9.91/4.65	YES
12.08/5.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.08/5.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.08/5.21	
12.08/5.21	
12.08/5.21	H-Termination with start terms of the given HASKELL could be proven:
12.08/5.21	
12.08/5.21	(0) HASKELL
12.08/5.21	(1) LR [EQUIVALENT, 0 ms]
12.08/5.21	(2) HASKELL
12.08/5.21	(3) BR [EQUIVALENT, 0 ms]
12.08/5.21	(4) HASKELL
12.08/5.21	(5) COR [EQUIVALENT, 0 ms]
12.08/5.21	(6) HASKELL
12.08/5.21	(7) Narrow [SOUND, 0 ms]
12.08/5.21	(8) AND
12.08/5.21	    (9) QDP
12.08/5.21	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.08/5.21	        (11) YES
12.08/5.21	    (12) QDP
12.08/5.21	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.08/5.21	        (14) YES
12.08/5.21	
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(0)
12.08/5.21	Obligation:
12.08/5.21	mainModule Main 
12.08/5.21	module FiniteMap where {
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.08/5.21	
12.08/5.21	eltsFM :: FiniteMap b a  ->  [a];
12.08/5.21	eltsFM fm = foldFM (\key elt rest ->elt : rest) [] fm;
12.08/5.21	
12.08/5.21	foldFM :: (c  ->  a  ->  b  ->  b)  ->  b  ->  FiniteMap c a  ->  b;
12.08/5.21	foldFM k z EmptyFM = z;
12.08/5.21	foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
12.08/5.21	
12.08/5.21	}
12.08/5.21	module Maybe where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	module Main where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(1) LR (EQUIVALENT)
12.08/5.21	Lambda Reductions:
12.08/5.21	The following Lambda expression
12.08/5.21	"\keyeltrest->elt : rest"
12.08/5.21	is transformed to
12.08/5.21	"eltsFM0 key elt rest = elt : rest;
12.08/5.21	"
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(2)
12.08/5.21	Obligation:
12.08/5.21	mainModule Main 
12.08/5.21	module FiniteMap where {
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.08/5.21	
12.08/5.21	eltsFM :: FiniteMap b a  ->  [a];
12.08/5.21	eltsFM fm = foldFM eltsFM0 [] fm;
12.08/5.21	
12.08/5.21	eltsFM0 key elt rest = elt : rest;
12.08/5.21	
12.08/5.21	foldFM :: (a  ->  b  ->  c  ->  c)  ->  c  ->  FiniteMap a b  ->  c;
12.08/5.21	foldFM k z EmptyFM = z;
12.08/5.21	foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
12.08/5.21	
12.08/5.21	}
12.08/5.21	module Maybe where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	module Main where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(3) BR (EQUIVALENT)
12.08/5.21	Replaced joker patterns by fresh variables and removed binding patterns.
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(4)
12.08/5.21	Obligation:
12.08/5.21	mainModule Main 
12.08/5.21	module FiniteMap where {
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.08/5.21	
12.08/5.21	eltsFM :: FiniteMap a b  ->  [b];
12.08/5.21	eltsFM fm = foldFM eltsFM0 [] fm;
12.08/5.21	
12.08/5.21	eltsFM0 key elt rest = elt : rest;
12.08/5.21	
12.08/5.21	foldFM :: (c  ->  b  ->  a  ->  a)  ->  a  ->  FiniteMap c b  ->  a;
12.08/5.21	foldFM k z EmptyFM = z;
12.08/5.21	foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
12.08/5.21	
12.08/5.21	}
12.08/5.21	module Maybe where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	module Main where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(5) COR (EQUIVALENT)
12.08/5.21	Cond Reductions:
12.08/5.21	The following Function with conditions
12.08/5.21	"undefined |Falseundefined;
12.08/5.21	"
12.08/5.21	is transformed to
12.08/5.21	"undefined  = undefined1;
12.08/5.21	"
12.08/5.21	"undefined0 True = undefined;
12.08/5.21	"
12.08/5.21	"undefined1  = undefined0 False;
12.08/5.21	"
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(6)
12.08/5.21	Obligation:
12.08/5.21	mainModule Main 
12.08/5.21	module FiniteMap where {
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
12.08/5.21	
12.08/5.21	eltsFM :: FiniteMap a b  ->  [b];
12.08/5.21	eltsFM fm = foldFM eltsFM0 [] fm;
12.08/5.21	
12.08/5.21	eltsFM0 key elt rest = elt : rest;
12.08/5.21	
12.08/5.21	foldFM :: (c  ->  b  ->  a  ->  a)  ->  a  ->  FiniteMap c b  ->  a;
12.08/5.21	foldFM k z EmptyFM = z;
12.08/5.21	foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
12.08/5.21	
12.08/5.21	}
12.08/5.21	module Maybe where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Main;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	module Main where {
12.08/5.21	import qualified FiniteMap;
12.08/5.21	import qualified Maybe;
12.08/5.21	import qualified Prelude;
12.08/5.21	}
12.08/5.21	
12.08/5.21	----------------------------------------
12.08/5.21	
12.08/5.21	(7) Narrow (SOUND)
12.08/5.21	Haskell To QDPs
12.08/5.21	
12.08/5.21	digraph dp_graph {
12.08/5.21	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.eltsFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.08/5.21	3[label="FiniteMap.eltsFM vz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
12.08/5.21	4[label="FiniteMap.foldFM FiniteMap.eltsFM0 [] vz3",fontsize=16,color="burlywood",shape="triangle"];22[label="vz3/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];4 -> 22[label="",style="solid", color="burlywood", weight=9];
12.08/5.21	22 -> 5[label="",style="solid", color="burlywood", weight=3];
12.08/5.21	23[label="vz3/FiniteMap.Branch vz30 vz31 vz32 vz33 vz34",fontsize=10,color="white",style="solid",shape="box"];4 -> 23[label="",style="solid", color="burlywood", weight=9];
12.08/5.21	23 -> 6[label="",style="solid", color="burlywood", weight=3];
12.08/5.21	5[label="FiniteMap.foldFM FiniteMap.eltsFM0 [] FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.08/5.22	6[label="FiniteMap.foldFM FiniteMap.eltsFM0 [] (FiniteMap.Branch vz30 vz31 vz32 vz33 vz34)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
12.08/5.22	7[label="[]",fontsize=16,color="green",shape="box"];8 -> 9[label="",style="dashed", color="red", weight=0];
12.08/5.22	8[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 (FiniteMap.foldFM FiniteMap.eltsFM0 [] vz34)) vz33",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	10 -> 4[label="",style="dashed", color="red", weight=0];
12.08/5.22	10[label="FiniteMap.foldFM FiniteMap.eltsFM0 [] vz34",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	9[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 vz4) vz33",fontsize=16,color="burlywood",shape="triangle"];24[label="vz33/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
12.08/5.22	24 -> 12[label="",style="solid", color="burlywood", weight=3];
12.08/5.22	25[label="vz33/FiniteMap.Branch vz330 vz331 vz332 vz333 vz334",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
12.08/5.22	25 -> 13[label="",style="solid", color="burlywood", weight=3];
12.08/5.22	11[label="vz34",fontsize=16,color="green",shape="box"];12[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 vz4) FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
12.08/5.22	13[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 vz4) (FiniteMap.Branch vz330 vz331 vz332 vz333 vz334)",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
12.08/5.22	14[label="FiniteMap.eltsFM0 vz30 vz31 vz4",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.08/5.22	15 -> 9[label="",style="dashed", color="red", weight=0];
12.08/5.22	15[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz330 vz331 (FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 vz4) vz334)) vz333",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	15 -> 18[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	15 -> 19[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	15 -> 20[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	16[label="vz31 : vz4",fontsize=16,color="green",shape="box"];17[label="vz330",fontsize=16,color="green",shape="box"];18 -> 9[label="",style="dashed", color="red", weight=0];
12.08/5.22	18[label="FiniteMap.foldFM FiniteMap.eltsFM0 (FiniteMap.eltsFM0 vz30 vz31 vz4) vz334",fontsize=16,color="magenta"];18 -> 21[label="",style="dashed", color="magenta", weight=3];
12.08/5.22	19[label="vz331",fontsize=16,color="green",shape="box"];20[label="vz333",fontsize=16,color="green",shape="box"];21[label="vz334",fontsize=16,color="green",shape="box"];}
12.08/5.22	
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(8)
12.08/5.22	Complex Obligation (AND)
12.08/5.22	
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(9)
12.08/5.22	Obligation:
12.08/5.22	Q DP problem:
12.08/5.22	The TRS P consists of the following rules:
12.08/5.22	
12.08/5.22	   new_foldFM1(Branch(vz30, vz31, vz32, vz33, vz34), h, ba) -> new_foldFM1(vz34, h, ba)
12.08/5.22	
12.08/5.22	R is empty.
12.08/5.22	Q is empty.
12.08/5.22	We have to consider all minimal (P,Q,R)-chains.
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(10) QDPSizeChangeProof (EQUIVALENT)
12.08/5.22	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. 
12.08/5.22	
12.08/5.22	From the DPs we obtained the following set of size-change graphs:
12.08/5.22	*new_foldFM1(Branch(vz30, vz31, vz32, vz33, vz34), h, ba) -> new_foldFM1(vz34, h, ba)
12.08/5.22	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
12.08/5.22	
12.08/5.22	
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(11)
12.08/5.22	YES
12.08/5.22	
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(12)
12.08/5.22	Obligation:
12.08/5.22	Q DP problem:
12.08/5.22	The TRS P consists of the following rules:
12.08/5.22	
12.08/5.22	   new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz30, vz31, vz4, vz334, h, ba)
12.08/5.22	   new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)
12.08/5.22	
12.08/5.22	The TRS R consists of the following rules:
12.08/5.22	
12.08/5.22	   new_foldFM0(vz30, vz31, vz4, EmptyFM, h, ba) -> :(vz31, vz4)
12.08/5.22	   new_foldFM0(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM0(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)
12.08/5.22	
12.08/5.22	The set Q consists of the following terms:
12.08/5.22	
12.08/5.22	   new_foldFM0(x0, x1, x2, Branch(x3, x4, x5, x6, x7), x8, x9)
12.08/5.22	   new_foldFM0(x0, x1, x2, EmptyFM, x3, x4)
12.08/5.22	
12.08/5.22	We have to consider all minimal (P,Q,R)-chains.
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(13) QDPSizeChangeProof (EQUIVALENT)
12.08/5.22	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. 
12.08/5.22	
12.08/5.22	From the DPs we obtained the following set of size-change graphs:
12.08/5.22	*new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz30, vz31, vz4, vz334, h, ba)
12.08/5.22	The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5, 6 >= 6
12.08/5.22	
12.08/5.22	
12.08/5.22	*new_foldFM(vz30, vz31, vz4, Branch(vz330, vz331, vz332, vz333, vz334), h, ba) -> new_foldFM(vz330, vz331, new_foldFM0(vz30, vz31, vz4, vz334, h, ba), vz333, h, ba)
12.08/5.22	The graph contains the following edges 4 > 1, 4 > 2, 4 > 4, 5 >= 5, 6 >= 6
12.08/5.22	
12.08/5.22	
12.08/5.22	----------------------------------------
12.08/5.22	
12.08/5.22	(14)
12.08/5.22	YES
12.08/5.25	EOF