10.13/4.50	YES
12.53/5.13	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.53/5.13	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.53/5.13	
12.53/5.13	
12.53/5.13	H-Termination with start terms of the given HASKELL could be proven:
12.53/5.13	
12.53/5.13	(0) HASKELL
12.53/5.13	(1) LR [EQUIVALENT, 0 ms]
12.53/5.13	(2) HASKELL
12.53/5.13	(3) BR [EQUIVALENT, 0 ms]
12.53/5.13	(4) HASKELL
12.53/5.13	(5) COR [EQUIVALENT, 0 ms]
12.53/5.13	(6) HASKELL
12.53/5.13	(7) Narrow [SOUND, 0 ms]
12.53/5.13	(8) AND
12.53/5.13	    (9) QDP
12.53/5.13	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.53/5.13	        (11) YES
12.53/5.13	    (12) QDP
12.53/5.13	        (13) TransformationProof [EQUIVALENT, 0 ms]
12.53/5.13	        (14) QDP
12.53/5.13	        (15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.53/5.13	        (16) YES
12.53/5.13	
12.53/5.13	
12.53/5.13	----------------------------------------
12.53/5.13	
12.53/5.13	(0)
12.53/5.13	Obligation:
12.53/5.13	mainModule Main 
12.53/5.13	module FiniteMap where {
12.53/5.13	import qualified Main;
12.53/5.13	import qualified Maybe;
12.53/5.13	import qualified Prelude;
12.53/5.13	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.53/5.13	
12.53/5.13	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.53/5.13	}
12.53/5.13	fmToList_LE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
12.53/5.13	fmToList_LE fm fr = foldFM_LE (\key elt rest ->(key,elt) : rest) [] fr fm;
12.53/5.14	
12.53/5.14	foldFM_LE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
12.53/5.14	foldFM_LE k z fr EmptyFM = z;
12.53/5.14	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
12.53/5.14	 | otherwise = foldFM_LE k z fr fm_l;
12.53/5.14	
12.53/5.14	}
12.53/5.14	module Maybe where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	module Main where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(1) LR (EQUIVALENT)
12.53/5.14	Lambda Reductions:
12.53/5.14	The following Lambda expression
12.53/5.14	"\keyeltrest->(key,elt) : rest"
12.53/5.14	is transformed to
12.53/5.14	"fmToList_LE0 key elt rest = (key,elt) : rest;
12.53/5.14	"
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(2)
12.53/5.14	Obligation:
12.53/5.14	mainModule Main 
12.53/5.14	module FiniteMap where {
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.53/5.14	
12.53/5.14	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.53/5.14	}
12.53/5.14	fmToList_LE :: Ord a => FiniteMap a b  ->  a  ->  [(a,b)];
12.53/5.14	fmToList_LE fm fr = foldFM_LE fmToList_LE0 [] fr fm;
12.53/5.14	
12.53/5.14	fmToList_LE0 key elt rest = (key,elt) : rest;
12.53/5.14	
12.53/5.14	foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
12.53/5.14	foldFM_LE k z fr EmptyFM = z;
12.53/5.14	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
12.53/5.14	 | otherwise = foldFM_LE k z fr fm_l;
12.53/5.14	
12.53/5.14	}
12.53/5.14	module Maybe where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	module Main where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(3) BR (EQUIVALENT)
12.53/5.14	Replaced joker patterns by fresh variables and removed binding patterns.
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(4)
12.53/5.14	Obligation:
12.53/5.14	mainModule Main 
12.53/5.14	module FiniteMap where {
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.53/5.14	
12.53/5.14	instance (Eq a, Eq b) => Eq FiniteMap a b where { 
12.53/5.14	}
12.53/5.14	fmToList_LE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
12.53/5.14	fmToList_LE fm fr = foldFM_LE fmToList_LE0 [] fr fm;
12.53/5.14	
12.53/5.14	fmToList_LE0 key elt rest = (key,elt) : rest;
12.53/5.14	
12.53/5.14	foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
12.53/5.14	foldFM_LE k z fr EmptyFM = z;
12.53/5.14	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
12.53/5.14	 | otherwise = foldFM_LE k z fr fm_l;
12.53/5.14	
12.53/5.14	}
12.53/5.14	module Maybe where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	module Main where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(5) COR (EQUIVALENT)
12.53/5.14	Cond Reductions:
12.53/5.14	The following Function with conditions
12.53/5.14	"undefined |Falseundefined;
12.53/5.14	"
12.53/5.14	is transformed to
12.53/5.14	"undefined  = undefined1;
12.53/5.14	"
12.53/5.14	"undefined0 True = undefined;
12.53/5.14	"
12.53/5.14	"undefined1  = undefined0 False;
12.53/5.14	"
12.53/5.14	The following Function with conditions
12.53/5.14	"foldFM_LE k z fr EmptyFM = z;
12.53/5.14	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;
12.53/5.14	"
12.53/5.14	is transformed to
12.53/5.14	"foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.53/5.14	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);
12.53/5.14	"
12.53/5.14	"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;
12.53/5.14	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;
12.53/5.14	"
12.53/5.14	"foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.53/5.14	"
12.53/5.14	"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);
12.53/5.14	"
12.53/5.14	"foldFM_LE3 k z fr EmptyFM = z;
12.53/5.14	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.53/5.14	"
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(6)
12.53/5.14	Obligation:
12.53/5.14	mainModule Main 
12.53/5.14	module FiniteMap where {
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.53/5.14	
12.53/5.14	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.53/5.14	}
12.53/5.14	fmToList_LE :: Ord b => FiniteMap b a  ->  b  ->  [(b,a)];
12.53/5.14	fmToList_LE fm fr = foldFM_LE fmToList_LE0 [] fr fm;
12.53/5.14	
12.53/5.14	fmToList_LE0 key elt rest = (key,elt) : rest;
12.53/5.14	
12.53/5.14	foldFM_LE :: Ord b => (b  ->  c  ->  a  ->  a)  ->  a  ->  b  ->  FiniteMap b c  ->  a;
12.53/5.14	foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.53/5.14	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);
12.53/5.14	
12.53/5.14	foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.53/5.14	
12.53/5.14	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;
12.53/5.14	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;
12.53/5.14	
12.53/5.14	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);
12.53/5.14	
12.53/5.14	foldFM_LE3 k z fr EmptyFM = z;
12.53/5.14	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.53/5.14	
12.53/5.14	}
12.53/5.14	module Maybe where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Main;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	module Main where {
12.53/5.14	import qualified FiniteMap;
12.53/5.14	import qualified Maybe;
12.53/5.14	import qualified Prelude;
12.53/5.14	}
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(7) Narrow (SOUND)
12.53/5.14	Haskell To QDPs
12.53/5.14	
12.53/5.14	digraph dp_graph {
12.53/5.14	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.fmToList_LE",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.53/5.14	3[label="FiniteMap.fmToList_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.53/5.14	4[label="FiniteMap.fmToList_LE wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
12.53/5.14	5[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 [] 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];
12.53/5.14	54 -> 6[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	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];
12.53/5.14	55 -> 7[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	6[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.53/5.14	7[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.53/5.14	8[label="FiniteMap.foldFM_LE3 FiniteMap.fmToList_LE0 [] wz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.53/5.14	9[label="FiniteMap.foldFM_LE2 FiniteMap.fmToList_LE0 [] wz4 (FiniteMap.Branch wz30 wz31 wz32 wz33 wz34)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
12.53/5.14	10[label="[]",fontsize=16,color="green",shape="box"];11[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (wz30 <= wz4)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12.53/5.14	12[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (compare wz30 wz4 /= GT)",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.53/5.14	13[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] wz4 wz30 wz31 wz32 wz33 wz34 (not (compare wz30 wz4 == GT))",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];
12.53/5.14	56 -> 14[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	14[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] wz4 () wz31 wz32 wz33 wz34 (not (compare () wz4 == GT))",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];
12.53/5.14	57 -> 15[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	15[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] () () wz31 wz32 wz33 wz34 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
12.53/5.14	16[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] () () wz31 wz32 wz33 wz34 (not (EQ == GT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
12.53/5.14	17[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] () () wz31 wz32 wz33 wz34 (not False)",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
12.53/5.14	18[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 [] () () wz31 wz32 wz33 wz34 True",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
12.53/5.14	19 -> 20[label="",style="dashed", color="red", weight=0];
12.53/5.14	19[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 (FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 [] () wz33)) () wz34",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	21 -> 5[label="",style="dashed", color="red", weight=0];
12.53/5.14	21[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 [] () wz33",fontsize=16,color="magenta"];21 -> 22[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	21 -> 23[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	20[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () wz34",fontsize=16,color="burlywood",shape="triangle"];58[label="wz34/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];20 -> 58[label="",style="solid", color="burlywood", weight=9];
12.53/5.14	58 -> 24[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	59[label="wz34/FiniteMap.Branch wz340 wz341 wz342 wz343 wz344",fontsize=10,color="white",style="solid",shape="box"];20 -> 59[label="",style="solid", color="burlywood", weight=9];
12.53/5.14	59 -> 25[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	22[label="wz33",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3];
12.53/5.14	25[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3];
12.53/5.14	26[label="FiniteMap.foldFM_LE3 FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3];
12.53/5.14	27[label="FiniteMap.foldFM_LE2 FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () (FiniteMap.Branch wz340 wz341 wz342 wz343 wz344)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
12.53/5.14	28[label="FiniteMap.fmToList_LE0 () wz31 wz5",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
12.53/5.14	29 -> 31[label="",style="dashed", color="red", weight=0];
12.53/5.14	29[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz31 wz5) () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="magenta"];29 -> 32[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	30[label="((),wz31) : wz5",fontsize=16,color="green",shape="box"];32 -> 28[label="",style="dashed", color="red", weight=0];
12.53/5.14	32[label="FiniteMap.fmToList_LE0 () wz31 wz5",fontsize=16,color="magenta"];31[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (wz340 <= ())",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3];
12.53/5.14	33[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (compare wz340 () /= GT)",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3];
12.53/5.14	34[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () wz340 wz341 wz342 wz343 wz344 (not (compare wz340 () == GT))",fontsize=16,color="burlywood",shape="box"];60[label="wz340/()",fontsize=10,color="white",style="solid",shape="box"];34 -> 60[label="",style="solid", color="burlywood", weight=9];
12.53/5.14	60 -> 35[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	35[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3];
12.53/5.14	36[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () () wz341 wz342 wz343 wz344 (not (EQ == GT))",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3];
12.53/5.14	37[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () () wz341 wz342 wz343 wz344 (not False)",fontsize=16,color="black",shape="box"];37 -> 38[label="",style="solid", color="black", weight=3];
12.53/5.14	38[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () () wz341 wz342 wz343 wz344 True",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3];
12.53/5.14	39 -> 20[label="",style="dashed", color="red", weight=0];
12.53/5.14	39[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 (FiniteMap.fmToList_LE0 () wz341 (FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 wz6 () wz343)) () wz344",fontsize=16,color="magenta"];39 -> 40[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	39 -> 41[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	39 -> 42[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	40[label="wz341",fontsize=16,color="green",shape="box"];41[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 wz6 () wz343",fontsize=16,color="burlywood",shape="box"];61[label="wz343/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];41 -> 61[label="",style="solid", color="burlywood", weight=9];
12.53/5.14	61 -> 43[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	62[label="wz343/FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434",fontsize=10,color="white",style="solid",shape="box"];41 -> 62[label="",style="solid", color="burlywood", weight=9];
12.53/5.14	62 -> 44[label="",style="solid", color="burlywood", weight=3];
12.53/5.14	42[label="wz344",fontsize=16,color="green",shape="box"];43[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
12.53/5.14	44[label="FiniteMap.foldFM_LE FiniteMap.fmToList_LE0 wz6 () (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
12.53/5.14	45[label="FiniteMap.foldFM_LE3 FiniteMap.fmToList_LE0 wz6 () FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];45 -> 47[label="",style="solid", color="black", weight=3];
12.53/5.14	46[label="FiniteMap.foldFM_LE2 FiniteMap.fmToList_LE0 wz6 () (FiniteMap.Branch wz3430 wz3431 wz3432 wz3433 wz3434)",fontsize=16,color="black",shape="box"];46 -> 48[label="",style="solid", color="black", weight=3];
12.53/5.14	47[label="wz6",fontsize=16,color="green",shape="box"];48 -> 31[label="",style="dashed", color="red", weight=0];
12.53/5.14	48[label="FiniteMap.foldFM_LE1 FiniteMap.fmToList_LE0 wz6 () wz3430 wz3431 wz3432 wz3433 wz3434 (wz3430 <= ())",fontsize=16,color="magenta"];48 -> 49[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	48 -> 50[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	48 -> 51[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	48 -> 52[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	48 -> 53[label="",style="dashed", color="magenta", weight=3];
12.53/5.14	49[label="wz3430",fontsize=16,color="green",shape="box"];50[label="wz3434",fontsize=16,color="green",shape="box"];51[label="wz3433",fontsize=16,color="green",shape="box"];52[label="wz3432",fontsize=16,color="green",shape="box"];53[label="wz3431",fontsize=16,color="green",shape="box"];}
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(8)
12.53/5.14	Complex Obligation (AND)
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(9)
12.53/5.14	Obligation:
12.53/5.14	Q DP problem:
12.53/5.14	The TRS P consists of the following rules:
12.53/5.14	
12.53/5.14	   new_foldFM_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE3(@0, wz33, h)
12.53/5.14	
12.53/5.14	R is empty.
12.53/5.14	Q is empty.
12.53/5.14	We have to consider all minimal (P,Q,R)-chains.
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(10) QDPSizeChangeProof (EQUIVALENT)
12.53/5.14	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. 
12.53/5.14	
12.53/5.14	From the DPs we obtained the following set of size-change graphs:
12.53/5.14	*new_foldFM_LE3(@0, Branch(@0, wz31, wz32, wz33, wz34), h) -> new_foldFM_LE3(@0, wz33, h)
12.53/5.14	The graph contains the following edges 1 >= 1, 2 > 1, 2 > 2, 3 >= 3
12.53/5.14	
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(11)
12.53/5.14	YES
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(12)
12.53/5.14	Obligation:
12.53/5.14	Q DP problem:
12.53/5.14	The TRS P consists of the following rules:
12.53/5.14	
12.53/5.14	   new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
12.53/5.14	   new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
12.53/5.14	   new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
12.53/5.14	
12.53/5.14	The TRS R consists of the following rules:
12.53/5.14	
12.53/5.14	   new_foldFM_LE2(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
12.53/5.14	   new_foldFM_LE0(wz6, EmptyFM, h) -> wz6
12.53/5.14	   new_foldFM_LE2(wz31, wz5, EmptyFM, h) -> new_fmToList_LE0(wz31, wz5, h)
12.53/5.14	   new_foldFM_LE0(wz6, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
12.53/5.14	   new_foldFM_LE10(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
12.53/5.14	   new_fmToList_LE0(wz31, wz5, h) -> :(@2(@0, wz31), wz5)
12.53/5.14	
12.53/5.14	The set Q consists of the following terms:
12.53/5.14	
12.53/5.14	   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
12.53/5.14	   new_foldFM_LE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.53/5.14	   new_foldFM_LE2(x0, x1, EmptyFM, x2)
12.53/5.14	   new_foldFM_LE0(x0, EmptyFM, x1)
12.53/5.14	   new_fmToList_LE0(x0, x1, x2)
12.53/5.14	   new_foldFM_LE10(x0, @0, x1, x2, x3, x4, x5)
12.53/5.14	
12.53/5.14	We have to consider all minimal (P,Q,R)-chains.
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(13) TransformationProof (EQUIVALENT)
12.53/5.14	By rewriting [LPAR04] the rule new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h) at position [0] we obtained the following new rules [LPAR04]:
12.53/5.14	
12.53/5.14	   (new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@2(@0, wz31), wz5), wz340, wz341, wz342, wz343, wz344, h),new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@2(@0, wz31), wz5), wz340, wz341, wz342, wz343, wz344, h))
12.53/5.14	
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(14)
12.53/5.14	Obligation:
12.53/5.14	Q DP problem:
12.53/5.14	The TRS P consists of the following rules:
12.53/5.14	
12.53/5.14	   new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
12.53/5.14	   new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
12.53/5.14	   new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@2(@0, wz31), wz5), wz340, wz341, wz342, wz343, wz344, h)
12.53/5.14	
12.53/5.14	The TRS R consists of the following rules:
12.53/5.14	
12.53/5.14	   new_foldFM_LE2(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE10(new_fmToList_LE0(wz31, wz5, h), wz340, wz341, wz342, wz343, wz344, h)
12.53/5.14	   new_foldFM_LE0(wz6, EmptyFM, h) -> wz6
12.53/5.14	   new_foldFM_LE2(wz31, wz5, EmptyFM, h) -> new_fmToList_LE0(wz31, wz5, h)
12.53/5.14	   new_foldFM_LE0(wz6, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), h) -> new_foldFM_LE10(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
12.53/5.14	   new_foldFM_LE10(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE2(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
12.53/5.14	   new_fmToList_LE0(wz31, wz5, h) -> :(@2(@0, wz31), wz5)
12.53/5.14	
12.53/5.14	The set Q consists of the following terms:
12.53/5.14	
12.53/5.14	   new_foldFM_LE0(x0, Branch(x1, x2, x3, x4, x5), x6)
12.53/5.14	   new_foldFM_LE2(x0, x1, Branch(x2, x3, x4, x5, x6), x7)
12.53/5.14	   new_foldFM_LE2(x0, x1, EmptyFM, x2)
12.53/5.14	   new_foldFM_LE0(x0, EmptyFM, x1)
12.53/5.14	   new_fmToList_LE0(x0, x1, x2)
12.53/5.14	   new_foldFM_LE10(x0, @0, x1, x2, x3, x4, x5)
12.53/5.14	
12.53/5.14	We have to consider all minimal (P,Q,R)-chains.
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(15) QDPSizeChangeProof (EQUIVALENT)
12.53/5.14	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. 
12.53/5.14	
12.53/5.14	From the DPs we obtained the following set of size-change graphs:
12.53/5.14	*new_foldFM_LE(wz31, wz5, Branch(wz340, wz341, wz342, wz343, wz344), h) -> new_foldFM_LE1(:(@2(@0, wz31), wz5), wz340, wz341, wz342, wz343, wz344, h)
12.53/5.14	The graph contains the following edges 3 > 2, 3 > 3, 3 > 4, 3 > 5, 3 > 6, 4 >= 7
12.53/5.14	
12.53/5.14	
12.53/5.14	*new_foldFM_LE1(wz6, @0, wz341, wz342, Branch(wz3430, wz3431, wz3432, wz3433, wz3434), wz344, h) -> new_foldFM_LE1(wz6, wz3430, wz3431, wz3432, wz3433, wz3434, h)
12.53/5.14	The graph contains the following edges 1 >= 1, 5 > 2, 5 > 3, 5 > 4, 5 > 5, 5 > 6, 7 >= 7
12.53/5.14	
12.53/5.14	
12.53/5.14	*new_foldFM_LE1(wz6, @0, wz341, wz342, wz343, wz344, h) -> new_foldFM_LE(wz341, new_foldFM_LE0(wz6, wz343, h), wz344, h)
12.53/5.14	The graph contains the following edges 3 >= 1, 6 >= 3, 7 >= 4
12.53/5.14	
12.53/5.14	
12.53/5.14	----------------------------------------
12.53/5.14	
12.53/5.14	(16)
12.53/5.14	YES
12.61/5.18	EOF