10.42/4.43	YES
12.05/4.93	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.05/4.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.05/4.93	
12.05/4.93	
12.05/4.93	H-Termination with start terms of the given HASKELL could be proven:
12.05/4.93	
12.05/4.93	(0) HASKELL
12.05/4.93	(1) BR [EQUIVALENT, 0 ms]
12.05/4.93	(2) HASKELL
12.05/4.93	(3) COR [EQUIVALENT, 0 ms]
12.05/4.93	(4) HASKELL
12.05/4.93	(5) Narrow [SOUND, 0 ms]
12.05/4.93	(6) AND
12.05/4.93	    (7) QDP
12.05/4.93	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.05/4.93	        (9) YES
12.05/4.93	    (10) QDP
12.05/4.93	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.05/4.93	        (12) YES
12.05/4.93	
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(0)
12.05/4.93	Obligation:
12.05/4.93	mainModule Main 
12.05/4.93	module Maybe where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	module List where {
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	inits :: [a]  ->  [[a]];
12.05/4.93	inits [] = [] : [];
12.05/4.93	inits (x : xs) = ([] : []) ++ map (: x) (inits xs);
12.05/4.93	
12.05/4.93	}
12.05/4.93	module Main where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(1) BR (EQUIVALENT)
12.05/4.93	Replaced joker patterns by fresh variables and removed binding patterns.
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(2)
12.05/4.93	Obligation:
12.05/4.93	mainModule Main 
12.05/4.93	module Maybe where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	module List where {
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	inits :: [a]  ->  [[a]];
12.05/4.93	inits [] = [] : [];
12.05/4.93	inits (x : xs) = ([] : []) ++ map (: x) (inits xs);
12.05/4.93	
12.05/4.93	}
12.05/4.93	module Main where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(3) COR (EQUIVALENT)
12.05/4.93	Cond Reductions:
12.05/4.93	The following Function with conditions
12.05/4.93	"undefined |Falseundefined;
12.05/4.93	"
12.05/4.93	is transformed to
12.05/4.93	"undefined  = undefined1;
12.05/4.93	"
12.05/4.93	"undefined0 True = undefined;
12.05/4.93	"
12.05/4.93	"undefined1  = undefined0 False;
12.05/4.93	"
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(4)
12.05/4.93	Obligation:
12.05/4.93	mainModule Main 
12.05/4.93	module Maybe where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	module List where {
12.05/4.93	import qualified Main;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	inits :: [a]  ->  [[a]];
12.05/4.93	inits [] = [] : [];
12.05/4.93	inits (x : xs) = ([] : []) ++ map (: x) (inits xs);
12.05/4.93	
12.05/4.93	}
12.05/4.93	module Main where {
12.05/4.93	import qualified List;
12.05/4.93	import qualified Maybe;
12.05/4.93	import qualified Prelude;
12.05/4.93	}
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(5) Narrow (SOUND)
12.05/4.93	Haskell To QDPs
12.05/4.93	
12.05/4.93	digraph dp_graph {
12.05/4.93	node [outthreshold=100, inthreshold=100];1[label="List.inits",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.05/4.93	3[label="List.inits vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];3 -> 20[label="",style="solid", color="burlywood", weight=9];
12.05/4.93	20 -> 4[label="",style="solid", color="burlywood", weight=3];
12.05/4.93	21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 21[label="",style="solid", color="burlywood", weight=9];
12.05/4.93	21 -> 5[label="",style="solid", color="burlywood", weight=3];
12.05/4.93	4[label="List.inits (vy30 : vy31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
12.05/4.93	5[label="List.inits []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.05/4.93	6 -> 8[label="",style="dashed", color="red", weight=0];
12.05/4.93	6[label="([] : []) ++ map (vy30 :) (List.inits vy31)",fontsize=16,color="magenta"];6 -> 9[label="",style="dashed", color="magenta", weight=3];
12.05/4.93	7[label="[] : []",fontsize=16,color="green",shape="box"];9 -> 3[label="",style="dashed", color="red", weight=0];
12.05/4.93	9[label="List.inits vy31",fontsize=16,color="magenta"];9 -> 10[label="",style="dashed", color="magenta", weight=3];
12.05/4.93	8[label="([] : []) ++ map (vy30 :) vy4",fontsize=16,color="black",shape="triangle"];8 -> 11[label="",style="solid", color="black", weight=3];
12.05/4.93	10[label="vy31",fontsize=16,color="green",shape="box"];11[label="[] : [] ++ map (vy30 :) vy4",fontsize=16,color="green",shape="box"];11 -> 12[label="",style="dashed", color="green", weight=3];
12.05/4.93	12[label="[] ++ map (vy30 :) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
12.05/4.93	13[label="map (vy30 :) vy4",fontsize=16,color="burlywood",shape="triangle"];22[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];13 -> 22[label="",style="solid", color="burlywood", weight=9];
12.05/4.93	22 -> 14[label="",style="solid", color="burlywood", weight=3];
12.05/4.93	23[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 23[label="",style="solid", color="burlywood", weight=9];
12.05/4.93	23 -> 15[label="",style="solid", color="burlywood", weight=3];
12.05/4.93	14[label="map (vy30 :) (vy40 : vy41)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
12.05/4.93	15[label="map (vy30 :) []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
12.05/4.93	16[label="(vy30 : vy40) : map (vy30 :) vy41",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
12.05/4.93	17[label="[]",fontsize=16,color="green",shape="box"];18 -> 13[label="",style="dashed", color="red", weight=0];
12.05/4.93	18[label="map (vy30 :) vy41",fontsize=16,color="magenta"];18 -> 19[label="",style="dashed", color="magenta", weight=3];
12.05/4.93	19[label="vy41",fontsize=16,color="green",shape="box"];}
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(6)
12.05/4.93	Complex Obligation (AND)
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(7)
12.05/4.93	Obligation:
12.05/4.93	Q DP problem:
12.05/4.93	The TRS P consists of the following rules:
12.05/4.93	
12.05/4.93	   new_inits(:(vy30, vy31), ba) -> new_inits(vy31, ba)
12.05/4.93	
12.05/4.93	R is empty.
12.05/4.93	Q is empty.
12.05/4.93	We have to consider all minimal (P,Q,R)-chains.
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(8) QDPSizeChangeProof (EQUIVALENT)
12.05/4.93	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. 
12.05/4.93	
12.05/4.93	From the DPs we obtained the following set of size-change graphs:
12.05/4.93	*new_inits(:(vy30, vy31), ba) -> new_inits(vy31, ba)
12.05/4.93	The graph contains the following edges 1 > 1, 2 >= 2
12.05/4.93	
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(9)
12.05/4.93	YES
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(10)
12.05/4.93	Obligation:
12.05/4.93	Q DP problem:
12.05/4.93	The TRS P consists of the following rules:
12.05/4.93	
12.05/4.93	   new_map(vy30, :(vy40, vy41), ba) -> new_map(vy30, vy41, ba)
12.05/4.93	
12.05/4.93	R is empty.
12.05/4.93	Q is empty.
12.05/4.93	We have to consider all minimal (P,Q,R)-chains.
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(11) QDPSizeChangeProof (EQUIVALENT)
12.05/4.93	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. 
12.05/4.93	
12.05/4.93	From the DPs we obtained the following set of size-change graphs:
12.05/4.93	*new_map(vy30, :(vy40, vy41), ba) -> new_map(vy30, vy41, ba)
12.05/4.93	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3
12.05/4.93	
12.05/4.93	
12.05/4.93	----------------------------------------
12.05/4.93	
12.05/4.93	(12)
12.05/4.93	YES
12.28/5.00	EOF