9.94/4.47	YES
11.77/5.04	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
11.77/5.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.77/5.04	
11.77/5.04	
11.77/5.04	H-Termination with start terms of the given HASKELL could be proven:
11.77/5.04	
11.77/5.04	(0) HASKELL
11.77/5.04	(1) CR [EQUIVALENT, 0 ms]
11.77/5.04	(2) HASKELL
11.77/5.04	(3) BR [EQUIVALENT, 0 ms]
11.77/5.04	(4) HASKELL
11.77/5.04	(5) COR [EQUIVALENT, 0 ms]
11.77/5.04	(6) HASKELL
11.77/5.04	(7) Narrow [SOUND, 0 ms]
11.77/5.04	(8) QDP
11.77/5.04	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.77/5.04	(10) YES
11.77/5.04	
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(0)
11.77/5.04	Obligation:
11.77/5.04	mainModule Main 
11.77/5.04	module FiniteMap where {
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.77/5.04	
11.77/5.04	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.77/5.04	minFM EmptyFM = Nothing;
11.77/5.04	minFM (Branch key _ _ fm_l _) = case minFM fm_l of { 
11.77/5.04	Nothing-> Just key; 
11.77/5.04	Just key1-> Just key1; 
11.77/5.04	 } ;
11.77/5.04	
11.77/5.04	}
11.77/5.04	module Maybe where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	module Main where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(1) CR (EQUIVALENT)
11.77/5.04	Case Reductions:
11.77/5.04	The following Case expression
11.77/5.04	"case minFM fm_l of {
11.77/5.04	Nothing  -> Just key;
11.77/5.04	Just key1  -> Just key1}
11.77/5.04	"
11.77/5.04	is transformed to
11.77/5.04	"minFM0 key Nothing = Just key;
11.77/5.04	minFM0 key (Just key1) = Just key1;
11.77/5.04	"
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(2)
11.77/5.04	Obligation:
11.77/5.04	mainModule Main 
11.77/5.04	module FiniteMap where {
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.77/5.04	
11.77/5.04	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.77/5.04	minFM EmptyFM = Nothing;
11.77/5.04	minFM (Branch key _ _ fm_l _) = minFM0 key (minFM fm_l);
11.77/5.04	
11.77/5.04	minFM0 key Nothing = Just key;
11.77/5.04	minFM0 key (Just key1) = Just key1;
11.77/5.04	
11.77/5.04	}
11.77/5.04	module Maybe where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	module Main where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(3) BR (EQUIVALENT)
11.77/5.04	Replaced joker patterns by fresh variables and removed binding patterns.
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(4)
11.77/5.04	Obligation:
11.77/5.04	mainModule Main 
11.77/5.04	module FiniteMap where {
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.77/5.04	
11.77/5.04	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.77/5.04	minFM EmptyFM = Nothing;
11.77/5.04	minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);
11.77/5.04	
11.77/5.04	minFM0 key Nothing = Just key;
11.77/5.04	minFM0 key (Just key1) = Just key1;
11.77/5.04	
11.77/5.04	}
11.77/5.04	module Maybe where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	module Main where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(5) COR (EQUIVALENT)
11.77/5.04	Cond Reductions:
11.77/5.04	The following Function with conditions
11.77/5.04	"undefined |Falseundefined;
11.77/5.04	"
11.77/5.04	is transformed to
11.77/5.04	"undefined  = undefined1;
11.77/5.04	"
11.77/5.04	"undefined0 True = undefined;
11.77/5.04	"
11.77/5.04	"undefined1  = undefined0 False;
11.77/5.04	"
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(6)
11.77/5.04	Obligation:
11.77/5.04	mainModule Main 
11.77/5.04	module FiniteMap where {
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.77/5.04	
11.77/5.04	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
11.77/5.04	minFM EmptyFM = Nothing;
11.77/5.04	minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);
11.77/5.04	
11.77/5.04	minFM0 key Nothing = Just key;
11.77/5.04	minFM0 key (Just key1) = Just key1;
11.77/5.04	
11.77/5.04	}
11.77/5.04	module Maybe where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Main;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	module Main where {
11.77/5.04	import qualified FiniteMap;
11.77/5.04	import qualified Maybe;
11.77/5.04	import qualified Prelude;
11.77/5.04	}
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(7) Narrow (SOUND)
11.77/5.04	Haskell To QDPs
11.77/5.04	
11.77/5.04	digraph dp_graph {
11.77/5.04	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.minFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.77/5.04	3[label="FiniteMap.minFM 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];
11.77/5.04	15 -> 4[label="",style="solid", color="burlywood", weight=3];
11.77/5.04	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];
11.77/5.04	16 -> 5[label="",style="solid", color="burlywood", weight=3];
11.77/5.04	4[label="FiniteMap.minFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
11.77/5.04	5[label="FiniteMap.minFM (FiniteMap.Branch wv30 wv31 wv32 wv33 wv34)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.77/5.04	6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
11.77/5.04	7[label="FiniteMap.minFM0 wv30 (FiniteMap.minFM wv33)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
11.77/5.04	9 -> 3[label="",style="dashed", color="red", weight=0];
11.77/5.04	9[label="FiniteMap.minFM wv33",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
11.77/5.04	8[label="FiniteMap.minFM0 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];
11.77/5.04	17 -> 11[label="",style="solid", color="burlywood", weight=3];
11.77/5.04	18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
11.77/5.04	18 -> 12[label="",style="solid", color="burlywood", weight=3];
11.77/5.04	10[label="wv33",fontsize=16,color="green",shape="box"];11[label="FiniteMap.minFM0 wv30 Nothing",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
11.77/5.04	12[label="FiniteMap.minFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
11.77/5.04	13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(8)
11.77/5.04	Obligation:
11.77/5.04	Q DP problem:
11.77/5.04	The TRS P consists of the following rules:
11.77/5.04	
11.77/5.04	   new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h)
11.77/5.04	
11.77/5.04	R is empty.
11.77/5.04	Q is empty.
11.77/5.04	We have to consider all minimal (P,Q,R)-chains.
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(9) QDPSizeChangeProof (EQUIVALENT)
11.77/5.04	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. 
11.77/5.04	
11.77/5.04	From the DPs we obtained the following set of size-change graphs:
11.77/5.04	*new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h) -> new_minFM(wv33, h)
11.77/5.04	The graph contains the following edges 1 > 1, 2 >= 2
11.77/5.04	
11.77/5.04	
11.77/5.04	----------------------------------------
11.77/5.04	
11.77/5.04	(10)
11.77/5.04	YES
12.04/7.81	EOF