.28	17[label="List.minimumByMin0 wv3 wv410 wv40 (wv3 wv40 wv410)",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
11.27/5.28	19[label="wv3 wv40 wv410",fontsize=16,color="green",shape="box"];19 -> 25[label="",style="dashed", color="green", weight=3];
11.27/5.28	19 -> 26[label="",style="dashed", color="green", weight=3];
11.27/5.28	18[label="List.minimumByMin0 wv3 wv410 wv40 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];
11.27/5.28	34 -> 22[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	35[label="wv5/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 35[label="",style="solid", color="burlywood", weight=9];
11.27/5.28	35 -> 23[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	36[label="wv5/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 36[label="",style="solid", color="burlywood", weight=9];
11.27/5.28	36 -> 24[label="",style="solid", color="burlywood", weight=3];
11.27/5.28	25[label="wv40",fontsize=16,color="green",shape="box"];26[label="wv410",fontsize=16,color="green",shape="box"];22[label="List.minimumByMin0 wv3 wv410 wv40 LT",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
11.27/5.28	23[label="List.minimumByMin0 wv3 wv410 wv40 EQ",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
11.27/5.28	24[label="List.minimumByMin0 wv3 wv410 wv40 GT",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
11.27/5.28	27[label="wv40",fontsize=16,color="green",shape="box"];28[label="wv40",fontsize=16,color="green",shape="box"];29[label="wv410",fontsize=16,color="green",shape="box"];}
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(10)
11.27/5.28	Obligation:
11.27/5.28	Q DP problem:
11.27/5.28	The TRS P consists of the following rules:
11.27/5.28	
11.27/5.28	   new_foldl(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)
11.27/5.28	
11.27/5.28	R is empty.
11.27/5.28	Q is empty.
11.27/5.28	We have to consider all minimal (P,Q,R)-chains.
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(11) QDPSizeChangeProof (EQUIVALENT)
11.27/5.28	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. 
11.27/5.28	
11.27/5.28	From the DPs we obtained the following set of size-change graphs:
11.27/5.28	*new_foldl(wv3, :(wv410, wv411), ba) -> new_foldl(wv3, wv411, ba)
11.27/5.28	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
11.27/5.28	
11.27/5.28	
11.27/5.28	----------------------------------------
11.27/5.28	
11.27/5.28	(12)
11.27/5.28	YES
11.47/5.36	EOF