10.01/4.54	YES
12.25/5.12	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.25/5.12	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.25/5.12	
12.25/5.12	
12.25/5.12	H-Termination with start terms of the given HASKELL could be proven:
12.25/5.12	
12.25/5.12	(0) HASKELL
12.25/5.12	(1) CR [EQUIVALENT, 0 ms]
12.25/5.12	(2) HASKELL
12.25/5.12	(3) BR [EQUIVALENT, 0 ms]
12.25/5.12	(4) HASKELL
12.25/5.12	(5) COR [EQUIVALENT, 0 ms]
12.25/5.12	(6) HASKELL
12.25/5.12	(7) Narrow [SOUND, 0 ms]
12.25/5.12	(8) QDP
12.25/5.12	(9) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.25/5.12	(10) YES
12.25/5.12	
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(0)
12.25/5.12	Obligation:
12.25/5.12	mainModule Main 
12.25/5.12	module FiniteMap where {
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.25/5.12	
12.25/5.12	minFM :: Ord a => FiniteMap a b  ->  Maybe a;
12.25/5.12	minFM EmptyFM = Nothing;
12.25/5.12	minFM (Branch key _ _ fm_l _) = case minFM fm_l of { 
12.25/5.12	Nothing-> Just key; 
12.25/5.12	Just key1-> Just key1; 
12.25/5.12	 } ;
12.25/5.12	
12.25/5.12	}
12.25/5.12	module Maybe where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	module Main where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(1) CR (EQUIVALENT)
12.25/5.12	Case Reductions:
12.25/5.12	The following Case expression
12.25/5.12	"case minFM fm_l of {
12.25/5.12	Nothing  -> Just key;
12.25/5.12	Just key1  -> Just key1}
12.25/5.12	"
12.25/5.12	is transformed to
12.25/5.12	"minFM0 key Nothing = Just key;
12.25/5.12	minFM0 key (Just key1) = Just key1;
12.25/5.12	"
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(2)
12.25/5.12	Obligation:
12.25/5.12	mainModule Main 
12.25/5.12	module FiniteMap where {
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.25/5.12	
12.25/5.12	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
12.25/5.12	minFM EmptyFM = Nothing;
12.25/5.12	minFM (Branch key _ _ fm_l _) = minFM0 key (minFM fm_l);
12.25/5.12	
12.25/5.12	minFM0 key Nothing = Just key;
12.25/5.12	minFM0 key (Just key1) = Just key1;
12.25/5.12	
12.25/5.12	}
12.25/5.12	module Maybe where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	module Main where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(3) BR (EQUIVALENT)
12.25/5.12	Replaced joker patterns by fresh variables and removed binding patterns.
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(4)
12.25/5.12	Obligation:
12.25/5.12	mainModule Main 
12.25/5.12	module FiniteMap where {
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.25/5.12	
12.25/5.12	minFM :: Ord b => FiniteMap b a  ->  Maybe b;
12.25/5.12	minFM EmptyFM = Nothing;
12.25/5.12	minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);
12.25/5.12	
12.25/5.12	minFM0 key Nothing = Just key;
12.25/5.12	minFM0 key (Just key1) = Just key1;
12.25/5.12	
12.25/5.12	}
12.25/5.12	module Maybe where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	module Main where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(5) COR (EQUIVALENT)
12.25/5.12	Cond Reductions:
12.25/5.12	The following Function with conditions
12.25/5.12	"undefined |Falseundefined;
12.25/5.12	"
12.25/5.12	is transformed to
12.25/5.12	"undefined  = undefined1;
12.25/5.12	"
12.25/5.12	"undefined0 True = undefined;
12.25/5.12	"
12.25/5.12	"undefined1  = undefined0 False;
12.25/5.12	"
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(6)
12.25/5.12	Obligation:
12.25/5.12	mainModule Main 
12.25/5.12	module FiniteMap where {
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.25/5.12	
12.25/5.12	minFM :: Ord a => FiniteMap a b  ->  Maybe a;
12.25/5.12	minFM EmptyFM = Nothing;
12.25/5.12	minFM (Branch key vy vz fm_l wu) = minFM0 key (minFM fm_l);
12.25/5.12	
12.25/5.12	minFM0 key Nothing = Just key;
12.25/5.12	minFM0 key (Just key1) = Just key1;
12.25/5.12	
12.25/5.12	}
12.25/5.12	module Maybe where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Main;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	module Main where {
12.25/5.12	import qualified FiniteMap;
12.25/5.12	import qualified Maybe;
12.25/5.12	import qualified Prelude;
12.25/5.12	}
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(7) Narrow (SOUND)
12.25/5.12	Haskell To QDPs
12.25/5.12	
12.25/5.12	digraph dp_graph {
12.25/5.12	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.minFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.25/5.12	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];
12.25/5.12	15 -> 4[label="",style="solid", color="burlywood", weight=3];
12.25/5.12	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];
12.25/5.12	16 -> 5[label="",style="solid", color="burlywood", weight=3];
12.25/5.12	4[label="FiniteMap.minFM FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
12.25/5.12	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];
12.25/5.12	6[label="Nothing",fontsize=16,color="green",shape="box"];7 -> 8[label="",style="dashed", color="red", weight=0];
12.25/5.12	7[label="FiniteMap.minFM0 wv30 (FiniteMap.minFM wv33)",fontsize=16,color="magenta"];7 -> 9[label="",style="dashed", color="magenta", weight=3];
12.25/5.12	9 -> 3[label="",style="dashed", color="red", weight=0];
12.25/5.12	9[label="FiniteMap.minFM wv33",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
12.25/5.12	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];
12.25/5.12	17 -> 11[label="",style="solid", color="burlywood", weight=3];
12.25/5.12	18[label="wv4/Just wv40",fontsize=10,color="white",style="solid",shape="box"];8 -> 18[label="",style="solid", color="burlywood", weight=9];
12.25/5.12	18 -> 12[label="",style="solid", color="burlywood", weight=3];
12.25/5.12	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];
12.25/5.12	12[label="FiniteMap.minFM0 wv30 (Just wv40)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
12.25/5.12	13[label="Just wv30",fontsize=16,color="green",shape="box"];14[label="Just wv40",fontsize=16,color="green",shape="box"];}
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(8)
12.25/5.12	Obligation:
12.25/5.12	Q DP problem:
12.25/5.12	The TRS P consists of the following rules:
12.25/5.12	
12.25/5.12	   new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h, ba, bb) -> new_minFM(wv33, h, ba, bb)
12.25/5.12	
12.25/5.12	R is empty.
12.25/5.12	Q is empty.
12.25/5.12	We have to consider all minimal (P,Q,R)-chains.
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(9) QDPSizeChangeProof (EQUIVALENT)
12.25/5.12	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. 
12.25/5.12	
12.25/5.12	From the DPs we obtained the following set of size-change graphs:
12.25/5.12	*new_minFM(Branch(wv30, wv31, wv32, wv33, wv34), h, ba, bb) -> new_minFM(wv33, h, ba, bb)
12.25/5.12	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
12.25/5.12	
12.25/5.12	
12.25/5.12	----------------------------------------
12.25/5.12	
12.25/5.12	(10)
12.25/5.12	YES
12.25/5.17	EOF