10.06/4.56	YES
12.25/5.15	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.25/5.15	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.25/5.15	
12.25/5.15	
12.25/5.15	H-Termination with start terms of the given HASKELL could be proven:
12.25/5.15	
12.25/5.15	(0) HASKELL
12.25/5.15	(1) CR [EQUIVALENT, 0 ms]
12.25/5.15	(2) HASKELL
12.25/5.15	(3) BR [EQUIVALENT, 0 ms]
12.25/5.15	(4) HASKELL
12.25/5.15	(5) COR [EQUIVALENT, 0 ms]
12.25/5.15	(6) HASKELL
12.25/5.15	(7) Narrow [SOUND, 0 ms]
12.25/5.15	(8) QDP
12.25/5.15	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.25/5.15	(10) YES
12.25/5.15	
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(0)
12.25/5.15	Obligation:
12.25/5.15	mainModule Main 
12.25/5.15	module FiniteMap where {
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.25/5.15	
12.25/5.15	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
12.25/5.15	maxFM EmptyFM = Nothing;
12.25/5.15	maxFM (Branch key _ _ _ fm_r) = case maxFM fm_r of { 
12.25/5.15	Nothing-> Just key; 
12.25/5.15	Just key1-> Just key1; 
12.25/5.15	 } ;
12.25/5.15	
12.25/5.15	}
12.25/5.15	module Maybe where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	module Main where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(1) CR (EQUIVALENT)
12.25/5.15	Case Reductions:
12.25/5.15	The following Case expression
12.25/5.15	"case maxFM fm_r of {
12.25/5.15	Nothing  -> Just key;
12.25/5.15	Just key1  -> Just key1}
12.25/5.15	"
12.25/5.15	is transformed to
12.25/5.15	"maxFM0 key Nothing = Just key;
12.25/5.15	maxFM0 key (Just key1) = Just key1;
12.25/5.15	"
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(2)
12.25/5.15	Obligation:
12.25/5.15	mainModule Main 
12.25/5.15	module FiniteMap where {
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.25/5.15	
12.25/5.15	maxFM :: Ord a => FiniteMap a b  ->  Maybe a;
12.25/5.15	maxFM EmptyFM = Nothing;
12.25/5.15	maxFM (Branch key _ _ _ fm_r) = maxFM0 key (maxFM fm_r);
12.25/5.15	
12.25/5.15	maxFM0 key Nothing = Just key;
12.25/5.15	maxFM0 key (Just key1) = Just key1;
12.25/5.15	
12.25/5.15	}
12.25/5.15	module Maybe where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	module Main where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(3) BR (EQUIVALENT)
12.25/5.15	Replaced joker patterns by fresh variables and removed binding patterns.
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(4)
12.25/5.15	Obligation:
12.25/5.15	mainModule Main 
12.25/5.15	module FiniteMap where {
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.25/5.15	
12.25/5.15	maxFM :: Ord a => FiniteMap a b  ->  Maybe a;
12.25/5.15	maxFM EmptyFM = Nothing;
12.25/5.15	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
12.25/5.15	
12.25/5.15	maxFM0 key Nothing = Just key;
12.25/5.15	maxFM0 key (Just key1) = Just key1;
12.25/5.15	
12.25/5.15	}
12.25/5.15	module Maybe where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	module Main where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(5) COR (EQUIVALENT)
12.25/5.15	Cond Reductions:
12.25/5.15	The following Function with conditions
12.25/5.15	"undefined |Falseundefined;
12.25/5.15	"
12.25/5.15	is transformed to
12.25/5.15	"undefined  = undefined1;
12.25/5.15	"
12.25/5.15	"undefined0 True = undefined;
12.25/5.15	"
12.25/5.15	"undefined1  = undefined0 False;
12.25/5.15	"
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(6)
12.25/5.15	Obligation:
12.25/5.15	mainModule Main 
12.25/5.15	module FiniteMap where {
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.25/5.15	
12.25/5.15	maxFM :: Ord a => FiniteMap a b  ->  Maybe a;
12.25/5.15	maxFM EmptyFM = Nothing;
12.25/5.15	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
12.25/5.15	
12.25/5.15	maxFM0 key Nothing = Just key;
12.25/5.15	maxFM0 key (Just key1) = Just key1;
12.25/5.15	
12.25/5.15	}
12.25/5.15	module Maybe where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Main;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	module Main where {
12.25/5.15	import qualified FiniteMap;
12.25/5.15	import qualified Maybe;
12.25/5.15	import qualified Prelude;
12.25/5.15	}
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(7) Narrow (SOUND)
12.25/5.15	Haskell To QDPs
12.25/5.15	
12.25/5.15	digraph dp_graph {
12.25/5.15	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.maxFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.25/5.15	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];
12.25/5.15	15 -> 4[label="",style="solid", color="burlywood", weight=3];
12.25/5.15	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];
12.25/5.15	16 -> 5[label="",style="solid", color="burlywood", weight=3];
12.25/5.15	4[label="FiniteMap.maxFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
12.25/5.15	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];
12.25/5.15	6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
12.25/5.15	7[label="FiniteMap.maxFM0 wv30 (FiniteMap.maxFM wv34)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
12.25/5.15	9 -> 3[label="",style="dashed", color="red", weight=0];
12.25/5.15	9[label="FiniteMap.maxFM wv34",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
12.25/5.15	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];
12.25/5.15	17 -> 11[label="",style="solid", color="burlywood", weight=3];
12.25/5.15	18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
12.25/5.15	18 -> 12[label="",style="solid", color="burlywood", weight=3];
12.25/5.15	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];
12.25/5.15	12[label="FiniteMap.maxFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
12.25/5.15	13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(8)
12.25/5.15	Obligation:
12.25/5.15	Q DP problem:
12.25/5.15	The TRS P consists of the following rules:
12.25/5.15	
12.25/5.15	   new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h, ba) -> new_maxFM(wv34, h, ba)
12.25/5.15	
12.25/5.15	R is empty.
12.25/5.15	Q is empty.
12.25/5.15	We have to consider all minimal (P,Q,R)-chains.
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(9) QDPSizeChangeProof (EQUIVALENT)
12.25/5.15	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. 
12.25/5.15	
12.25/5.15	From the DPs we obtained the following set of size-change graphs:
12.25/5.15	*new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h, ba) -> new_maxFM(wv34, h, ba)
12.25/5.15	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.25/5.15	
12.25/5.15	
12.25/5.15	----------------------------------------
12.25/5.15	
12.25/5.15	(10)
12.25/5.15	YES
12.36/7.48	EOF