10.21/4.55	YES
12.23/5.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.23/5.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.23/5.14	
12.23/5.14	
12.23/5.14	H-Termination with start terms of the given HASKELL could be proven:
12.23/5.14	
12.23/5.14	(0) HASKELL
12.23/5.14	(1) LR [EQUIVALENT, 0 ms]
12.23/5.14	(2) HASKELL
12.23/5.14	(3) BR [EQUIVALENT, 0 ms]
12.23/5.14	(4) HASKELL
12.23/5.14	(5) COR [EQUIVALENT, 0 ms]
12.23/5.14	(6) HASKELL
12.23/5.14	(7) Narrow [SOUND, 0 ms]
12.23/5.14	(8) AND
12.23/5.14	    (9) QDP
12.23/5.14	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.23/5.14	        (11) YES
12.23/5.14	    (12) QDP
12.23/5.14	        (13) TransformationProof [EQUIVALENT, 0 ms]
12.23/5.14	        (14) QDP
12.23/5.14	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.23/5.14	        (16) YES
12.23/5.14	    (17) QDP
12.23/5.14	        (18) TransformationProof [EQUIVALENT, 0 ms]
12.23/5.14	        (19) QDP
12.23/5.14	        (20) TransformationProof [EQUIVALENT, 0 ms]
12.23/5.14	        (21) QDP
12.23/5.14	        (22) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.23/5.14	        (23) YES
12.23/5.14	
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(0)
12.23/5.14	Obligation:
12.23/5.14	mainModule Main 
12.23/5.14	module FiniteMap where {
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.23/5.14	
12.23/5.14	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.23/5.14	}
12.23/5.14	foldFM_GE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
12.23/5.14	foldFM_GE k z fr EmptyFM = z;
12.23/5.14	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
12.23/5.14	 | otherwise = foldFM_GE k z fr fm_r;
12.23/5.14	
12.23/5.14	keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
12.23/5.14	keysFM_GE fm fr = foldFM_GE (\key elt rest ->key : rest) [] fr fm;
12.23/5.14	
12.23/5.14	}
12.23/5.14	module Maybe where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	module Main where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(1) LR (EQUIVALENT)
12.23/5.14	Lambda Reductions:
12.23/5.14	The following Lambda expression
12.23/5.14	"\keyeltrest->key : rest"
12.23/5.14	is transformed to
12.23/5.14	"keysFM_GE0 key elt rest = key : rest;
12.23/5.14	"
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(2)
12.23/5.14	Obligation:
12.23/5.14	mainModule Main 
12.23/5.14	module FiniteMap where {
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.23/5.14	
12.23/5.14	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.23/5.14	}
12.23/5.14	foldFM_GE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
12.23/5.14	foldFM_GE k z fr EmptyFM = z;
12.23/5.14	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
12.23/5.14	 | otherwise = foldFM_GE k z fr fm_r;
12.23/5.14	
12.23/5.14	keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
12.23/5.14	keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;
12.23/5.14	
12.23/5.14	keysFM_GE0 key elt rest = key : rest;
12.23/5.14	
12.23/5.14	}
12.23/5.14	module Maybe where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	module Main where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(3) BR (EQUIVALENT)
12.23/5.14	Replaced joker patterns by fresh variables and removed binding patterns.
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(4)
12.23/5.14	Obligation:
12.23/5.14	mainModule Main 
12.23/5.14	module FiniteMap where {
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.23/5.14	
12.23/5.14	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.23/5.14	}
12.23/5.14	foldFM_GE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
12.23/5.14	foldFM_GE k z fr EmptyFM = z;
12.23/5.14	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
12.23/5.14	 | otherwise = foldFM_GE k z fr fm_r;
12.23/5.14	
12.23/5.14	keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
12.23/5.14	keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;
12.23/5.14	
12.23/5.14	keysFM_GE0 key elt rest = key : rest;
12.23/5.14	
12.23/5.14	}
12.23/5.14	module Maybe where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	module Main where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(5) COR (EQUIVALENT)
12.23/5.14	Cond Reductions:
12.23/5.14	The following Function with conditions
12.23/5.14	"compare x y|x == yEQ|x <= yLT|otherwiseGT;
12.23/5.14	"
12.23/5.14	is transformed to
12.23/5.14	"compare x y = compare3 x y;
12.23/5.14	"
12.23/5.14	"compare1 x y True = LT;
12.23/5.14	compare1 x y False = compare0 x y otherwise;
12.23/5.14	"
12.23/5.14	"compare0 x y True = GT;
12.23/5.14	"
12.23/5.14	"compare2 x y True = EQ;
12.23/5.14	compare2 x y False = compare1 x y (x <= y);
12.23/5.14	"
12.23/5.14	"compare3 x y = compare2 x y (x == y);
12.23/5.14	"
12.23/5.14	The following Function with conditions
12.23/5.14	"undefined |Falseundefined;
12.23/5.14	"
12.23/5.14	is transformed to
12.23/5.14	"undefined  = undefined1;
12.23/5.14	"
12.23/5.14	"undefined0 True = undefined;
12.23/5.14	"
12.23/5.14	"undefined1  = undefined0 False;
12.23/5.14	"
12.23/5.14	The following Function with conditions
12.23/5.14	"foldFM_GE k z fr EmptyFM = z;
12.23/5.14	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;
12.23/5.14	"
12.23/5.14	is transformed to
12.23/5.14	"foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
12.23/5.14	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);
12.23/5.14	"
12.23/5.14	"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;
12.23/5.14	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;
12.23/5.14	"
12.23/5.14	"foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
12.23/5.14	"
12.23/5.14	"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);
12.23/5.14	"
12.23/5.14	"foldFM_GE3 k z fr EmptyFM = z;
12.23/5.14	foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
12.23/5.14	"
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(6)
12.23/5.14	Obligation:
12.23/5.14	mainModule Main 
12.23/5.14	module FiniteMap where {
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.23/5.14	
12.23/5.14	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.23/5.14	}
12.23/5.14	foldFM_GE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
12.23/5.14	foldFM_GE k z fr EmptyFM = foldFM_GE3 k z fr EmptyFM;
12.23/5.14	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);
12.23/5.14	
12.23/5.14	foldFM_GE0 k z fr key elt vy fm_l fm_r True = foldFM_GE k z fr fm_r;
12.23/5.14	
12.23/5.14	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;
12.23/5.14	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;
12.23/5.14	
12.23/5.14	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);
12.23/5.14	
12.23/5.14	foldFM_GE3 k z fr EmptyFM = z;
12.23/5.14	foldFM_GE3 wv ww wx wy = foldFM_GE2 wv ww wx wy;
12.23/5.14	
12.23/5.14	keysFM_GE :: Ord a => FiniteMap a b  ->  a  ->  [a];
12.23/5.14	keysFM_GE fm fr = foldFM_GE keysFM_GE0 [] fr fm;
12.23/5.14	
12.23/5.14	keysFM_GE0 key elt rest = key : rest;
12.23/5.14	
12.23/5.14	}
12.23/5.14	module Maybe where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Main;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	module Main where {
12.23/5.14	import qualified FiniteMap;
12.23/5.14	import qualified Maybe;
12.23/5.14	import qualified Prelude;
12.23/5.14	}
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(7) Narrow (SOUND)
12.23/5.14	Haskell To QDPs
12.23/5.14	
12.23/5.14	digraph dp_graph {
12.23/5.14	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.keysFM_GE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.23/5.14	3[label="FiniteMap.keysFM_GE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.23/5.14	4[label="FiniteMap.keysFM_GE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.23/5.14	5[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 wz3",fontsize=16,color="burlywood",shape="triangle"];170[label="wz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 170[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	170 -> 6[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	171[label="wz3/FiniteMap.Branch wz30 wz31 wz32 wz33 wz34",fontsize=10,color="white",style="solid",shape="box"];5 -> 171[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	171 -> 7[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	6[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.23/5.14	7[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.23/5.14	8[label="FiniteMap.foldFM_GE3 FiniteMap.keysFM_GE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.23/5.14	9[label="FiniteMap.foldFM_GE2 FiniteMap.keysFM_GE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.23/5.14	10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 >= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.23/5.14	12[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_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];
12.23/5.14	13[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == LT))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
12.23/5.14	14[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare3 wz30 wz4 == LT))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
12.23/5.14	15[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare2 wz30 wz4 (wz30 == wz4) == LT))",fontsize=16,color="burlywood",shape="box"];172[label="wz30/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 172[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	172 -> 16[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	173[label="wz30/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 173[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	173 -> 17[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	16[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 False wz31 wz32 wz33 wz34 (not (compare2 False wz4 (False == wz4) == LT))",fontsize=16,color="burlywood",shape="box"];174[label="wz4/False",fontsize=10,color="white",style="solid",shape="box"];16 -> 174[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	174 -> 18[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	175[label="wz4/True",fontsize=10,color="white",style="solid",shape="box"];16 -> 175[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	175 -> 19[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	17[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] wz4 True wz31 wz32 wz33 wz34 (not (compare2 True wz4 (True == wz4) == LT))",fontsize=16,color="burlywood",shape="box"];176[label="wz4/False",fontsize=10,color="white",style="solid",shape="box"];17 -> 176[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	176 -> 20[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	177[label="wz4/True",fontsize=10,color="white",style="solid",shape="box"];17 -> 177[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	177 -> 21[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	18[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not (compare2 False False (False == False) == LT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
12.23/5.14	19[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare2 False True (False == True) == LT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
12.23/5.14	20[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare2 True False (True == False) == LT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
12.23/5.14	21[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (compare2 True True (True == True) == LT))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
12.23/5.14	22[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not (compare2 False False True == LT))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.23/5.14	23[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare2 False True False == LT))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.23/5.14	24[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare2 True False False == LT))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
12.23/5.14	25[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (compare2 True True True == LT))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
12.23/5.14	26[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
12.23/5.14	27[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True False wz31 wz32 wz33 wz34 (not (compare1 False True (False <= True) == LT))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
12.23/5.14	28[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False True wz31 wz32 wz33 wz34 (not (compare1 True False (True <= False) == LT))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
12.23/5.14	29[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] True True wz31 wz32 wz33 wz34 (not (EQ == LT))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
12.23/5.14	30[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 [] False False wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
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12.23/5.14	131 -> 137[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	154[label="wz331",fontsize=16,color="green",shape="box"];155 -> 83[label="",style="dashed", color="red", weight=0];
12.23/5.14	155[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False wz334",fontsize=16,color="magenta"];155 -> 162[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	156[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz10 False True wz331 wz332 wz333 wz334 (not False)",fontsize=16,color="black",shape="box"];156 -> 163[label="",style="solid", color="black", weight=3];
12.23/5.14	134[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz9 True False wz331 wz332 wz333 wz334 False",fontsize=16,color="black",shape="box"];134 -> 140[label="",style="solid", color="black", weight=3];
12.23/5.14	135[label="wz331",fontsize=16,color="green",shape="box"];136[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz9 True wz334",fontsize=16,color="burlywood",shape="triangle"];186[label="wz334/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];136 -> 186[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	186 -> 141[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	187[label="wz334/FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344",fontsize=10,color="white",style="solid",shape="box"];136 -> 187[label="",style="solid", color="burlywood", weight=9];
12.23/5.14	187 -> 142[label="",style="solid", color="burlywood", weight=3];
12.23/5.14	137[label="wz333",fontsize=16,color="green",shape="box"];162[label="wz334",fontsize=16,color="green",shape="box"];163[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz10 False True wz331 wz332 wz333 wz334 True",fontsize=16,color="black",shape="box"];163 -> 164[label="",style="solid", color="black", weight=3];
12.23/5.14	140[label="FiniteMap.foldFM_GE0 FiniteMap.keysFM_GE0 wz9 True False wz331 wz332 wz333 wz334 otherwise",fontsize=16,color="black",shape="box"];140 -> 145[label="",style="solid", color="black", weight=3];
12.23/5.14	141[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz9 True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];141 -> 146[label="",style="solid", color="black", weight=3];
12.23/5.14	142[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz9 True (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];142 -> 147[label="",style="solid", color="black", weight=3];
12.23/5.14	164 -> 83[label="",style="dashed", color="red", weight=0];
12.23/5.14	164[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 (FiniteMap.keysFM_GE0 True wz331 (FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False wz334)) False wz333",fontsize=16,color="magenta"];164 -> 165[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	164 -> 166[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	145[label="FiniteMap.foldFM_GE0 FiniteMap.keysFM_GE0 wz9 True False wz331 wz332 wz333 wz334 True",fontsize=16,color="black",shape="box"];145 -> 151[label="",style="solid", color="black", weight=3];
12.23/5.14	146[label="FiniteMap.foldFM_GE3 FiniteMap.keysFM_GE0 wz9 True FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];146 -> 152[label="",style="solid", color="black", weight=3];
12.23/5.14	147[label="FiniteMap.foldFM_GE2 FiniteMap.keysFM_GE0 wz9 True (FiniteMap.Branch wz3340 wz3341 wz3342 wz3343 wz3344)",fontsize=16,color="black",shape="box"];147 -> 153[label="",style="solid", color="black", weight=3];
12.23/5.14	165 -> 68[label="",style="dashed", color="red", weight=0];
12.23/5.14	165[label="FiniteMap.keysFM_GE0 True wz331 (FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False wz334)",fontsize=16,color="magenta"];165 -> 167[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	165 -> 168[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	166[label="wz333",fontsize=16,color="green",shape="box"];151 -> 136[label="",style="dashed", color="red", weight=0];
12.23/5.14	151[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz9 True wz334",fontsize=16,color="magenta"];152[label="wz9",fontsize=16,color="green",shape="box"];153 -> 78[label="",style="dashed", color="red", weight=0];
12.23/5.14	153[label="FiniteMap.foldFM_GE1 FiniteMap.keysFM_GE0 wz9 True wz3340 wz3341 wz3342 wz3343 wz3344 (wz3340 >= True)",fontsize=16,color="magenta"];153 -> 157[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	153 -> 158[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	153 -> 159[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	153 -> 160[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	153 -> 161[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	167[label="wz331",fontsize=16,color="green",shape="box"];168 -> 83[label="",style="dashed", color="red", weight=0];
12.23/5.14	168[label="FiniteMap.foldFM_GE FiniteMap.keysFM_GE0 wz10 False wz334",fontsize=16,color="magenta"];168 -> 169[label="",style="dashed", color="magenta", weight=3];
12.23/5.14	157[label="wz3341",fontsize=16,color="green",shape="box"];158[label="wz3343",fontsize=16,color="green",shape="box"];159[label="wz3344",fontsize=16,color="green",shape="box"];160[label="wz3340",fontsize=16,color="green",shape="box"];161[label="wz3342",fontsize=16,color="green",shape="box"];169[label="wz334",fontsize=16,color="green",shape="box"];}
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(8)
12.23/5.14	Complex Obligation (AND)
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(9)
12.23/5.14	Obligation:
12.23/5.14	Q DP problem:
12.23/5.14	The TRS P consists of the following rules:
12.23/5.14	
12.23/5.14	   new_foldFM_GE6(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
12.23/5.14	   new_foldFM_GE6(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
12.23/5.14	
12.23/5.14	R is empty.
12.23/5.14	Q is empty.
12.23/5.14	We have to consider all minimal (P,Q,R)-chains.
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(10) QDPSizeChangeProof (EQUIVALENT)
12.23/5.14	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. 
12.23/5.14	
12.23/5.14	From the DPs we obtained the following set of size-change graphs:
12.23/5.14	*new_foldFM_GE6(True, Branch(True, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
12.23/5.14	The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
12.23/5.14	
12.23/5.14	
12.23/5.14	*new_foldFM_GE6(True, Branch(False, wz31, wz32, wz33, wz34), h) -> new_foldFM_GE6(True, wz34, h)
12.23/5.14	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.23/5.14	
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(11)
12.23/5.14	YES
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(12)
12.23/5.14	Obligation:
12.23/5.14	Q DP problem:
12.23/5.14	The TRS P consists of the following rules:
12.23/5.14	
12.23/5.14	   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE1(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
12.23/5.14	   new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
12.23/5.14	   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
12.23/5.14	
12.23/5.14	The TRS R consists of the following rules:
12.23/5.14	
12.23/5.14	   new_foldFM_GE2(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE3(wz31, wz6, EmptyFM, h) -> new_keysFM_GE0(wz31, wz6, h)
12.23/5.14	   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
12.23/5.14	   new_foldFM_GE10(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE3(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
12.23/5.14	   new_foldFM_GE10(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz9, wz334, h)
12.23/5.14	   new_foldFM_GE2(wz9, EmptyFM, h) -> wz9
12.23/5.14	   new_foldFM_GE3(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
12.23/5.14	
12.23/5.14	The set Q consists of the following terms:
12.23/5.14	
12.23/5.14	   new_foldFM_GE10(x0, True, x1, x2, x3, x4, x5)
12.23/5.14	   new_foldFM_GE2(x0, Branch(x1, x2, x3, x4, x5), x6)
12.23/5.14	   new_foldFM_GE10(x0, False, x1, x2, x3, x4, x5)
12.23/5.14	   new_keysFM_GE0(x0, x1, x2)
12.23/5.14	   new_foldFM_GE3(x0, x1, EmptyFM, x2)
12.23/5.14	   new_foldFM_GE2(x0, EmptyFM, x1)
12.23/5.14	   new_foldFM_GE3(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.23/5.14	
12.23/5.14	We have to consider all minimal (P,Q,R)-chains.
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(13) TransformationProof (EQUIVALENT)
12.23/5.14	By rewriting [LPAR04] the rule new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h) at position [0] we obtained the following new rules [LPAR04]:
12.23/5.14	
12.23/5.14	   (new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h),new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h))
12.23/5.14	
12.23/5.14	
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(14)
12.23/5.14	Obligation:
12.23/5.14	Q DP problem:
12.23/5.14	The TRS P consists of the following rules:
12.23/5.14	
12.23/5.14	   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE1(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
12.23/5.14	   new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
12.23/5.14	   new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h)
12.23/5.14	
12.23/5.14	The TRS R consists of the following rules:
12.23/5.14	
12.23/5.14	   new_foldFM_GE2(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE10(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.14	   new_foldFM_GE3(wz31, wz6, EmptyFM, h) -> new_keysFM_GE0(wz31, wz6, h)
12.23/5.14	   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
12.23/5.14	   new_foldFM_GE10(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE3(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
12.23/5.14	   new_foldFM_GE10(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE2(wz9, wz334, h)
12.23/5.14	   new_foldFM_GE2(wz9, EmptyFM, h) -> wz9
12.23/5.14	   new_foldFM_GE3(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE10(new_keysFM_GE0(wz31, wz6, h), wz330, wz331, wz332, wz333, wz334, h)
12.23/5.14	
12.23/5.14	The set Q consists of the following terms:
12.23/5.14	
12.23/5.14	   new_foldFM_GE10(x0, True, x1, x2, x3, x4, x5)
12.23/5.14	   new_foldFM_GE2(x0, Branch(x1, x2, x3, x4, x5), x6)
12.23/5.14	   new_foldFM_GE10(x0, False, x1, x2, x3, x4, x5)
12.23/5.14	   new_keysFM_GE0(x0, x1, x2)
12.23/5.14	   new_foldFM_GE3(x0, x1, EmptyFM, x2)
12.23/5.14	   new_foldFM_GE2(x0, EmptyFM, x1)
12.23/5.14	   new_foldFM_GE3(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.23/5.14	
12.23/5.14	We have to consider all minimal (P,Q,R)-chains.
12.23/5.14	----------------------------------------
12.23/5.14	
12.23/5.14	(15) QDPSizeChangeProof (EQUIVALENT)
12.23/5.14	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. 
12.23/5.15	
12.23/5.15	From the DPs we obtained the following set of size-change graphs:
12.23/5.15	*new_foldFM_GE1(wz9, True, wz331, wz332, wz333, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.15	The graph contains the following edges 1 >= 1, 6 > 2, 6 > 3, 6 > 4, 6 > 5, 6 > 6, 7 >= 7
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE1(wz9, False, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE(wz9, wz334, h)
12.23/5.15	The graph contains the following edges 1 >= 1, 6 >= 2, 7 >= 3
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE1(wz9, True, wz331, wz332, wz333, wz334, h) -> new_foldFM_GE0(wz331, new_foldFM_GE2(wz9, wz334, h), wz333, h)
12.23/5.15	The graph contains the following edges 3 >= 1, 5 >= 3, 7 >= 4
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE(wz9, Branch(wz3340, wz3341, wz3342, wz3343, wz3344), h) -> new_foldFM_GE1(wz9, wz3340, wz3341, wz3342, wz3343, wz3344, h)
12.23/5.15	The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 2 > 4, 2 > 5, 2 > 6, 3 >= 7
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE0(wz31, wz6, Branch(wz330, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE1(:(True, wz6), wz330, wz331, wz332, wz333, wz334, h)
12.23/5.15	The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
12.23/5.15	
12.23/5.15	
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(16)
12.23/5.15	YES
12.23/5.15	
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(17)
12.23/5.15	Obligation:
12.23/5.15	Q DP problem:
12.23/5.15	The TRS P consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	
12.23/5.15	The TRS R consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
12.23/5.15	   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
12.23/5.15	   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10
12.23/5.15	
12.23/5.15	The set Q consists of the following terms:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
12.23/5.15	   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
12.23/5.15	   new_keysFM_GE0(x0, x1, x2)
12.23/5.15	   new_foldFM_GE5(x0, EmptyFM, x1)
12.23/5.15	   new_keysFM_GE00(x0, x1, x2)
12.23/5.15	
12.23/5.15	We have to consider all minimal (P,Q,R)-chains.
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(18) TransformationProof (EQUIVALENT)
12.23/5.15	By rewriting [LPAR04] the rule new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h) at position [0] we obtained the following new rules [LPAR04]:
12.23/5.15	
12.23/5.15	   (new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h),new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h))
12.23/5.15	
12.23/5.15	
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(19)
12.23/5.15	Obligation:
12.23/5.15	Q DP problem:
12.23/5.15	The TRS P consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
12.23/5.15	
12.23/5.15	The TRS R consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
12.23/5.15	   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
12.23/5.15	   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10
12.23/5.15	
12.23/5.15	The set Q consists of the following terms:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
12.23/5.15	   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
12.23/5.15	   new_keysFM_GE0(x0, x1, x2)
12.23/5.15	   new_foldFM_GE5(x0, EmptyFM, x1)
12.23/5.15	   new_keysFM_GE00(x0, x1, x2)
12.23/5.15	
12.23/5.15	We have to consider all minimal (P,Q,R)-chains.
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(20) TransformationProof (EQUIVALENT)
12.23/5.15	By rewriting [LPAR04] the rule new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h) at position [0] we obtained the following new rules [LPAR04]:
12.23/5.15	
12.23/5.15	   (new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h),new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h))
12.23/5.15	
12.23/5.15	
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12.23/5.15	
12.23/5.15	(21)
12.23/5.15	Obligation:
12.23/5.15	Q DP problem:
12.23/5.15	The TRS P consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
12.23/5.15	   new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
12.23/5.15	
12.23/5.15	The TRS R consists of the following rules:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE0(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_foldFM_GE5(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE5(new_keysFM_GE00(wz331, new_foldFM_GE5(wz10, wz334, h), h), wz333, h)
12.23/5.15	   new_keysFM_GE00(wz31, wz11, h) -> :(False, wz11)
12.23/5.15	   new_keysFM_GE0(wz31, wz6, h) -> :(True, wz6)
12.23/5.15	   new_foldFM_GE5(wz10, EmptyFM, h) -> wz10
12.23/5.15	
12.23/5.15	The set Q consists of the following terms:
12.23/5.15	
12.23/5.15	   new_foldFM_GE5(x0, Branch(True, x1, x2, x3, x4), x5)
12.23/5.15	   new_foldFM_GE5(x0, Branch(False, x1, x2, x3, x4), x5)
12.23/5.15	   new_keysFM_GE0(x0, x1, x2)
12.23/5.15	   new_foldFM_GE5(x0, EmptyFM, x1)
12.23/5.15	   new_keysFM_GE00(x0, x1, x2)
12.23/5.15	
12.23/5.15	We have to consider all minimal (P,Q,R)-chains.
12.23/5.15	----------------------------------------
12.23/5.15	
12.23/5.15	(22) QDPSizeChangeProof (EQUIVALENT)
12.23/5.15	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. 
12.23/5.15	
12.23/5.15	From the DPs we obtained the following set of size-change graphs:
12.23/5.15	*new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(wz10, wz334, h)
12.23/5.15	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE4(wz10, Branch(True, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(True, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
12.23/5.15	The graph contains the following edges 2 > 2, 3 >= 3
12.23/5.15	
12.23/5.15	
12.23/5.15	*new_foldFM_GE4(wz10, Branch(False, wz331, wz332, wz333, wz334), h) -> new_foldFM_GE4(:(False, new_foldFM_GE5(wz10, wz334, h)), wz333, h)
12.23/5.15	The graph contains the following edges 2 > 2, 3 >= 3
12.23/5.15	
12.23/5.15	
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12.23/5.15	
12.23/5.15	(23)
12.23/5.15	YES
12.50/5.23	EOF