7.91/3.53	YES
10.23/4.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.23/4.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.23/4.14	
10.23/4.14	
10.23/4.14	H-Termination with start terms of the given HASKELL could be proven:
10.23/4.14	
10.23/4.14	(0) HASKELL
10.23/4.14	(1) LR [EQUIVALENT, 0 ms]
10.23/4.14	(2) HASKELL
10.23/4.14	(3) CR [EQUIVALENT, 0 ms]
10.23/4.14	(4) HASKELL
10.23/4.14	(5) IFR [EQUIVALENT, 0 ms]
10.23/4.14	(6) HASKELL
10.23/4.14	(7) BR [EQUIVALENT, 0 ms]
10.23/4.14	(8) HASKELL
10.23/4.14	(9) COR [EQUIVALENT, 0 ms]
10.23/4.14	(10) HASKELL
10.23/4.14	(11) Narrow [SOUND, 0 ms]
10.23/4.14	(12) QDP
10.23/4.14	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.23/4.14	(14) YES
10.23/4.14	
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(0)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(1) LR (EQUIVALENT)
10.23/4.14	Lambda Reductions:
10.23/4.14	The following Lambda expression
10.23/4.14	"\vu39->case vu39 of {
10.23/4.14	x  -> if p x then x : [] else [];
10.23/4.14	_  -> []}
10.23/4.14	"
10.23/4.14	is transformed to
10.23/4.14	"filter0 p vu39 = case vu39 of {
10.23/4.14	x  -> if p x then x : [] else [];
10.23/4.14	_  -> []}
10.23/4.14	;
10.23/4.14	"
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(2)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(3) CR (EQUIVALENT)
10.23/4.14	Case Reductions:
10.23/4.14	The following Case expression
10.23/4.14	"case vu39 of {
10.23/4.14	x  -> if p x then x : [] else [];
10.23/4.14	_  -> []}
10.23/4.14	"
10.23/4.14	is transformed to
10.23/4.14	"filter00 p x = if p x then x : [] else [];
10.23/4.14	filter00 p _ = [];
10.23/4.14	"
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(4)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(5) IFR (EQUIVALENT)
10.23/4.14	If Reductions:
10.23/4.14	The following If expression
10.23/4.14	"if p x then x : [] else []"
10.23/4.14	is transformed to
10.23/4.14	"filter000 x True = x : [];
10.23/4.14	filter000 x False = [];
10.23/4.14	"
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(6)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(7) BR (EQUIVALENT)
10.23/4.14	Replaced joker patterns by fresh variables and removed binding patterns.
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(8)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(9) COR (EQUIVALENT)
10.23/4.14	Cond Reductions:
10.23/4.14	The following Function with conditions
10.23/4.14	"undefined |Falseundefined;
10.23/4.14	"
10.23/4.14	is transformed to
10.23/4.14	"undefined  = undefined1;
10.23/4.14	"
10.23/4.14	"undefined0 True = undefined;
10.23/4.14	"
10.23/4.14	"undefined1  = undefined0 False;
10.23/4.14	"
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(10)
10.23/4.14	Obligation:
10.23/4.14	mainModule Main 
10.23/4.14	module Main where {
10.23/4.14	import qualified Prelude;
10.23/4.14	}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(11) Narrow (SOUND)
10.23/4.14	Haskell To QDPs
10.23/4.14	
10.23/4.14	digraph dp_graph {
10.23/4.14	node [outthreshold=100, inthreshold=100];1[label="filter",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.23/4.14	3[label="filter vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.23/4.14	4[label="filter vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.23/4.14	5[label="concatMap (filter0 vy3) vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.23/4.14	6[label="concat . map (filter0 vy3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.23/4.14	7[label="concat (map (filter0 vy3) vy4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.23/4.14	8[label="foldr (++) [] (map (filter0 vy3) vy4)",fontsize=16,color="burlywood",shape="triangle"];31[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];8 -> 31[label="",style="solid", color="burlywood", weight=9];
10.23/4.14	31 -> 9[label="",style="solid", color="burlywood", weight=3];
10.23/4.14	32[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 32[label="",style="solid", color="burlywood", weight=9];
10.23/4.14	32 -> 10[label="",style="solid", color="burlywood", weight=3];
10.23/4.14	9[label="foldr (++) [] (map (filter0 vy3) (vy40 : vy41))",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10.23/4.14	10[label="foldr (++) [] (map (filter0 vy3) [])",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.23/4.14	11[label="foldr (++) [] (filter0 vy3 vy40 : map (filter0 vy3) vy41)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.23/4.14	12[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.23/4.14	13 -> 15[label="",style="dashed", color="red", weight=0];
10.23/4.14	13[label="(++) filter0 vy3 vy40 foldr (++) [] (map (filter0 vy3) vy41)",fontsize=16,color="magenta"];13 -> 16[label="",style="dashed", color="magenta", weight=3];
10.23/4.14	14[label="[]",fontsize=16,color="green",shape="box"];16 -> 8[label="",style="dashed", color="red", weight=0];
10.23/4.14	16[label="foldr (++) [] (map (filter0 vy3) vy41)",fontsize=16,color="magenta"];16 -> 17[label="",style="dashed", color="magenta", weight=3];
10.23/4.14	15[label="(++) filter0 vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];15 -> 18[label="",style="solid", color="black", weight=3];
10.23/4.14	17[label="vy41",fontsize=16,color="green",shape="box"];18[label="(++) filter00 vy3 vy40 vy5",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
10.23/4.14	19 -> 20[label="",style="dashed", color="red", weight=0];
10.23/4.14	19[label="(++) filter000 vy40 (vy3 vy40) vy5",fontsize=16,color="magenta"];19 -> 21[label="",style="dashed", color="magenta", weight=3];
10.23/4.14	21[label="vy3 vy40",fontsize=16,color="green",shape="box"];21 -> 25[label="",style="dashed", color="green", weight=3];
10.23/4.14	20[label="(++) filter000 vy40 vy6 vy5",fontsize=16,color="burlywood",shape="triangle"];33[label="vy6/False",fontsize=10,color="white",style="solid",shape="box"];20 -> 33[label="",style="solid", color="burlywood", weight=9];
10.23/4.14	33 -> 23[label="",style="solid", color="burlywood", weight=3];
10.23/4.14	34[label="vy6/True",fontsize=10,color="white",style="solid",shape="box"];20 -> 34[label="",style="solid", color="burlywood", weight=9];
10.23/4.14	34 -> 24[label="",style="solid", color="burlywood", weight=3];
10.23/4.14	25[label="vy40",fontsize=16,color="green",shape="box"];23[label="(++) filter000 vy40 False vy5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
10.23/4.14	24[label="(++) filter000 vy40 True vy5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
10.23/4.14	26[label="(++) [] vy5",fontsize=16,color="black",shape="triangle"];26 -> 28[label="",style="solid", color="black", weight=3];
10.23/4.14	27[label="(++) (vy40 : []) vy5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
10.23/4.14	28[label="vy5",fontsize=16,color="green",shape="box"];29[label="vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];29 -> 30[label="",style="dashed", color="green", weight=3];
10.23/4.14	30 -> 26[label="",style="dashed", color="red", weight=0];
10.23/4.14	30[label="[] ++ vy5",fontsize=16,color="magenta"];}
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(12)
10.23/4.14	Obligation:
10.23/4.14	Q DP problem:
10.23/4.14	The TRS P consists of the following rules:
10.23/4.14	
10.23/4.14	   new_foldr(vy3, :(vy40, vy41), h) -> new_foldr(vy3, vy41, h)
10.23/4.14	
10.23/4.14	R is empty.
10.23/4.14	Q is empty.
10.23/4.14	We have to consider all minimal (P,Q,R)-chains.
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(13) QDPSizeChangeProof (EQUIVALENT)
10.23/4.14	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. 
10.23/4.14	
10.23/4.14	From the DPs we obtained the following set of size-change graphs:
10.23/4.14	*new_foldr(vy3, :(vy40, vy41), h) -> new_foldr(vy3, vy41, h)
10.23/4.14	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
10.23/4.14	
10.23/4.14	
10.23/4.14	----------------------------------------
10.23/4.14	
10.23/4.14	(14)
10.23/4.14	YES
10.23/4.18	EOF