9.84/4.43	YES
12.06/4.98	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.06/4.98	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.06/4.98	
12.06/4.98	
12.06/4.98	H-Termination with start terms of the given HASKELL could be proven:
12.06/4.98	
12.06/4.98	(0) HASKELL
12.06/4.98	(1) BR [EQUIVALENT, 0 ms]
12.06/4.98	(2) HASKELL
12.06/4.98	(3) COR [EQUIVALENT, 0 ms]
12.06/4.98	(4) HASKELL
12.06/4.98	(5) Narrow [SOUND, 0 ms]
12.06/4.98	(6) QDP
12.06/4.98	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.06/4.98	(8) YES
12.06/4.98	
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(0)
12.06/4.98	Obligation:
12.06/4.98	mainModule Main 
12.06/4.98	module FiniteMap where {
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.06/4.98	
12.06/4.98	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.06/4.98	}
12.06/4.98	foldFM_LE :: Ord a => (a  ->  c  ->  b  ->  b)  ->  b  ->  a  ->  FiniteMap a c  ->  b;
12.06/4.98	foldFM_LE k z fr EmptyFM = z;
12.06/4.98	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
12.06/4.98	 | otherwise = foldFM_LE k z fr fm_l;
12.06/4.98	
12.06/4.98	}
12.06/4.98	module Maybe where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	module Main where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(1) BR (EQUIVALENT)
12.06/4.98	Replaced joker patterns by fresh variables and removed binding patterns.
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(2)
12.06/4.98	Obligation:
12.06/4.98	mainModule Main 
12.06/4.98	module FiniteMap where {
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.06/4.98	
12.06/4.98	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.06/4.98	}
12.06/4.98	foldFM_LE :: Ord c => (c  ->  a  ->  b  ->  b)  ->  b  ->  c  ->  FiniteMap c a  ->  b;
12.06/4.98	foldFM_LE k z fr EmptyFM = z;
12.06/4.98	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
12.06/4.98	 | otherwise = foldFM_LE k z fr fm_l;
12.06/4.98	
12.06/4.98	}
12.06/4.98	module Maybe where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	module Main where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(3) COR (EQUIVALENT)
12.06/4.98	Cond Reductions:
12.06/4.98	The following Function with conditions
12.06/4.98	"undefined |Falseundefined;
12.06/4.98	"
12.06/4.98	is transformed to
12.06/4.98	"undefined  = undefined1;
12.06/4.98	"
12.06/4.98	"undefined0 True = undefined;
12.06/4.98	"
12.06/4.98	"undefined1  = undefined0 False;
12.06/4.98	"
12.06/4.98	The following Function with conditions
12.06/4.98	"foldFM_LE k z fr EmptyFM = z;
12.06/4.98	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;
12.06/4.98	"
12.06/4.98	is transformed to
12.06/4.98	"foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.06/4.98	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);
12.06/4.98	"
12.06/4.98	"foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.06/4.98	"
12.06/4.98	"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;
12.06/4.98	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;
12.06/4.98	"
12.06/4.98	"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);
12.06/4.98	"
12.06/4.98	"foldFM_LE3 k z fr EmptyFM = z;
12.06/4.98	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.06/4.98	"
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(4)
12.06/4.98	Obligation:
12.06/4.98	mainModule Main 
12.06/4.98	module FiniteMap where {
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	data FiniteMap a b = EmptyFM  | Branch a b Int (FiniteMap a b) (FiniteMap a b) ;
12.06/4.98	
12.06/4.98	instance (Eq a, Eq b) => Eq FiniteMap b a where { 
12.06/4.98	}
12.06/4.98	foldFM_LE :: Ord b => (b  ->  a  ->  c  ->  c)  ->  c  ->  b  ->  FiniteMap b a  ->  c;
12.06/4.98	foldFM_LE k z fr EmptyFM = foldFM_LE3 k z fr EmptyFM;
12.06/4.98	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);
12.06/4.98	
12.06/4.98	foldFM_LE0 k z fr key elt vy fm_l fm_r True = foldFM_LE k z fr fm_l;
12.06/4.98	
12.06/4.98	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;
12.06/4.98	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;
12.06/4.98	
12.06/4.98	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);
12.06/4.98	
12.06/4.98	foldFM_LE3 k z fr EmptyFM = z;
12.06/4.98	foldFM_LE3 wv ww wx wy = foldFM_LE2 wv ww wx wy;
12.06/4.98	
12.06/4.98	}
12.06/4.98	module Maybe where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Main;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	module Main where {
12.06/4.98	import qualified FiniteMap;
12.06/4.98	import qualified Maybe;
12.06/4.98	import qualified Prelude;
12.06/4.98	}
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(5) Narrow (SOUND)
12.06/4.98	Haskell To QDPs
12.06/4.98	
12.06/4.98	digraph dp_graph {
12.06/4.98	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];
12.06/4.98	3[label="FiniteMap.foldFM_LE wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.06/4.98	4[label="FiniteMap.foldFM_LE wz3 wz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
12.06/4.98	5[label="FiniteMap.foldFM_LE wz3 wz4 wz5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
12.06/4.98	6[label="FiniteMap.foldFM_LE wz3 wz4 wz5 wz6",fontsize=16,color="burlywood",shape="triangle"];29[label="wz6/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];6 -> 29[label="",style="solid", color="burlywood", weight=9];
12.06/4.98	29 -> 7[label="",style="solid", color="burlywood", weight=3];
12.06/4.98	30[label="wz6/FiniteMap.Branch wz60 wz61 wz62 wz63 wz64",fontsize=10,color="white",style="solid",shape="box"];6 -> 30[label="",style="solid", color="burlywood", weight=9];
12.06/4.98	30 -> 8[label="",style="solid", color="burlywood", weight=3];
12.06/4.98	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];
12.06/4.98	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];
12.06/4.98	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];
12.06/4.98	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];
12.06/4.98	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="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.06/4.98	13[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (compare wz60 wz5 /= GT)",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
12.06/4.98	14[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 wz60 wz61 wz62 wz63 wz64 (not (compare wz60 wz5 == GT))",fontsize=16,color="burlywood",shape="box"];31[label="wz60/()",fontsize=10,color="white",style="solid",shape="box"];14 -> 31[label="",style="solid", color="burlywood", weight=9];
12.06/4.98	31 -> 15[label="",style="solid", color="burlywood", weight=3];
12.06/4.98	15[label="FiniteMap.foldFM_LE1 wz3 wz4 wz5 () wz61 wz62 wz63 wz64 (not (compare () wz5 == GT))",fontsize=16,color="burlywood",shape="box"];32[label="wz5/()",fontsize=10,color="white",style="solid",shape="box"];15 -> 32[label="",style="solid", color="burlywood", weight=9];
12.06/4.98	32 -> 16[label="",style="solid", color="burlywood", weight=3];
12.06/4.98	16[label="FiniteMap.foldFM_LE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (compare () () == GT))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
12.06/4.98	17[label="FiniteMap.foldFM_LE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3];
12.06/4.98	18[label="FiniteMap.foldFM_LE1 wz3 wz4 () () wz61 wz62 wz63 wz64 (not False)",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
12.06/4.98	19[label="FiniteMap.foldFM_LE1 wz3 wz4 () () wz61 wz62 wz63 wz64 True",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
12.06/4.98	20 -> 6[label="",style="dashed", color="red", weight=0];
12.06/4.98	20[label="FiniteMap.foldFM_LE wz3 (wz3 () wz61 (FiniteMap.foldFM_LE wz3 wz4 () wz63)) () wz64",fontsize=16,color="magenta"];20 -> 21[label="",style="dashed", color="magenta", weight=3];
12.06/4.98	20 -> 22[label="",style="dashed", color="magenta", weight=3];
12.06/4.98	20 -> 23[label="",style="dashed", color="magenta", weight=3];
12.06/4.98	21[label="wz3 () wz61 (FiniteMap.foldFM_LE wz3 wz4 () wz63)",fontsize=16,color="green",shape="box"];21 -> 24[label="",style="dashed", color="green", weight=3];
12.06/4.98	21 -> 25[label="",style="dashed", color="green", weight=3];
12.06/4.98	21 -> 26[label="",style="dashed", color="green", weight=3];
12.06/4.98	22[label="wz64",fontsize=16,color="green",shape="box"];23[label="()",fontsize=16,color="green",shape="box"];24[label="()",fontsize=16,color="green",shape="box"];25[label="wz61",fontsize=16,color="green",shape="box"];26 -> 6[label="",style="dashed", color="red", weight=0];
12.06/4.98	26[label="FiniteMap.foldFM_LE wz3 wz4 () wz63",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
12.06/4.98	26 -> 28[label="",style="dashed", color="magenta", weight=3];
12.06/4.98	27[label="wz63",fontsize=16,color="green",shape="box"];28[label="()",fontsize=16,color="green",shape="box"];}
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(6)
12.06/4.98	Obligation:
12.06/4.98	Q DP problem:
12.06/4.98	The TRS P consists of the following rules:
12.06/4.98	
12.06/4.98	   new_foldFM_LE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, @0, wz63, h, ba)
12.06/4.98	   new_foldFM_LE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, @0, wz64, h, ba)
12.06/4.98	
12.06/4.98	R is empty.
12.06/4.98	Q is empty.
12.06/4.98	We have to consider all minimal (P,Q,R)-chains.
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(7) QDPSizeChangeProof (EQUIVALENT)
12.06/4.98	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. 
12.06/4.98	
12.06/4.98	From the DPs we obtained the following set of size-change graphs:
12.06/4.98	*new_foldFM_LE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, @0, wz63, h, ba)
12.06/4.98	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.06/4.98	
12.06/4.98	
12.06/4.98	*new_foldFM_LE(wz3, @0, Branch(@0, wz61, wz62, wz63, wz64), h, ba) -> new_foldFM_LE(wz3, @0, wz64, h, ba)
12.06/4.98	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5
12.06/4.98	
12.06/4.98	
12.06/4.98	----------------------------------------
12.06/4.98	
12.06/4.98	(8)
12.06/4.98	YES
12.06/5.01	EOF