9.61/4.31	YES
11.05/4.77	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.05/4.77	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.05/4.77	
11.05/4.77	
11.05/4.77	H-Termination with start terms of the given HASKELL could be proven:
11.05/4.77	
11.05/4.77	(0) HASKELL
11.05/4.77	(1) BR [EQUIVALENT, 0 ms]
11.05/4.77	(2) HASKELL
11.05/4.77	(3) COR [EQUIVALENT, 0 ms]
11.05/4.77	(4) HASKELL
11.05/4.77	(5) Narrow [SOUND, 0 ms]
11.05/4.77	(6) QDP
11.05/4.77	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.05/4.77	(8) YES
11.05/4.77	
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(0)
11.05/4.77	Obligation:
11.05/4.77	mainModule Main 
11.05/4.77	module FiniteMap where {
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
11.05/4.77	
11.05/4.77	emptyFM :: FiniteMap b a;
11.05/4.77	emptyFM = EmptyFM;
11.05/4.77	
11.05/4.77	mapFM :: (a  ->  c  ->  b)  ->  FiniteMap a c  ->  FiniteMap a b;
11.05/4.77	mapFM f EmptyFM = emptyFM;
11.05/4.77	mapFM f (Branch key elt size fm_l fm_r) = Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r);
11.05/4.77	
11.05/4.77	}
11.05/4.77	module Maybe where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	module Main where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(1) BR (EQUIVALENT)
11.05/4.77	Replaced joker patterns by fresh variables and removed binding patterns.
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(2)
11.05/4.77	Obligation:
11.05/4.77	mainModule Main 
11.05/4.77	module FiniteMap where {
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.05/4.77	
11.05/4.77	emptyFM :: FiniteMap a b;
11.05/4.77	emptyFM = EmptyFM;
11.05/4.77	
11.05/4.77	mapFM :: (b  ->  a  ->  c)  ->  FiniteMap b a  ->  FiniteMap b c;
11.05/4.77	mapFM f EmptyFM = emptyFM;
11.05/4.77	mapFM f (Branch key elt size fm_l fm_r) = Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r);
11.05/4.77	
11.05/4.77	}
11.05/4.77	module Maybe where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	module Main where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(3) COR (EQUIVALENT)
11.05/4.77	Cond Reductions:
11.05/4.77	The following Function with conditions
11.05/4.77	"undefined |Falseundefined;
11.05/4.77	"
11.05/4.77	is transformed to
11.05/4.77	"undefined  = undefined1;
11.05/4.77	"
11.05/4.77	"undefined0 True = undefined;
11.05/4.77	"
11.05/4.77	"undefined1  = undefined0 False;
11.05/4.77	"
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(4)
11.05/4.77	Obligation:
11.05/4.77	mainModule Main 
11.05/4.77	module FiniteMap where {
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.05/4.77	
11.05/4.77	emptyFM :: FiniteMap b a;
11.05/4.77	emptyFM = EmptyFM;
11.05/4.77	
11.05/4.77	mapFM :: (c  ->  a  ->  b)  ->  FiniteMap c a  ->  FiniteMap c b;
11.05/4.77	mapFM f EmptyFM = emptyFM;
11.05/4.77	mapFM f (Branch key elt size fm_l fm_r) = Branch key (f key elt) size (mapFM f fm_l) (mapFM f fm_r);
11.05/4.77	
11.05/4.77	}
11.05/4.77	module Maybe where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Main;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	module Main where {
11.05/4.77	import qualified FiniteMap;
11.05/4.77	import qualified Maybe;
11.05/4.77	import qualified Prelude;
11.05/4.77	}
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(5) Narrow (SOUND)
11.05/4.77	Haskell To QDPs
11.05/4.77	
11.05/4.77	digraph dp_graph {
11.05/4.77	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.mapFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.05/4.77	3[label="FiniteMap.mapFM vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.05/4.77	4[label="FiniteMap.mapFM vy3 vy4",fontsize=16,color="burlywood",shape="triangle"];17[label="vy4/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 17[label="",style="solid", color="burlywood", weight=9];
11.05/4.77	17 -> 5[label="",style="solid", color="burlywood", weight=3];
11.05/4.77	18[label="vy4/FiniteMap.Branch vy40 vy41 vy42 vy43 vy44",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
11.05/4.77	18 -> 6[label="",style="solid", color="burlywood", weight=3];
11.05/4.77	5[label="FiniteMap.mapFM vy3 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
11.05/4.77	6[label="FiniteMap.mapFM vy3 (FiniteMap.Branch vy40 vy41 vy42 vy43 vy44)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.05/4.77	7[label="FiniteMap.emptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.05/4.77	8[label="FiniteMap.Branch vy40 (vy3 vy40 vy41) vy42 (FiniteMap.mapFM vy3 vy43) (FiniteMap.mapFM vy3 vy44)",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
11.05/4.77	8 -> 11[label="",style="dashed", color="green", weight=3];
11.05/4.77	8 -> 12[label="",style="dashed", color="green", weight=3];
11.05/4.77	9[label="FiniteMap.EmptyFM",fontsize=16,color="green",shape="box"];10[label="vy3 vy40 vy41",fontsize=16,color="green",shape="box"];10 -> 13[label="",style="dashed", color="green", weight=3];
11.05/4.77	10 -> 14[label="",style="dashed", color="green", weight=3];
11.05/4.77	11 -> 4[label="",style="dashed", color="red", weight=0];
11.05/4.77	11[label="FiniteMap.mapFM vy3 vy43",fontsize=16,color="magenta"];11 -> 15[label="",style="dashed", color="magenta", weight=3];
11.05/4.77	12 -> 4[label="",style="dashed", color="red", weight=0];
11.05/4.77	12[label="FiniteMap.mapFM vy3 vy44",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3];
11.05/4.77	13[label="vy40",fontsize=16,color="green",shape="box"];14[label="vy41",fontsize=16,color="green",shape="box"];15[label="vy43",fontsize=16,color="green",shape="box"];16[label="vy44",fontsize=16,color="green",shape="box"];}
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(6)
11.05/4.77	Obligation:
11.05/4.77	Q DP problem:
11.05/4.77	The TRS P consists of the following rules:
11.05/4.77	
11.05/4.77	   new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) -> new_mapFM(vy3, vy44, h, ba, bb)
11.05/4.77	   new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) -> new_mapFM(vy3, vy43, h, ba, bb)
11.05/4.77	
11.05/4.77	R is empty.
11.05/4.77	Q is empty.
11.05/4.77	We have to consider all minimal (P,Q,R)-chains.
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(7) QDPSizeChangeProof (EQUIVALENT)
11.05/4.77	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. 
11.05/4.77	
11.05/4.77	From the DPs we obtained the following set of size-change graphs:
11.05/4.77	*new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) -> new_mapFM(vy3, vy44, h, ba, bb)
11.05/4.77	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
11.05/4.77	
11.05/4.77	
11.05/4.77	*new_mapFM(vy3, Branch(vy40, vy41, vy42, vy43, vy44), h, ba, bb) -> new_mapFM(vy3, vy43, h, ba, bb)
11.05/4.77	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
11.05/4.77	
11.05/4.77	
11.05/4.77	----------------------------------------
11.05/4.77	
11.05/4.77	(8)
11.05/4.77	YES
11.05/4.81	EOF