10.52/4.47	YES
12.62/5.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.62/5.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.62/5.03	
12.62/5.03	
12.62/5.03	H-Termination with start terms of the given HASKELL could be proven:
12.62/5.03	
12.62/5.03	(0) HASKELL
12.62/5.03	(1) IFR [EQUIVALENT, 0 ms]
12.62/5.03	(2) HASKELL
12.62/5.03	(3) BR [EQUIVALENT, 0 ms]
12.62/5.03	(4) HASKELL
12.62/5.03	(5) COR [EQUIVALENT, 0 ms]
12.62/5.03	(6) HASKELL
12.62/5.03	(7) Narrow [SOUND, 0 ms]
12.62/5.03	(8) AND
12.62/5.03	    (9) QDP
12.62/5.03	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.62/5.03	        (11) YES
12.62/5.03	    (12) QDP
12.62/5.03	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.62/5.03	        (14) YES
12.62/5.03	
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(0)
12.62/5.03	Obligation:
12.62/5.03	mainModule Main 
12.62/5.03	module Maybe where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	module List where {
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.62/5.03	deleteBy _ _ [] = [];
12.62/5.03	deleteBy eq x (y : ys) =  if x `eq` y then ys else y : deleteBy eq x ys;
12.62/5.03	
12.62/5.03	deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.62/5.03	deleteFirstsBy eq = foldl (flip (deleteBy eq));
12.62/5.03	
12.62/5.03	}
12.62/5.03	module Main where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(1) IFR (EQUIVALENT)
12.62/5.03	If Reductions:
12.62/5.03	The following If expression
12.62/5.03	"if eq x y then ys else y : deleteBy eq x ys"
12.62/5.03	is transformed to
12.62/5.03	"deleteBy0 ys y eq x True = ys;
12.62/5.03	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.62/5.03	"
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(2)
12.62/5.03	Obligation:
12.62/5.03	mainModule Main 
12.62/5.03	module Maybe where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	module List where {
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.62/5.03	deleteBy _ _ [] = [];
12.62/5.03	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.62/5.03	
12.62/5.03	deleteBy0 ys y eq x True = ys;
12.62/5.03	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.62/5.03	
12.62/5.03	deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.62/5.03	deleteFirstsBy eq = foldl (flip (deleteBy eq));
12.62/5.03	
12.62/5.03	}
12.62/5.03	module Main where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(3) BR (EQUIVALENT)
12.62/5.03	Replaced joker patterns by fresh variables and removed binding patterns.
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(4)
12.62/5.03	Obligation:
12.62/5.03	mainModule Main 
12.62/5.03	module Maybe where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	module List where {
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.62/5.03	deleteBy vy vz [] = [];
12.62/5.03	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.62/5.03	
12.62/5.03	deleteBy0 ys y eq x True = ys;
12.62/5.03	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.62/5.03	
12.62/5.03	deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.62/5.03	deleteFirstsBy eq = foldl (flip (deleteBy eq));
12.62/5.03	
12.62/5.03	}
12.62/5.03	module Main where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(5) COR (EQUIVALENT)
12.62/5.03	Cond Reductions:
12.62/5.03	The following Function with conditions
12.62/5.03	"undefined |Falseundefined;
12.62/5.03	"
12.62/5.03	is transformed to
12.62/5.03	"undefined  = undefined1;
12.62/5.03	"
12.62/5.03	"undefined0 True = undefined;
12.62/5.03	"
12.62/5.03	"undefined1  = undefined0 False;
12.62/5.03	"
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(6)
12.62/5.03	Obligation:
12.62/5.03	mainModule Main 
12.62/5.03	module Maybe where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	module List where {
12.62/5.03	import qualified Main;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	deleteBy :: (a  ->  a  ->  Bool)  ->  a  ->  [a]  ->  [a];
12.62/5.03	deleteBy vy vz [] = [];
12.62/5.03	deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y);
12.62/5.03	
12.62/5.03	deleteBy0 ys y eq x True = ys;
12.62/5.03	deleteBy0 ys y eq x False = y : deleteBy eq x ys;
12.62/5.03	
12.62/5.03	deleteFirstsBy :: (a  ->  a  ->  Bool)  ->  [a]  ->  [a]  ->  [a];
12.62/5.03	deleteFirstsBy eq = foldl (flip (deleteBy eq));
12.62/5.03	
12.62/5.03	}
12.62/5.03	module Main where {
12.62/5.03	import qualified List;
12.62/5.03	import qualified Maybe;
12.62/5.03	import qualified Prelude;
12.62/5.03	}
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(7) Narrow (SOUND)
12.62/5.03	Haskell To QDPs
12.62/5.03	
12.62/5.03	digraph dp_graph {
12.62/5.03	node [outthreshold=100, inthreshold=100];1[label="List.deleteFirstsBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.62/5.03	3[label="List.deleteFirstsBy wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
12.62/5.03	4[label="List.deleteFirstsBy wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
12.62/5.03	5[label="List.deleteFirstsBy wu3 wu4 wu5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
12.62/5.03	6[label="foldl (flip (List.deleteBy wu3)) wu4 wu5",fontsize=16,color="burlywood",shape="triangle"];30[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];6 -> 30[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	30 -> 7[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	31[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 31[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	31 -> 8[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	7[label="foldl (flip (List.deleteBy wu3)) wu4 (wu50 : wu51)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
12.62/5.03	8[label="foldl (flip (List.deleteBy wu3)) wu4 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
12.62/5.03	9 -> 6[label="",style="dashed", color="red", weight=0];
12.62/5.03	9[label="foldl (flip (List.deleteBy wu3)) (flip (List.deleteBy wu3) wu4 wu50) wu51",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3];
12.62/5.03	9 -> 12[label="",style="dashed", color="magenta", weight=3];
12.62/5.03	10[label="wu4",fontsize=16,color="green",shape="box"];11[label="flip (List.deleteBy wu3) wu4 wu50",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12.62/5.03	12[label="wu51",fontsize=16,color="green",shape="box"];13[label="List.deleteBy wu3 wu50 wu4",fontsize=16,color="burlywood",shape="triangle"];32[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];13 -> 32[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	32 -> 14[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	33[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 33[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	33 -> 15[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	14[label="List.deleteBy wu3 wu50 (wu40 : wu41)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.62/5.03	15[label="List.deleteBy wu3 wu50 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.62/5.03	16 -> 18[label="",style="dashed", color="red", weight=0];
12.62/5.03	16[label="List.deleteBy0 wu41 wu40 wu3 wu50 (wu3 wu50 wu40)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3];
12.62/5.03	17[label="[]",fontsize=16,color="green",shape="box"];19[label="wu3 wu50 wu40",fontsize=16,color="green",shape="box"];19 -> 24[label="",style="dashed", color="green", weight=3];
12.62/5.03	19 -> 25[label="",style="dashed", color="green", weight=3];
12.62/5.03	18[label="List.deleteBy0 wu41 wu40 wu3 wu50 wu6",fontsize=16,color="burlywood",shape="triangle"];34[label="wu6/False",fontsize=10,color="white",style="solid",shape="box"];18 -> 34[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	34 -> 22[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	35[label="wu6/True",fontsize=10,color="white",style="solid",shape="box"];18 -> 35[label="",style="solid", color="burlywood", weight=9];
12.62/5.03	35 -> 23[label="",style="solid", color="burlywood", weight=3];
12.62/5.03	24[label="wu50",fontsize=16,color="green",shape="box"];25[label="wu40",fontsize=16,color="green",shape="box"];22[label="List.deleteBy0 wu41 wu40 wu3 wu50 False",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
12.62/5.03	23[label="List.deleteBy0 wu41 wu40 wu3 wu50 True",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
12.62/5.03	26[label="wu40 : List.deleteBy wu3 wu50 wu41",fontsize=16,color="green",shape="box"];26 -> 28[label="",style="dashed", color="green", weight=3];
12.62/5.03	27[label="wu41",fontsize=16,color="green",shape="box"];28 -> 13[label="",style="dashed", color="red", weight=0];
12.62/5.03	28[label="List.deleteBy wu3 wu50 wu41",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
12.62/5.03	29[label="wu41",fontsize=16,color="green",shape="box"];}
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(8)
12.62/5.03	Complex Obligation (AND)
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(9)
12.62/5.03	Obligation:
12.62/5.03	Q DP problem:
12.62/5.03	The TRS P consists of the following rules:
12.62/5.03	
12.62/5.03	   new_foldl(wu3, :(wu50, wu51), ba) -> new_foldl(wu3, wu51, ba)
12.62/5.03	
12.62/5.03	R is empty.
12.62/5.03	Q is empty.
12.62/5.03	We have to consider all minimal (P,Q,R)-chains.
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(10) QDPSizeChangeProof (EQUIVALENT)
12.62/5.03	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. 
12.62/5.03	
12.62/5.03	From the DPs we obtained the following set of size-change graphs:
12.62/5.03	*new_foldl(wu3, :(wu50, wu51), ba) -> new_foldl(wu3, wu51, ba)
12.62/5.03	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.62/5.03	
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(11)
12.62/5.03	YES
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(12)
12.62/5.03	Obligation:
12.62/5.03	Q DP problem:
12.62/5.03	The TRS P consists of the following rules:
12.62/5.03	
12.62/5.03	   new_deleteBy(wu3, wu50, :(wu40, wu41), ba) -> new_deleteBy0(wu41, wu40, wu3, wu50, ba)
12.62/5.03	   new_deleteBy0(wu41, wu40, wu3, wu50, ba) -> new_deleteBy(wu3, wu50, wu41, ba)
12.62/5.03	
12.62/5.03	R is empty.
12.62/5.03	Q is empty.
12.62/5.03	We have to consider all minimal (P,Q,R)-chains.
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(13) QDPSizeChangeProof (EQUIVALENT)
12.62/5.03	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. 
12.62/5.03	
12.62/5.03	From the DPs we obtained the following set of size-change graphs:
12.62/5.03	*new_deleteBy0(wu41, wu40, wu3, wu50, ba) -> new_deleteBy(wu3, wu50, wu41, ba)
12.62/5.03	The graph contains the following edges 3 >= 1, 4 >= 2, 1 >= 3, 5 >= 4
12.62/5.03	
12.62/5.03	
12.62/5.03	*new_deleteBy(wu3, wu50, :(wu40, wu41), ba) -> new_deleteBy0(wu41, wu40, wu3, wu50, ba)
12.62/5.03	The graph contains the following edges 3 > 1, 3 > 2, 1 >= 3, 2 >= 4, 4 >= 5
12.62/5.03	
12.62/5.03	
12.62/5.03	----------------------------------------
12.62/5.03	
12.62/5.03	(14)
12.62/5.03	YES
12.66/5.07	EOF