10.30/4.53	YES
12.13/5.04	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.13/5.04	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.13/5.04	
12.13/5.04	
12.13/5.04	H-Termination with start terms of the given HASKELL could be proven:
12.13/5.04	
12.13/5.04	(0) HASKELL
12.13/5.04	(1) BR [EQUIVALENT, 0 ms]
12.13/5.04	(2) HASKELL
12.13/5.04	(3) COR [EQUIVALENT, 0 ms]
12.13/5.04	(4) HASKELL
12.13/5.04	(5) Narrow [SOUND, 0 ms]
12.13/5.04	(6) QDP
12.13/5.04	(7) DependencyGraphProof [EQUIVALENT, 0 ms]
12.13/5.04	(8) AND
12.13/5.04	    (9) QDP
12.13/5.04	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.13/5.04	        (11) YES
12.13/5.04	    (12) QDP
12.13/5.04	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.13/5.04	        (14) YES
12.13/5.04	    (15) QDP
12.13/5.04	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.13/5.04	        (17) YES
12.13/5.04	
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(0)
12.13/5.04	Obligation:
12.13/5.04	mainModule Main 
12.13/5.04	module FiniteMap where {
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.13/5.04	
12.13/5.04	foldFM_LE :: Ord a => (a  ->  b  ->  c  ->  c)  ->  c  ->  a  ->  FiniteMap a b  ->  c;
12.13/5.04	foldFM_LE k z fr EmptyFM = z;
12.13/5.04	foldFM_LE k z fr (Branch key elt _ fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
12.13/5.04	 | otherwise = foldFM_LE k z fr fm_l;
12.13/5.04	
12.13/5.04	}
12.13/5.04	module Maybe where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	module Main where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(1) BR (EQUIVALENT)
12.13/5.04	Replaced joker patterns by fresh variables and removed binding patterns.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(2)
12.13/5.04	Obligation:
12.13/5.04	mainModule Main 
12.13/5.04	module FiniteMap where {
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.13/5.04	
12.13/5.04	foldFM_LE :: Ord c => (c  ->  a  ->  b  ->  b)  ->  b  ->  c  ->  FiniteMap c a  ->  b;
12.13/5.04	foldFM_LE k z fr EmptyFM = z;
12.13/5.04	foldFM_LE k z fr (Branch key elt vy fm_l fm_r)  | key <= fr = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r
12.13/5.04	 | otherwise = foldFM_LE k z fr fm_l;
12.13/5.04	
12.13/5.04	}
12.13/5.04	module Maybe where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	module Main where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(3) COR (EQUIVALENT)
12.13/5.04	Cond Reductions:
12.13/5.04	The following Function with conditions
12.13/5.04	"undefined |Falseundefined;
12.13/5.04	"
12.13/5.04	is transformed to
12.13/5.04	"undefined  = undefined1;
12.13/5.04	"
12.13/5.04	"undefined0 True = undefined;
12.13/5.04	"
12.13/5.04	"undefined1  = undefined0 False;
12.13/5.04	"
12.13/5.04	The following Function with conditions
12.13/5.04	"foldFM_LE k z fr EmptyFM = z;
12.13/5.04	foldFM_LE k z fr (Branch key elt vy fm_l fm_r)|key <= frfoldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r|otherwisefoldFM_LE k z fr fm_l;
12.13/5.04	"
12.13/5.04	is transformed to
12.13/5.04	"foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.13/5.04	foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);
12.13/5.04	"
12.13/5.04	"foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
12.13/5.04	foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;
12.13/5.04	"
12.13/5.04	"foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.13/5.04	"
12.13/5.04	"foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);
12.13/5.04	"
12.13/5.04	"foldFM_LE3 k z fr EmptyFM = z;
12.13/5.04	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.13/5.04	"
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(4)
12.13/5.04	Obligation:
12.13/5.04	mainModule Main 
12.13/5.04	module FiniteMap where {
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.13/5.04	
12.13/5.04	foldFM_LE :: Ord c => (c  ->  a  ->  b  ->  b)  ->  b  ->  c  ->  FiniteMap c a  ->  b;
12.13/5.04	foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.13/5.04	foldFM_LE k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r);
12.13/5.04	
12.13/5.04	foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.13/5.04	
12.13/5.04	foldFM_LE1 k z fr key elt vy fm_l fm_r True = foldFM_LE k (k key elt (foldFM_LE k z fr fm_l)) fr fm_r;
12.13/5.04	foldFM_LE1 k z fr key elt vy fm_l fm_r False = foldFM_LE0 k z fr key elt vy fm_l fm_r otherwise;
12.13/5.04	
12.13/5.04	foldFM_LE2 k z fr (Branch key elt vy fm_l fm_r) = foldFM_LE1 k z fr key elt vy fm_l fm_r (key <= fr);
12.13/5.04	
12.13/5.04	foldFM_LE3 k z fr EmptyFM = z;
12.13/5.04	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.13/5.04	
12.13/5.04	}
12.13/5.04	module Maybe where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Main;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	module Main where {
12.13/5.04	import qualified FiniteMap;
12.13/5.04	import qualified Maybe;
12.13/5.04	import qualified Prelude;
12.13/5.04	}
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(5) Narrow (SOUND)
12.13/5.04	Haskell To QDPs
12.13/5.04	
12.13/5.04	digraph dp_graph {
12.13/5.04	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.13/5.04	3[label="FiniteMap.foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.13/5.04	4[label="FiniteMap.foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
12.13/5.04	5[label="FiniteMap.foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
12.13/5.04	6[label="FiniteMap.foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];103[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 103[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	103 -> 7[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	104[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 104[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	104 -> 8[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	7[label="FiniteMap.foldFM_LE wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.13/5.04	8[label="FiniteMap.foldFM_LE wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.13/5.04	9[label="FiniteMap.foldFM_LE3 wz3 wz4 wz5 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.13/5.04	10[label="FiniteMap.foldFM_LE2 wz3 wz4 wz5 (FiniteMap.Branch wz60 wz61 wz62 wz63 wz64)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
12.13/5.04	11[label="wz4",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (wz60 <= wz5)",fontsize=16,color="burlywood",shape="box"];105[label="wz60/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 105[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	105 -> 13[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	106[label="wz60/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 106[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	106 -> 14[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	107[label="wz60/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 107[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	107 -> 15[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	13[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 LT wz61 wz62 wz63 wz64 (LT <= wz5)",fontsize=16,color="burlywood",shape="box"];108[label="wz5/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 108[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	108 -> 16[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	109[label="wz5/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 109[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	109 -> 17[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	110[label="wz5/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 110[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	110 -> 18[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	14[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 EQ wz61 wz62 wz63 wz64 (EQ <= wz5)",fontsize=16,color="burlywood",shape="box"];111[label="wz5/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 111[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	111 -> 19[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	112[label="wz5/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 112[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	112 -> 20[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	113[label="wz5/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 113[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	113 -> 21[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	15[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 GT wz61 wz62 wz63 wz64 (GT <= wz5)",fontsize=16,color="burlywood",shape="box"];114[label="wz5/LT",fontsize=10,color="white",style="solid",shape="box"];15 -> 114[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	114 -> 22[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	115[label="wz5/EQ",fontsize=10,color="white",style="solid",shape="box"];15 -> 115[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	115 -> 23[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	116[label="wz5/GT",fontsize=10,color="white",style="solid",shape="box"];15 -> 116[label="",style="solid", color="burlywood", weight=9];
12.13/5.04	116 -> 24[label="",style="solid", color="burlywood", weight=3];
12.13/5.04	16[label="FiniteMap.foldFM_LE1 wz3 wz4 LT LT wz61 wz62 wz63 wz64 (LT <= LT)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3];
12.13/5.04	17[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 (LT <= EQ)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3];
12.13/5.04	18[label="FiniteMap.foldFM_LE1 wz3 wz4 GT LT wz61 wz62 wz63 wz64 (LT <= GT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3];
12.13/5.04	19[label="FiniteMap.foldFM_LE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 (EQ <= LT)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3];
12.13/5.04	20[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ EQ wz61 wz62 wz63 wz64 (EQ <= EQ)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
12.13/5.04	21[label="FiniteMap.foldFM_LE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 (EQ <= GT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3];
12.13/5.04	22[label="FiniteMap.foldFM_LE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 (GT <= LT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3];
12.13/5.04	23[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 (GT <= EQ)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3];
12.13/5.04	24[label="FiniteMap.foldFM_LE1 wz3 wz4 GT GT wz61 wz62 wz63 wz64 (GT <= GT)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3];
12.13/5.04	25[label="FiniteMap.foldFM_LE1 wz3 wz4 LT LT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3];
12.13/5.04	26[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ LT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3];
12.13/5.04	27[label="FiniteMap.foldFM_LE1 wz3 wz4 GT LT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
12.13/5.04	28[label="FiniteMap.foldFM_LE1 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
12.13/5.04	29[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3];
12.13/5.04	30[label="FiniteMap.foldFM_LE1 wz3 wz4 GT EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
12.13/5.04	31[label="FiniteMap.foldFM_LE1 wz3 wz4 LT GT wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3];
12.13/5.04	32[label="FiniteMap.foldFM_LE1 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 False",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3];
12.13/5.04	33[label="FiniteMap.foldFM_LE1 wz3 wz4 GT GT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3];
12.13/5.04	34 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	34[label="FiniteMap.foldFM_LE wz3 (wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 LT wz63)) LT wz64",fontsize=16,color="magenta"];34 -> 43[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	34 -> 44[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	34 -> 45[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	35 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	35[label="FiniteMap.foldFM_LE wz3 (wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 EQ wz63)) EQ wz64",fontsize=16,color="magenta"];35 -> 46[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	35 -> 47[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	35 -> 48[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	36 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	36[label="FiniteMap.foldFM_LE wz3 (wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)) GT wz64",fontsize=16,color="magenta"];36 -> 49[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	36 -> 50[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	36 -> 51[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	37[label="FiniteMap.foldFM_LE0 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];37 -> 52[label="",style="solid", color="black", weight=3];
12.13/5.04	38 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	38[label="FiniteMap.foldFM_LE wz3 (wz3 EQ wz61 (FiniteMap.foldFM_LE wz3 wz4 EQ wz63)) EQ wz64",fontsize=16,color="magenta"];38 -> 53[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	38 -> 54[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	38 -> 55[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	39 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	39[label="FiniteMap.foldFM_LE wz3 (wz3 EQ wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)) GT wz64",fontsize=16,color="magenta"];39 -> 56[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	39 -> 57[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	39 -> 58[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	40[label="FiniteMap.foldFM_LE0 wz3 wz4 LT GT wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];40 -> 59[label="",style="solid", color="black", weight=3];
12.13/5.04	41[label="FiniteMap.foldFM_LE0 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 otherwise",fontsize=16,color="black",shape="box"];41 -> 60[label="",style="solid", color="black", weight=3];
12.13/5.04	42 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	42[label="FiniteMap.foldFM_LE wz3 (wz3 GT wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)) GT wz64",fontsize=16,color="magenta"];42 -> 61[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	42 -> 62[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	42 -> 63[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	43[label="LT",fontsize=16,color="green",shape="box"];44[label="wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 LT wz63)",fontsize=16,color="green",shape="box"];44 -> 64[label="",style="dashed", color="green", weight=3];
12.13/5.04	44 -> 65[label="",style="dashed", color="green", weight=3];
12.13/5.04	44 -> 66[label="",style="dashed", color="green", weight=3];
12.13/5.04	45[label="wz64",fontsize=16,color="green",shape="box"];46[label="EQ",fontsize=16,color="green",shape="box"];47[label="wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 EQ wz63)",fontsize=16,color="green",shape="box"];47 -> 67[label="",style="dashed", color="green", weight=3];
12.13/5.04	47 -> 68[label="",style="dashed", color="green", weight=3];
12.13/5.04	47 -> 69[label="",style="dashed", color="green", weight=3];
12.13/5.04	48[label="wz64",fontsize=16,color="green",shape="box"];49[label="GT",fontsize=16,color="green",shape="box"];50[label="wz3 LT wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)",fontsize=16,color="green",shape="box"];50 -> 70[label="",style="dashed", color="green", weight=3];
12.13/5.04	50 -> 71[label="",style="dashed", color="green", weight=3];
12.13/5.04	50 -> 72[label="",style="dashed", color="green", weight=3];
12.13/5.04	51[label="wz64",fontsize=16,color="green",shape="box"];52[label="FiniteMap.foldFM_LE0 wz3 wz4 LT EQ wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];52 -> 73[label="",style="solid", color="black", weight=3];
12.13/5.04	53[label="EQ",fontsize=16,color="green",shape="box"];54[label="wz3 EQ wz61 (FiniteMap.foldFM_LE wz3 wz4 EQ wz63)",fontsize=16,color="green",shape="box"];54 -> 74[label="",style="dashed", color="green", weight=3];
12.13/5.04	54 -> 75[label="",style="dashed", color="green", weight=3];
12.13/5.04	54 -> 76[label="",style="dashed", color="green", weight=3];
12.13/5.04	55[label="wz64",fontsize=16,color="green",shape="box"];56[label="GT",fontsize=16,color="green",shape="box"];57[label="wz3 EQ wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)",fontsize=16,color="green",shape="box"];57 -> 77[label="",style="dashed", color="green", weight=3];
12.13/5.04	57 -> 78[label="",style="dashed", color="green", weight=3];
12.13/5.04	57 -> 79[label="",style="dashed", color="green", weight=3];
12.13/5.04	58[label="wz64",fontsize=16,color="green",shape="box"];59[label="FiniteMap.foldFM_LE0 wz3 wz4 LT GT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];59 -> 80[label="",style="solid", color="black", weight=3];
12.13/5.04	60[label="FiniteMap.foldFM_LE0 wz3 wz4 EQ GT wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];60 -> 81[label="",style="solid", color="black", weight=3];
12.13/5.04	61[label="GT",fontsize=16,color="green",shape="box"];62[label="wz3 GT wz61 (FiniteMap.foldFM_LE wz3 wz4 GT wz63)",fontsize=16,color="green",shape="box"];62 -> 82[label="",style="dashed", color="green", weight=3];
12.13/5.04	62 -> 83[label="",style="dashed", color="green", weight=3];
12.13/5.04	62 -> 84[label="",style="dashed", color="green", weight=3];
12.13/5.04	63[label="wz64",fontsize=16,color="green",shape="box"];64[label="LT",fontsize=16,color="green",shape="box"];65[label="wz61",fontsize=16,color="green",shape="box"];66 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	66[label="FiniteMap.foldFM_LE wz3 wz4 LT wz63",fontsize=16,color="magenta"];66 -> 85[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	66 -> 86[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	67[label="LT",fontsize=16,color="green",shape="box"];68[label="wz61",fontsize=16,color="green",shape="box"];69 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	69[label="FiniteMap.foldFM_LE wz3 wz4 EQ wz63",fontsize=16,color="magenta"];69 -> 87[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	69 -> 88[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	70[label="LT",fontsize=16,color="green",shape="box"];71[label="wz61",fontsize=16,color="green",shape="box"];72 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	72[label="FiniteMap.foldFM_LE wz3 wz4 GT wz63",fontsize=16,color="magenta"];72 -> 89[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	72 -> 90[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	73 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	73[label="FiniteMap.foldFM_LE wz3 wz4 LT wz63",fontsize=16,color="magenta"];73 -> 91[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	73 -> 92[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	74[label="EQ",fontsize=16,color="green",shape="box"];75[label="wz61",fontsize=16,color="green",shape="box"];76 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	76[label="FiniteMap.foldFM_LE wz3 wz4 EQ wz63",fontsize=16,color="magenta"];76 -> 93[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	76 -> 94[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	77[label="EQ",fontsize=16,color="green",shape="box"];78[label="wz61",fontsize=16,color="green",shape="box"];79 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	79[label="FiniteMap.foldFM_LE wz3 wz4 GT wz63",fontsize=16,color="magenta"];79 -> 95[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	79 -> 96[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	80 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	80[label="FiniteMap.foldFM_LE wz3 wz4 LT wz63",fontsize=16,color="magenta"];80 -> 97[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	80 -> 98[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	81 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	81[label="FiniteMap.foldFM_LE wz3 wz4 EQ wz63",fontsize=16,color="magenta"];81 -> 99[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	81 -> 100[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	82[label="GT",fontsize=16,color="green",shape="box"];83[label="wz61",fontsize=16,color="green",shape="box"];84 -> 6[label="",style="dashed", color="red", weight=0];
12.13/5.04	84[label="FiniteMap.foldFM_LE wz3 wz4 GT wz63",fontsize=16,color="magenta"];84 -> 101[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	84 -> 102[label="",style="dashed", color="magenta", weight=3];
12.13/5.04	85[label="LT",fontsize=16,color="green",shape="box"];86[label="wz63",fontsize=16,color="green",shape="box"];87[label="EQ",fontsize=16,color="green",shape="box"];88[label="wz63",fontsize=16,color="green",shape="box"];89[label="GT",fontsize=16,color="green",shape="box"];90[label="wz63",fontsize=16,color="green",shape="box"];91[label="LT",fontsize=16,color="green",shape="box"];92[label="wz63",fontsize=16,color="green",shape="box"];93[label="EQ",fontsize=16,color="green",shape="box"];94[label="wz63",fontsize=16,color="green",shape="box"];95[label="GT",fontsize=16,color="green",shape="box"];96[label="wz63",fontsize=16,color="green",shape="box"];97[label="LT",fontsize=16,color="green",shape="box"];98[label="wz63",fontsize=16,color="green",shape="box"];99[label="EQ",fontsize=16,color="green",shape="box"];100[label="wz63",fontsize=16,color="green",shape="box"];101[label="GT",fontsize=16,color="green",shape="box"];102[label="wz63",fontsize=16,color="green",shape="box"];}
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(6)
12.13/5.04	Obligation:
12.13/5.04	Q DP problem:
12.13/5.04	The TRS P consists of the following rules:
12.13/5.04	
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	
12.13/5.04	R is empty.
12.13/5.04	Q is empty.
12.13/5.04	We have to consider all minimal (P,Q,R)-chains.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(7) DependencyGraphProof (EQUIVALENT)
12.13/5.04	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(8)
12.13/5.04	Complex Obligation (AND)
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(9)
12.13/5.04	Obligation:
12.13/5.04	Q DP problem:
12.13/5.04	The TRS P consists of the following rules:
12.13/5.04	
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, LT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	
12.13/5.04	R is empty.
12.13/5.04	Q is empty.
12.13/5.04	We have to consider all minimal (P,Q,R)-chains.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(10) QDPSizeChangeProof (EQUIVALENT)
12.13/5.04	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. 
12.13/5.04	
12.13/5.04	From the DPs we obtained the following set of size-change graphs:
12.13/5.04	*new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, LT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, LT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, LT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, LT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(11)
12.13/5.04	YES
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(12)
12.13/5.04	Obligation:
12.13/5.04	Q DP problem:
12.13/5.04	The TRS P consists of the following rules:
12.13/5.04	
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	
12.13/5.04	R is empty.
12.13/5.04	Q is empty.
12.13/5.04	We have to consider all minimal (P,Q,R)-chains.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(13) QDPSizeChangeProof (EQUIVALENT)
12.13/5.04	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. 
12.13/5.04	
12.13/5.04	From the DPs we obtained the following set of size-change graphs:
12.13/5.04	*new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, EQ, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, EQ, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, EQ, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, EQ, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(14)
12.13/5.04	YES
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(15)
12.13/5.04	Obligation:
12.13/5.04	Q DP problem:
12.13/5.04	The TRS P consists of the following rules:
12.13/5.04	
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	   new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	
12.13/5.04	R is empty.
12.13/5.04	Q is empty.
12.13/5.04	We have to consider all minimal (P,Q,R)-chains.
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(16) QDPSizeChangeProof (EQUIVALENT)
12.13/5.04	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. 
12.13/5.04	
12.13/5.04	From the DPs we obtained the following set of size-change graphs:
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(EQ, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz64, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(LT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	*new_foldFM_LE(wz3, GT, Branch(GT, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, GT, wz63, h, ba)
12.13/5.04	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.13/5.04	
12.13/5.04	
12.13/5.04	----------------------------------------
12.13/5.04	
12.13/5.04	(17)
12.13/5.04	YES
12.13/5.09	EOF