9.56/4.54	YES
11.55/5.08	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.55/5.08	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.55/5.08	
11.55/5.08	
11.55/5.08	H-Termination with start terms of the given HASKELL could be proven:
11.55/5.08	
11.55/5.08	(0) HASKELL
11.55/5.08	(1) CR [EQUIVALENT, 0 ms]
11.55/5.08	(2) HASKELL
11.55/5.08	(3) BR [EQUIVALENT, 0 ms]
11.55/5.08	(4) HASKELL
11.55/5.08	(5) COR [EQUIVALENT, 0 ms]
11.55/5.08	(6) HASKELL
11.55/5.08	(7) Narrow [SOUND, 0 ms]
11.55/5.08	(8) QDP
11.55/5.08	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.55/5.08	(10) YES
11.55/5.08	
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(0)
11.55/5.08	Obligation:
11.55/5.08	mainModule Main 
11.55/5.08	module FiniteMap where {
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.55/5.08	
11.55/5.08	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.55/5.08	maxFM EmptyFM = Nothing;
11.55/5.08	maxFM (Branch key _ _ _ fm_r) = case maxFM fm_r of { 
11.55/5.08	Nothing-> Just key; 
11.55/5.08	Just key1-> Just key1; 
11.55/5.08	 } ;
11.55/5.08	
11.55/5.08	}
11.55/5.08	module Maybe where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	module Main where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(1) CR (EQUIVALENT)
11.55/5.08	Case Reductions:
11.55/5.08	The following Case expression
11.55/5.08	"case maxFM fm_r of {
11.55/5.08	Nothing  -> Just key;
11.55/5.08	Just key1  -> Just key1}
11.55/5.08	"
11.55/5.08	is transformed to
11.55/5.08	"maxFM0 key Nothing = Just key;
11.55/5.08	maxFM0 key (Just key1) = Just key1;
11.55/5.08	"
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(2)
11.55/5.08	Obligation:
11.55/5.08	mainModule Main 
11.55/5.08	module FiniteMap where {
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.55/5.08	
11.55/5.08	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.55/5.08	maxFM EmptyFM = Nothing;
11.55/5.08	maxFM (Branch key _ _ _ fm_r) = maxFM0 key (maxFM fm_r);
11.55/5.08	
11.55/5.08	maxFM0 key Nothing = Just key;
11.55/5.08	maxFM0 key (Just key1) = Just key1;
11.55/5.08	
11.55/5.08	}
11.55/5.08	module Maybe where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	module Main where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(3) BR (EQUIVALENT)
11.55/5.08	Replaced joker patterns by fresh variables and removed binding patterns.
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(4)
11.55/5.08	Obligation:
11.55/5.08	mainModule Main 
11.55/5.08	module FiniteMap where {
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.55/5.08	
11.55/5.08	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.55/5.08	maxFM EmptyFM = Nothing;
11.55/5.08	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
11.55/5.08	
11.55/5.08	maxFM0 key Nothing = Just key;
11.55/5.08	maxFM0 key (Just key1) = Just key1;
11.55/5.08	
11.55/5.08	}
11.55/5.08	module Maybe where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	module Main where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(5) COR (EQUIVALENT)
11.55/5.08	Cond Reductions:
11.55/5.08	The following Function with conditions
11.55/5.08	"undefined |Falseundefined;
11.55/5.08	"
11.55/5.08	is transformed to
11.55/5.08	"undefined  = undefined1;
11.55/5.08	"
11.55/5.08	"undefined0 True = undefined;
11.55/5.08	"
11.55/5.08	"undefined1  = undefined0 False;
11.55/5.08	"
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(6)
11.55/5.08	Obligation:
11.55/5.08	mainModule Main 
11.55/5.08	module FiniteMap where {
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.55/5.08	
11.55/5.08	maxFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.55/5.08	maxFM EmptyFM = Nothing;
11.55/5.08	maxFM (Branch key vy vz wu fm_r) = maxFM0 key (maxFM fm_r);
11.55/5.08	
11.55/5.08	maxFM0 key Nothing = Just key;
11.55/5.08	maxFM0 key (Just key1) = Just key1;
11.55/5.08	
11.55/5.08	}
11.55/5.08	module Maybe where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Main;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	module Main where {
11.55/5.08	import qualified FiniteMap;
11.55/5.08	import qualified Maybe;
11.55/5.08	import qualified Prelude;
11.55/5.08	}
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(7) Narrow (SOUND)
11.55/5.08	Haskell To QDPs
11.55/5.08	
11.55/5.08	digraph dp_graph {
11.55/5.08	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.maxFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.55/5.08	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];
11.55/5.08	15 -> 4[label="",style="solid", color="burlywood", weight=3];
11.55/5.08	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];
11.55/5.08	16 -> 5[label="",style="solid", color="burlywood", weight=3];
11.55/5.08	4[label="FiniteMap.maxFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
11.55/5.08	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];
11.55/5.08	6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
11.55/5.08	7[label="FiniteMap.maxFM0 wv30 (FiniteMap.maxFM wv34)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
11.55/5.08	9 -> 3[label="",style="dashed", color="red", weight=0];
11.55/5.08	9[label="FiniteMap.maxFM wv34",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
11.55/5.08	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];
11.55/5.08	17 -> 11[label="",style="solid", color="burlywood", weight=3];
11.55/5.08	18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
11.55/5.08	18 -> 12[label="",style="solid", color="burlywood", weight=3];
11.55/5.08	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];
11.55/5.08	12[label="FiniteMap.maxFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.55/5.08	13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(8)
11.55/5.08	Obligation:
11.55/5.08	Q DP problem:
11.55/5.08	The TRS P consists of the following rules:
11.55/5.08	
11.55/5.08	   new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_maxFM(wv34, h)
11.55/5.08	
11.55/5.08	R is empty.
11.55/5.08	Q is empty.
11.55/5.08	We have to consider all minimal (P,Q,R)-chains.
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(9) QDPSizeChangeProof (EQUIVALENT)
11.55/5.08	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. 
11.55/5.08	
11.55/5.08	From the DPs we obtained the following set of size-change graphs:
11.55/5.08	*new_maxFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_maxFM(wv34, h)
11.55/5.08	The graph contains the following edges 1 > 1, 2 >= 2
11.55/5.08	
11.55/5.08	
11.55/5.08	----------------------------------------
11.55/5.08	
11.55/5.08	(10)
11.55/5.08	YES
11.69/7.40	EOF