9.92/4.38	YES
11.55/4.86	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.55/4.86	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.55/4.86	
11.55/4.86	
11.55/4.86	H-Termination with start terms of the given HASKELL could be proven:
11.55/4.86	
11.55/4.86	(0) HASKELL
11.55/4.86	(1) BR [EQUIVALENT, 0 ms]
11.55/4.86	(2) HASKELL
11.55/4.86	(3) COR [EQUIVALENT, 0 ms]
11.55/4.86	(4) HASKELL
11.55/4.86	(5) Narrow [SOUND, 0 ms]
11.55/4.86	(6) QDP
11.55/4.86	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.55/4.86	(8) YES
11.55/4.86	
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(0)
11.55/4.86	Obligation:
11.55/4.86	mainModule Main 
11.55/4.86	module FiniteMap where {
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.55/4.86	
11.55/4.86	foldFM :: (a  ->  c  ->  b  ->  b)  ->  b  ->  FiniteMap a c  ->  b;
11.55/4.86	foldFM k z EmptyFM = z;
11.55/4.86	foldFM k z (Branch key elt _ fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
11.55/4.86	
11.55/4.86	}
11.55/4.86	module Maybe where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	module Main where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(1) BR (EQUIVALENT)
11.55/4.86	Replaced joker patterns by fresh variables and removed binding patterns.
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(2)
11.55/4.86	Obligation:
11.55/4.86	mainModule Main 
11.55/4.86	module FiniteMap where {
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.55/4.86	
11.55/4.86	foldFM :: (a  ->  b  ->  c  ->  c)  ->  c  ->  FiniteMap a b  ->  c;
11.55/4.86	foldFM k z EmptyFM = z;
11.55/4.86	foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
11.55/4.86	
11.55/4.86	}
11.55/4.86	module Maybe where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	module Main where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(3) COR (EQUIVALENT)
11.55/4.86	Cond Reductions:
11.55/4.86	The following Function with conditions
11.55/4.86	"undefined |Falseundefined;
11.55/4.86	"
11.55/4.86	is transformed to
11.55/4.86	"undefined  = undefined1;
11.55/4.86	"
11.55/4.86	"undefined0 True = undefined;
11.55/4.86	"
11.55/4.86	"undefined1  = undefined0 False;
11.55/4.86	"
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(4)
11.55/4.86	Obligation:
11.55/4.86	mainModule Main 
11.55/4.86	module FiniteMap where {
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	data FiniteMap b a = EmptyFM  | Branch b a Int (FiniteMap b a) (FiniteMap b a) ;
11.55/4.86	
11.55/4.86	foldFM :: (c  ->  a  ->  b  ->  b)  ->  b  ->  FiniteMap c a  ->  b;
11.55/4.86	foldFM k z EmptyFM = z;
11.55/4.86	foldFM k z (Branch key elt vy fm_l fm_r) = foldFM k (k key elt (foldFM k z fm_r)) fm_l;
11.55/4.86	
11.55/4.86	}
11.55/4.86	module Maybe where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Main;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	module Main where {
11.55/4.86	import qualified FiniteMap;
11.55/4.86	import qualified Maybe;
11.55/4.86	import qualified Prelude;
11.55/4.86	}
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(5) Narrow (SOUND)
11.55/4.86	Haskell To QDPs
11.55/4.86	
11.55/4.86	digraph dp_graph {
11.55/4.86	node [outthreshold=100, inthreshold=100];1[label="FiniteMap.foldFM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.55/4.86	3[label="FiniteMap.foldFM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.55/4.86	4[label="FiniteMap.foldFM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
11.55/4.86	5[label="FiniteMap.foldFM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];16[label="vz5/FiniteMap.EmptyFM",fontsize=10,color="white",style="solid",shape="box"];5 -> 16[label="",style="solid", color="burlywood", weight=9];
11.55/4.86	16 -> 6[label="",style="solid", color="burlywood", weight=3];
11.55/4.86	17[label="vz5/FiniteMap.Branch vz50 vz51 vz52 vz53 vz54",fontsize=10,color="white",style="solid",shape="box"];5 -> 17[label="",style="solid", color="burlywood", weight=9];
11.55/4.86	17 -> 7[label="",style="solid", color="burlywood", weight=3];
11.55/4.86	6[label="FiniteMap.foldFM vz3 vz4 FiniteMap.EmptyFM",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
11.55/4.86	7[label="FiniteMap.foldFM vz3 vz4 (FiniteMap.Branch vz50 vz51 vz52 vz53 vz54)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
11.55/4.86	8[label="vz4",fontsize=16,color="green",shape="box"];9 -> 5[label="",style="dashed", color="red", weight=0];
11.55/4.86	9[label="FiniteMap.foldFM vz3 (vz3 vz50 vz51 (FiniteMap.foldFM vz3 vz4 vz54)) vz53",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
11.55/4.86	9 -> 11[label="",style="dashed", color="magenta", weight=3];
11.55/4.86	10[label="vz53",fontsize=16,color="green",shape="box"];11[label="vz3 vz50 vz51 (FiniteMap.foldFM vz3 vz4 vz54)",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
11.55/4.86	11 -> 13[label="",style="dashed", color="green", weight=3];
11.55/4.86	11 -> 14[label="",style="dashed", color="green", weight=3];
11.55/4.86	12[label="vz50",fontsize=16,color="green",shape="box"];13[label="vz51",fontsize=16,color="green",shape="box"];14 -> 5[label="",style="dashed", color="red", weight=0];
11.55/4.86	14[label="FiniteMap.foldFM vz3 vz4 vz54",fontsize=16,color="magenta"];14 -> 15[label="",style="dashed", color="magenta", weight=3];
11.55/4.86	15[label="vz54",fontsize=16,color="green",shape="box"];}
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(6)
11.55/4.86	Obligation:
11.55/4.86	Q DP problem:
11.55/4.86	The TRS P consists of the following rules:
11.55/4.86	
11.55/4.86	   new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz54, h, ba, bb)
11.55/4.86	   new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz53, h, ba, bb)
11.55/4.86	
11.55/4.86	R is empty.
11.55/4.86	Q is empty.
11.55/4.86	We have to consider all minimal (P,Q,R)-chains.
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(7) QDPSizeChangeProof (EQUIVALENT)
11.55/4.86	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. 
11.55/4.86	
11.55/4.86	From the DPs we obtained the following set of size-change graphs:
11.55/4.86	*new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz54, h, ba, bb)
11.55/4.86	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
11.55/4.86	
11.55/4.86	
11.55/4.86	*new_foldFM(vz3, Branch(vz50, vz51, vz52, vz53, vz54), h, ba, bb) -> new_foldFM(vz3, vz53, h, ba, bb)
11.55/4.86	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5
11.55/4.86	
11.55/4.86	
11.55/4.86	----------------------------------------
11.55/4.86	
11.55/4.86	(8)
11.55/4.86	YES
11.69/4.90	EOF