10.29/4.50	YES
12.63/5.11	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.63/5.11	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.63/5.11	
12.63/5.11	
12.63/5.11	H-Termination with start terms of the given HASKELL could be proven:
12.63/5.11	
12.63/5.11	(0) HASKELL
12.63/5.11	(1) LR [EQUIVALENT, 0 ms]
12.63/5.11	(2) HASKELL
12.63/5.11	(3) BR [EQUIVALENT, 0 ms]
12.63/5.11	(4) HASKELL
12.63/5.11	(5) COR [EQUIVALENT, 0 ms]
12.63/5.11	(6) HASKELL
12.63/5.11	(7) Narrow [SOUND, 0 ms]
12.63/5.11	(8) AND
12.63/5.11	    (9) QDP
12.63/5.11	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.63/5.11	        (11) YES
12.63/5.11	    (12) QDP
12.63/5.11	        (13) TransformationProof [EQUIVALENT, 0 ms]
12.63/5.11	        (14) QDP
12.63/5.11	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.63/5.11	        (16) YES
12.63/5.11	
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(0)
12.63/5.11	Obligation:
12.63/5.11	mainModule Main 
12.63/5.11	module FiniteMap where {
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.63/5.11	
12.63/5.11	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.63/5.11	}
12.63/5.11	eltsFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [a];
12.63/5.11	eltsFM_GE fm fr = foldFM_GE (\key elt rest ->elt : rest) [] fr fm;
12.63/5.11	
12.63/5.11	foldFM_GE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
12.63/5.11	foldFM_GE k z fr EmptyFM = z;
12.63/5.11	foldFM_GE k z fr (Branch key elt _ fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
12.63/5.11	 | otherwise = foldFM_GE k z fr fm_r;
12.63/5.11	
12.63/5.11	}
12.63/5.11	module Maybe where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	module Main where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(1) LR (EQUIVALENT)
12.63/5.11	Lambda Reductions:
12.63/5.11	The following Lambda expression
12.63/5.11	"\keyeltrest->elt : rest"
12.63/5.11	is transformed to
12.63/5.11	"eltsFM_GE0 key elt rest = elt : rest;
12.63/5.11	"
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(2)
12.63/5.11	Obligation:
12.63/5.11	mainModule Main 
12.63/5.11	module FiniteMap where {
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.63/5.11	
12.63/5.11	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.63/5.11	}
12.63/5.11	eltsFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [b];
12.63/5.11	eltsFM_GE fm fr = foldFM_GE eltsFM_GE0 [] fr fm;
12.63/5.11	
12.63/5.11	eltsFM_GE0 key elt rest = elt : rest;
12.63/5.11	
12.63/5.11	foldFM_GE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
12.63/5.11	foldFM_GE k z fr EmptyFM = z;
12.63/5.11	foldFM_GE k z fr (Branch key elt _ fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
12.63/5.11	 | otherwise = foldFM_GE k z fr fm_r;
12.63/5.11	
12.63/5.11	}
12.63/5.11	module Maybe where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	module Main where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(3) BR (EQUIVALENT)
12.63/5.11	Replaced joker patterns by fresh variables and removed binding patterns.
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(4)
12.63/5.11	Obligation:
12.63/5.11	mainModule Main 
12.63/5.11	module FiniteMap where {
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.63/5.11	
12.63/5.11	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.63/5.11	}
12.63/5.11	eltsFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [a];
12.63/5.11	eltsFM_GE fm fr = foldFM_GE eltsFM_GE0 [] fr fm;
12.63/5.11	
12.63/5.11	eltsFM_GE0 key elt rest = elt : rest;
12.63/5.11	
12.63/5.11	foldFM_GE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
12.63/5.11	foldFM_GE k z fr EmptyFM = z;
12.63/5.11	foldFM_GE k z fr (Branch key elt vy fm_l fm_r)  | key >= fr = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l
12.63/5.11	 | otherwise = foldFM_GE k z fr fm_r;
12.63/5.11	
12.63/5.11	}
12.63/5.11	module Maybe where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	module Main where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(5) COR (EQUIVALENT)
12.63/5.11	Cond Reductions:
12.63/5.11	The following Function with conditions
12.63/5.11	"undefined |Falseundefined;
12.63/5.11	"
12.63/5.11	is transformed to
12.63/5.11	"undefined  = undefined1;
12.63/5.11	"
12.63/5.11	"undefined0 True = undefined;
12.63/5.11	"
12.63/5.11	"undefined1  = undefined0 False;
12.63/5.11	"
12.63/5.11	The following Function with conditions
12.63/5.11	"foldFM_GE k z fr EmptyFM = z;
12.63/5.11	foldFM_GE k z fr (Branch key elt vy fm_l fm_r)|key >= frfoldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l|otherwisefoldFM_GE k z fr fm_r;
12.63/5.11	"
12.63/5.11	is transformed to
12.63/5.11	"foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
12.63/5.11	foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);
12.63/5.11	"
12.63/5.11	"foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
12.63/5.11	"
12.63/5.11	"foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
12.63/5.11	foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;
12.63/5.11	"
12.63/5.11	"foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);
12.63/5.11	"
12.63/5.11	"foldFM_GE3 k z fr EmptyFM = z;
12.63/5.11	foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
12.63/5.11	"
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(6)
12.63/5.11	Obligation:
12.63/5.11	mainModule Main 
12.63/5.11	module FiniteMap where {
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.63/5.11	
12.63/5.11	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.63/5.11	}
12.63/5.11	eltsFM_GE :: Ord b => FiniteMap b a  ->  b  ->  [a];
12.63/5.11	eltsFM_GE fm fr = foldFM_GE eltsFM_GE0 [] fr fm;
12.63/5.11	
12.63/5.11	eltsFM_GE0 key elt rest = elt : rest;
12.63/5.11	
12.63/5.11	foldFM_GE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
12.63/5.11	foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
12.63/5.11	foldFM_GE k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r);
12.63/5.11	
12.63/5.11	foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
12.63/5.11	
12.63/5.11	foldFM_GE1 k z fr key elt vy fm_l fm_r True = foldFM_GE k (k key elt (foldFM_GE k z fr fm_r)) fr fm_l;
12.63/5.11	foldFM_GE1 k z fr key elt vy fm_l fm_r False = foldFM_GE0 k z fr key elt vy fm_l fm_r otherwise;
12.63/5.11	
12.63/5.11	foldFM_GE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_GE1 k z fr key elt vy fm_l fm_r (key >= fr);
12.63/5.11	
12.63/5.11	foldFM_GE3 k z fr EmptyFM = z;
12.63/5.11	foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
12.63/5.11	
12.63/5.11	}
12.63/5.11	module Maybe where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Main;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	module Main where {
12.63/5.11	import qualified FiniteMap;
12.63/5.11	import qualified Maybe;
12.63/5.11	import qualified Prelude;
12.63/5.11	}
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(7) Narrow (SOUND)
12.63/5.11	Haskell To QDPs
12.63/5.11	
12.63/5.11	digraph dp_graph {
12.63/5.11	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.eltsFM_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.63/5.11	3[label="FiniteMap.eltsFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.63/5.11	4[label="FiniteMap.eltsFM_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.63/5.11	5[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];54[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 54[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	54 -> 6[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	55[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 55[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	55 -> 7[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	6[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.63/5.11	7[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.63/5.11	8[label="FiniteMap.foldFM_GE3 FiniteMap.eltsFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.63/5.11	9[label="FiniteMap.foldFM_GE2 FiniteMap.eltsFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.63/5.11	10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 >= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.63/5.11	12[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= LT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.63/5.11	13[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",fontsize=16,color="burlywood",shape="box"];56[label="wz30/()",fontsize=10,color="white",style="solid",shape="box"];13 -> 56[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	56 -> 14[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	14[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == LT))",fontsize=16,color="burlywood",shape="box"];57[label="wz4/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 57[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	57 -> 15[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	15[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
12.63/5.11	16[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
12.63/5.11	17[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
12.63/5.11	18[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 [] () () wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
12.63/5.11	19 -> 20[label="",style="dashed", color="red", weight=0];
12.63/5.11	19[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 (FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 [] () wz34)) () wz33",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	21 -> 5[label="",style="dashed", color="red", weight=0];
12.63/5.11	21[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 [] () wz34",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	21 -> 23[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	20[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () wz33",fontsize=16,color="burlywood",shape="triangle"];58[label="wz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];20 -> 58[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	58 -> 24[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	59[label="wz33/FiniteMap.Branch wz330 wz331 wz332 wz333 wz334",fontsize=10,color="white",style="solid",shape="box"];20 -> 59[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	59 -> 25[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	22[label="wz34",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
12.63/5.11	25[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
12.63/5.11	26[label="FiniteMap.foldFM_GE3 FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
12.63/5.11	27[label="FiniteMap.foldFM_GE2 FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () (FiniteMap.Branch wz330 wz331 wz332 wz333 wz334)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
12.63/5.11	28[label="FiniteMap.eltsFM_GE0 () wz31 wz5",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
12.63/5.11	29 -> 31[label="",style="dashed", color="red", weight=0];
12.63/5.11	29[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz31 wz5) () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	30[label="wz31 : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0];
12.63/5.11	32[label="FiniteMap.eltsFM_GE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (wz330 >= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
12.63/5.11	33[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (compare wz330 () /= LT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
12.63/5.11	34[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () wz330 wz331 wz332 wz333 wz334 (not (compare wz330 () == LT))",fontsize=16,color="burlywood",shape="box"];60[label="wz330/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 60[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	60 -> 35[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	35[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (compare () () == LT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
12.63/5.11	36[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not (EQ == LT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
12.63/5.11	37[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () () wz331 wz332 wz333 wz334 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
12.63/5.11	38[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () () wz331 wz332 wz333 wz334 True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
12.63/5.11	39 -> 20[label="",style="dashed", color="red", weight=0];
12.63/5.11	39[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 (FiniteMap.eltsFM_GE0 () wz331 (FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 wz6 () wz334)) () wz333",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	39 -> 41[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	39 -> 42[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	40[label="wz333",fontsize=16,color="green",shape="box"];41[label="wz331",fontsize=16,color="green",shape="box"];42[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 wz6 () wz334",fontsize=16,color="burlywood",shape="box"];61[label="wz334/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];42 -> 61[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	61 -> 43[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	62[label="wz334/FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344",fontsize=10,color="white",style="solid",shape="box"];42 -> 62[label="",style="solid", color="burlywood", weight=9];
12.63/5.11	62 -> 44[label="",style="solid", color="burlywood", weight=3];
12.63/5.11	43[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
12.63/5.11	44[label="FiniteMap.foldFM_GE FiniteMap.eltsFM_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
12.63/5.11	45[label="FiniteMap.foldFM_GE3 FiniteMap.eltsFM_GE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
12.63/5.11	46[label="FiniteMap.foldFM_GE2 FiniteMap.eltsFM_GE0 wz6 () (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
12.63/5.11	47[label="wz6",fontsize=16,color="green",shape="box"];48 -> 31[label="",style="dashed", color="red", weight=0];
12.63/5.11	48[label="FiniteMap.foldFM_GE1 FiniteMap.eltsFM_GE0 wz6 () wz3340 wz3341 wz3342 wz3343 wz3344 (wz3340 >= ())",fontsize=16,color="magenta"];48 -> 49[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	48 -> 50[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	48 -> 51[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	48 -> 52[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	48 -> 53[label="",style="dashed", color="magenta", weight=3];
12.63/5.11	49[label="wz3340",fontsize=16,color="green",shape="box"];50[label="wz3344",fontsize=16,color="green",shape="box"];51[label="wz3343",fontsize=16,color="green",shape="box"];52[label="wz3342",fontsize=16,color="green",shape="box"];53[label="wz3341",fontsize=16,color="green",shape="box"];}
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(8)
12.63/5.11	Complex Obligation (AND)
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(9)
12.63/5.11	Obligation:
12.63/5.11	Q DP problem:
12.63/5.11	The TRS P consists of the following rules:
12.63/5.11	
12.63/5.11	   new_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h)
12.63/5.11	
12.63/5.11	R is empty.
12.63/5.11	Q is empty.
12.63/5.11	We have to consider all minimal (P,Q,R)-chains.
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(10) QDPSizeChangeProof (EQUIVALENT)
12.63/5.11	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. 
12.63/5.11	
12.63/5.11	From the DPs we obtained the following set of size-change graphs:
12.63/5.11	*new_foldFM_GE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE3(@0, wz34, h)
12.63/5.11	The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
12.63/5.11	
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(11)
12.63/5.11	YES
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(12)
12.63/5.11	Obligation:
12.63/5.11	Q DP problem:
12.63/5.11	The TRS P consists of the following rules:
12.63/5.11	
12.63/5.11	   new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_eltsFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
12.63/5.11	   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.63/5.11	   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
12.63/5.11	
12.63/5.11	The TRS R consists of the following rules:
12.63/5.11	
12.63/5.11	   new_foldFM_GE0(wz6, EmptyFM, h) -> wz6
12.63/5.11	   new_eltsFM_GE0(wz31, wz5, h) -> :(wz31, wz5)
12.63/5.11	   new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.63/5.11	   new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_eltsFM_GE0(wz31, wz5, h)
12.63/5.11	   new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
12.63/5.11	   new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_eltsFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
12.63/5.11	
12.63/5.11	The set Q consists of the following terms:
12.63/5.11	
12.63/5.11	   new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6)
12.63/5.11	   new_foldFM_GE2(x0, x1, EmptyFM, x2)
12.63/5.11	   new_eltsFM_GE0(x0, x1, x2)
12.63/5.11	   new_foldFM_GE0(x0, EmptyFM, x1)
12.63/5.11	   new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5)
12.63/5.11	   new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.63/5.11	
12.63/5.11	We have to consider all minimal (P,Q,R)-chains.
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(13) TransformationProof (EQUIVALENT)
12.63/5.11	By rewriting [LPAR04] the rule new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_eltsFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h) at position [0] we obtained the following new rules [LPAR04]:
12.63/5.11	
12.63/5.11	   (new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(wz31, wz5), wz330, wz331, wz332, wz333, wz334, h),new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(wz31, wz5), wz330, wz331, wz332, wz333, wz334, h))
12.63/5.11	
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(14)
12.63/5.11	Obligation:
12.63/5.11	Q DP problem:
12.63/5.11	The TRS P consists of the following rules:
12.63/5.11	
12.63/5.11	   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.63/5.11	   new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
12.63/5.11	   new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(wz31, wz5), wz330, wz331, wz332, wz333, wz334, h)
12.63/5.11	
12.63/5.11	The TRS R consists of the following rules:
12.63/5.11	
12.63/5.11	   new_foldFM_GE0(wz6, EmptyFM, h) -> wz6
12.63/5.11	   new_eltsFM_GE0(wz31, wz5, h) -> :(wz31, wz5)
12.63/5.11	   new_foldFM_GE0(wz6, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.63/5.11	   new_foldFM_GE2(wz31, wz5, EmptyFM, h) -> new_eltsFM_GE0(wz31, wz5, h)
12.63/5.11	   new_foldFM_GE10(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
12.63/5.11	   new_foldFM_GE2(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_eltsFM_GE0(wz31, wz5, h), wz330, wz331, wz332, wz333, wz334, h)
12.63/5.11	
12.63/5.11	The set Q consists of the following terms:
12.63/5.11	
12.63/5.11	   new_foldFM_GE0(x0, Branch(x1, x2, x3, x4, x5), x6)
12.63/5.11	   new_foldFM_GE2(x0, x1, EmptyFM, x2)
12.63/5.11	   new_eltsFM_GE0(x0, x1, x2)
12.63/5.11	   new_foldFM_GE0(x0, EmptyFM, x1)
12.63/5.11	   new_foldFM_GE10(x0, @0, x1, x2, x3, x4, x5)
12.63/5.11	   new_foldFM_GE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.63/5.11	
12.63/5.11	We have to consider all minimal (P,Q,R)-chains.
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(15) QDPSizeChangeProof (EQUIVALENT)
12.63/5.11	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. 
12.63/5.11	
12.63/5.11	From the DPs we obtained the following set of size-change graphs:
12.63/5.11	*new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz6, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.63/5.11	The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7
12.63/5.11	
12.63/5.11	
12.63/5.11	*new_foldFM_GE1(wz6, @0, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz331, new_foldFM_GE0(wz6, wz334, h), wz333, h)
12.63/5.11	The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4
12.63/5.11	
12.63/5.11	
12.63/5.11	*new_foldFM_GE(wz31, wz5, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(wz31, wz5), wz330, wz331, wz332, wz333, wz334, h)
12.63/5.11	The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
12.63/5.11	
12.63/5.11	
12.63/5.11	----------------------------------------
12.63/5.11	
12.63/5.11	(16)
12.63/5.11	YES
12.68/5.15	EOF