9.55/4.37	YES
11.60/4.90	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
11.60/4.90	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.60/4.90	
11.60/4.90	
11.60/4.90	H-Termination with start terms of the given HASKELL could be proven:
11.60/4.90	
11.60/4.90	(0) HASKELL
11.60/4.90	(1) CR [EQUIVALENT, 0 ms]
11.60/4.90	(2) HASKELL
11.60/4.90	(3) BR [EQUIVALENT, 0 ms]
11.60/4.90	(4) HASKELL
11.60/4.90	(5) COR [EQUIVALENT, 0 ms]
11.60/4.90	(6) HASKELL
11.60/4.90	(7) Narrow [SOUND, 0 ms]
11.60/4.90	(8) QDP
11.60/4.90	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.60/4.90	(10) YES
11.60/4.90	
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(0)
11.60/4.90	Obligation:
11.60/4.90	mainModule Main 
11.60/4.90	module FiniteMap where {
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.60/4.90	
11.60/4.90	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.60/4.90	maxFM EmptyFM = Nothing;
11.60/4.90	maxFM (Branch key _ _ _ fm_r) = case maxFM fm_r of { 
11.60/4.90	Nothing-> Just key; 
11.60/4.90	Just key1-> Just key1; 
11.60/4.90	 } ;
11.60/4.90	
11.60/4.90	}
11.60/4.90	module Maybe where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	module Main where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(1) CR (EQUIVALENT)
11.60/4.90	Case Reductions:
11.60/4.90	The following Case expression
11.60/4.90	"case maxFM fm_r of {
11.60/4.90	Nothing  -> Just key;
11.60/4.90	Just key1  -> Just key1}
11.60/4.90	"
11.60/4.90	is transformed to
11.60/4.90	"maxFM0 key Nothing = Just key;
11.60/4.90	maxFM0 key (Just key1) = Just key1;
11.60/4.90	"
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(2)
11.60/4.90	Obligation:
11.60/4.90	mainModule Main 
11.60/4.90	module FiniteMap where {
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.60/4.90	
11.60/4.90	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.60/4.90	maxFM EmptyFM = Nothing;
11.60/4.90	maxFM (Branch key _ _ _ fm_r) = maxFM0 key (maxFM fm_r);
11.60/4.90	
11.60/4.90	maxFM0 key Nothing = Just key;
11.60/4.90	maxFM0 key (Just key1) = Just key1;
11.60/4.90	
11.60/4.90	}
11.60/4.90	module Maybe where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	module Main where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(3) BR (EQUIVALENT)
11.60/4.90	Replaced joker patterns by fresh variables and removed binding patterns.
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(4)
11.60/4.90	Obligation:
11.60/4.90	mainModule Main 
11.60/4.90	module FiniteMap where {
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.60/4.90	
11.60/4.90	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.60/4.90	maxFM EmptyFM = Nothing;
11.60/4.90	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
11.60/4.90	
11.60/4.90	maxFM0 key Nothing = Just key;
11.60/4.90	maxFM0 key (Just key1) = Just key1;
11.60/4.90	
11.60/4.90	}
11.60/4.90	module Maybe where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	module Main where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(5) COR (EQUIVALENT)
11.60/4.90	Cond Reductions:
11.60/4.90	The following Function with conditions
11.60/4.90	"undefined |Falseundefined;
11.60/4.90	"
11.60/4.90	is transformed to
11.60/4.90	"undefined  = undefined1;
11.60/4.90	"
11.60/4.90	"undefined0 True = undefined;
11.60/4.90	"
11.60/4.90	"undefined1  = undefined0 False;
11.60/4.90	"
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(6)
11.60/4.90	Obligation:
11.60/4.90	mainModule Main 
11.60/4.90	module FiniteMap where {
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.60/4.90	
11.60/4.90	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.60/4.90	maxFM EmptyFM = Nothing;
11.60/4.90	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
11.60/4.90	
11.60/4.90	maxFM0 key Nothing = Just key;
11.60/4.90	maxFM0 key (Just key1) = Just key1;
11.60/4.90	
11.60/4.90	}
11.60/4.90	module Maybe where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Main;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	module Main where {
11.60/4.90	import qualified FiniteMap;
11.60/4.90	import qualified Maybe;
11.60/4.90	import qualified Prelude;
11.60/4.90	}
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(7) Narrow (SOUND)
11.60/4.90	Haskell To QDPs
11.60/4.90	
11.60/4.90	digraph dp_graph {
11.60/4.90	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.maxFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.60/4.90	3[label="FiniteMap.maxFM wv3",fontsize=16,color="burlywood",shape="triangle"];15[label="wv3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];3 -> 15[label="",style="solid", color="burlywood", weight=9];
11.60/4.90	15 -> 4[label="",style="solid", color="burlywood", weight=3];
11.60/4.90	16[label="wv3/FiniteMap.Branch wv30 wv31 wv32 wv33 wv34",fontsize=10,color="white",style="solid",shape="box"];3 -> 16[label="",style="solid", color="burlywood", weight=9];
11.60/4.90	16 -> 5[label="",style="solid", color="burlywood", weight=3];
11.60/4.90	4[label="FiniteMap.maxFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
11.60/4.90	5[label="FiniteMap.maxFM (FiniteMap.Branch wv30 wv31 wv32 wv33 wv34)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.60/4.90	6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
11.60/4.90	7[label="FiniteMap.maxFM0 wv30 (FiniteMap.maxFM wv34)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
11.60/4.90	9 -> 3[label="",style="dashed", color="red", weight=0];
11.60/4.90	9[label="FiniteMap.maxFM wv34",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
11.60/4.90	8[label="FiniteMap.maxFM0 wv30 wv4",fontsize=16,color="burlywood",shape="triangle"];17[label="wv4/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 17[label="",style="solid", color="burlywood", weight=9];
11.60/4.90	17 -> 11[label="",style="solid", color="burlywood", weight=3];
11.60/4.90	18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
11.60/4.90	18 -> 12[label="",style="solid", color="burlywood", weight=3];
11.60/4.90	10[label="wv34",fontsize=16,color="green",shape="box"];11[label="FiniteMap.maxFM0 wv30 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.60/4.90	12[label="FiniteMap.maxFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.60/4.90	13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(8)
11.60/4.90	Obligation:
11.60/4.90	Q DP problem:
11.60/4.90	The TRS P consists of the following rules:
11.60/4.90	
11.60/4.90	   new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_maxFM(wv34, h)
11.60/4.90	
11.60/4.90	R is empty.
11.60/4.90	Q is empty.
11.60/4.90	We have to consider all minimal (P,Q,R)-chains.
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(9) QDPSizeChangeProof (EQUIVALENT)
11.60/4.90	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. 
11.60/4.90	
11.60/4.90	From the DPs we obtained the following set of size-change graphs:
11.60/4.90	*new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_maxFM(wv34, h)
11.60/4.90	The graph contains the following edges 1 > 1, 2 >= 2
11.60/4.90	
11.60/4.90	
11.60/4.90	----------------------------------------
11.60/4.90	
11.60/4.90	(10)
11.60/4.90	YES
11.60/4.95	EOF