10.59/4.34	YES
12.33/4.84	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
12.33/4.84	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
12.33/4.84	
12.33/4.84	
12.33/4.84	H-Termination with start terms of the given HASKELL could be proven:
12.33/4.84	
12.33/4.84	(0) HASKELL
12.33/4.84	(1) BR [EQUIVALENT, 0 ms]
12.33/4.84	(2) HASKELL
12.33/4.84	(3) COR [EQUIVALENT, 0 ms]
12.33/4.84	(4) HASKELL
12.33/4.84	(5) Narrow [SOUND, 0 ms]
12.33/4.84	(6) QDP
12.33/4.84	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
12.33/4.84	(8) YES
12.33/4.84	
12.33/4.84	
12.33/4.84	----------------------------------------
12.33/4.84	
12.33/4.84	(0)
12.33/4.84	Obligation:
12.33/4.84	mainModule Main 
12.33/4.84	module Maybe where {
12.33/4.84	import qualified List;
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Prelude;
12.33/4.84	}
12.33/4.84	module List where {
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Maybe;
12.33/4.84	import qualified Prelude;
12.33/4.84	tails :: [a]  ->  [[a]];
12.33/4.84	tails [] = [] : [];
12.33/4.84	tails xxs@(_ : xs) = xxs : tails xs;
12.33/4.84	
12.33/4.84	}
12.33/4.84	module Main where {
12.33/4.84	import qualified List;
12.33/4.84	import qualified Maybe;
12.33/4.84	import qualified Prelude;
12.33/4.84	}
12.33/4.84	
12.33/4.84	----------------------------------------
12.33/4.84	
12.33/4.84	(1) BR (EQUIVALENT)
12.33/4.84	Replaced joker patterns by fresh variables and removed binding patterns.
12.33/4.84	
12.33/4.84	Binding Reductions:
12.33/4.84	The bind variable of the following binding Pattern
12.33/4.84	"xxs@(vz : wu)"
12.33/4.84	is replaced by the following term
12.33/4.84	"vz : wu"
12.33/4.84	
12.33/4.84	----------------------------------------
12.33/4.84	
12.33/4.84	(2)
12.33/4.84	Obligation:
12.33/4.84	mainModule Main 
12.33/4.84	module Maybe where {
12.33/4.84	import qualified List;
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Prelude;
12.33/4.84	}
12.33/4.84	module List where {
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Maybe;
12.33/4.84	import qualified Prelude;
12.33/4.84	tails :: [a]  ->  [[a]];
12.33/4.84	tails [] = [] : [];
12.33/4.84	tails (vz : wu) = (vz : wu) : tails wu;
12.33/4.84	
12.33/4.84	}
12.33/4.84	module Main where {
12.33/4.84	import qualified List;
12.33/4.84	import qualified Maybe;
12.33/4.84	import qualified Prelude;
12.33/4.84	}
12.33/4.84	
12.33/4.84	----------------------------------------
12.33/4.84	
12.33/4.84	(3) COR (EQUIVALENT)
12.33/4.84	Cond Reductions:
12.33/4.84	The following Function with conditions
12.33/4.84	"undefined |Falseundefined;
12.33/4.84	"
12.33/4.84	is transformed to
12.33/4.84	"undefined  = undefined1;
12.33/4.84	"
12.33/4.84	"undefined0 True = undefined;
12.33/4.84	"
12.33/4.84	"undefined1  = undefined0 False;
12.33/4.84	"
12.33/4.84	
12.33/4.84	----------------------------------------
12.33/4.84	
12.33/4.84	(4)
12.33/4.84	Obligation:
12.33/4.84	mainModule Main 
12.33/4.84	module Maybe where {
12.33/4.84	import qualified List;
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Prelude;
12.33/4.84	}
12.33/4.84	module List where {
12.33/4.84	import qualified Main;
12.33/4.84	import qualified Maybe;
12.33/4.85	import qualified Prelude;
12.33/4.85	tails :: [a]  ->  [[a]];
12.33/4.85	tails [] = [] : [];
12.33/4.85	tails (vz : wu) = (vz : wu) : tails wu;
12.33/4.85	
12.33/4.85	}
12.33/4.85	module Main where {
12.33/4.85	import qualified List;
12.33/4.85	import qualified Maybe;
12.33/4.85	import qualified Prelude;
12.33/4.85	}
12.33/4.85	
12.33/4.85	----------------------------------------
12.33/4.85	
12.33/4.85	(5) Narrow (SOUND)
12.33/4.85	Haskell To QDPs
12.33/4.85	
12.33/4.85	digraph dp_graph {
12.33/4.85	node [outthreshold=100, inthreshold=100];1[label="List.tails",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
12.33/4.85	3[label="List.tails wv3",fontsize=16,color="burlywood",shape="triangle"];10[label="wv3/wv30 : wv31",fontsize=10,color="white",style="solid",shape="box"];3 -> 10[label="",style="solid", color="burlywood", weight=9];
12.33/4.85	10 -> 4[label="",style="solid", color="burlywood", weight=3];
12.33/4.85	11[label="wv3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 11[label="",style="solid", color="burlywood", weight=9];
12.33/4.85	11 -> 5[label="",style="solid", color="burlywood", weight=3];
12.33/4.85	4[label="List.tails (wv30 : wv31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
12.33/4.85	5[label="List.tails []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
12.33/4.85	6[label="(wv30 : wv31) : List.tails wv31",fontsize=16,color="green",shape="box"];6 -> 8[label="",style="dashed", color="green", weight=3];
12.33/4.85	7[label="[] : []",fontsize=16,color="green",shape="box"];8 -> 3[label="",style="dashed", color="red", weight=0];
12.33/4.85	8[label="List.tails wv31",fontsize=16,color="magenta"];8 -> 9[label="",style="dashed", color="magenta", weight=3];
12.33/4.85	9[label="wv31",fontsize=16,color="green",shape="box"];}
12.33/4.85	
12.33/4.85	----------------------------------------
12.33/4.85	
12.33/4.85	(6)
12.33/4.85	Obligation:
12.33/4.85	Q DP problem:
12.33/4.85	The TRS P consists of the following rules:
12.33/4.85	
12.33/4.85	   new_tails(:(wv30, wv31), ba) -> new_tails(wv31, ba)
12.33/4.85	
12.33/4.85	R is empty.
12.33/4.85	Q is empty.
12.33/4.85	We have to consider all minimal (P,Q,R)-chains.
12.33/4.85	----------------------------------------
12.33/4.85	
12.33/4.85	(7) QDPSizeChangeProof (EQUIVALENT)
12.33/4.85	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. 
12.33/4.85	
12.33/4.85	From the DPs we obtained the following set of size-change graphs:
12.33/4.85	*new_tails(:(wv30, wv31), ba) -> new_tails(wv31, ba)
12.33/4.85	The graph contains the following edges 1 > 1, 2 >= 2
12.33/4.85	
12.33/4.85	
12.33/4.85	----------------------------------------
12.33/4.85	
12.33/4.85	(8)
12.33/4.85	YES
12.51/4.93	EOF