7.87/3.51	YES
9.48/4.03	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.48/4.03	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.48/4.03	
9.48/4.03	
9.48/4.03	H-Termination with start terms of the given HASKELL could be proven:
9.48/4.03	
9.48/4.03	(0) HASKELL
9.48/4.03	(1) BR [EQUIVALENT, 0 ms]
9.48/4.03	(2) HASKELL
9.48/4.03	(3) COR [EQUIVALENT, 0 ms]
9.48/4.03	(4) HASKELL
9.48/4.03	(5) NumRed [SOUND, 0 ms]
9.48/4.03	(6) HASKELL
9.48/4.03	(7) Narrow [SOUND, 0 ms]
9.48/4.03	(8) AND
9.48/4.03	    (9) QDP
9.48/4.03	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.48/4.03	        (11) YES
9.48/4.03	    (12) QDP
9.48/4.03	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.48/4.03	        (14) YES
9.48/4.03	
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(0)
9.48/4.03	Obligation:
9.48/4.03	mainModule Main 
9.48/4.03	module Main where {
9.48/4.03	import qualified Prelude;
9.48/4.03	}
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(1) BR (EQUIVALENT)
9.48/4.03	Replaced joker patterns by fresh variables and removed binding patterns.
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(2)
9.48/4.03	Obligation:
9.48/4.03	mainModule Main 
9.48/4.03	module Main where {
9.48/4.03	import qualified Prelude;
9.48/4.03	}
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(3) COR (EQUIVALENT)
9.48/4.03	Cond Reductions:
9.48/4.03	The following Function with conditions
9.48/4.03	"undefined |Falseundefined;
9.48/4.03	"
9.48/4.03	is transformed to
9.48/4.03	"undefined  = undefined1;
9.48/4.03	"
9.48/4.03	"undefined0 True = undefined;
9.48/4.03	"
9.48/4.03	"undefined1  = undefined0 False;
9.48/4.03	"
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(4)
9.48/4.03	Obligation:
9.48/4.03	mainModule Main 
9.48/4.03	module Main where {
9.48/4.03	import qualified Prelude;
9.48/4.03	}
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(5) NumRed (SOUND)
9.48/4.03	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(6)
9.48/4.03	Obligation:
9.48/4.03	mainModule Main 
9.48/4.03	module Main where {
9.48/4.03	import qualified Prelude;
9.48/4.03	}
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(7) Narrow (SOUND)
9.48/4.03	Haskell To QDPs
9.48/4.03	
9.48/4.03	digraph dp_graph {
9.48/4.03	node [outthreshold=100, inthreshold=100];1[label="unwords",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.48/4.03	3[label="unwords vx3",fontsize=16,color="burlywood",shape="triangle"];28[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];3 -> 28[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	28 -> 4[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	29[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 29[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	29 -> 5[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	4[label="unwords (vx30 : vx31)",fontsize=16,color="burlywood",shape="box"];30[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	30 -> 6[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	31[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	31 -> 7[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	5[label="unwords []",fontsize=16,color="black",shape="box"];5 -> 8[label="",style="solid", color="black", weight=3];
9.48/4.03	6[label="unwords (vx30 : vx310 : vx311)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
9.48/4.03	7[label="unwords (vx30 : [])",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9.48/4.03	8[label="[]",fontsize=16,color="green",shape="box"];9 -> 16[label="",style="dashed", color="red", weight=0];
9.48/4.03	9[label="vx30 ++ Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))) : unwords (vx310 : vx311)",fontsize=16,color="magenta"];9 -> 17[label="",style="dashed", color="magenta", weight=3];
9.48/4.03	9 -> 18[label="",style="dashed", color="magenta", weight=3];
9.48/4.03	9 -> 19[label="",style="dashed", color="magenta", weight=3];
9.48/4.03	10[label="vx30",fontsize=16,color="green",shape="box"];17[label="vx30",fontsize=16,color="green",shape="box"];18 -> 3[label="",style="dashed", color="red", weight=0];
9.48/4.03	18[label="unwords (vx310 : vx311)",fontsize=16,color="magenta"];18 -> 21[label="",style="dashed", color="magenta", weight=3];
9.48/4.03	19[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];16[label="vx5 ++ Char (Succ vx6) : vx9",fontsize=16,color="burlywood",shape="triangle"];32[label="vx5/vx50 : vx51",fontsize=10,color="white",style="solid",shape="box"];16 -> 32[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	32 -> 22[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	33[label="vx5/[]",fontsize=10,color="white",style="solid",shape="box"];16 -> 33[label="",style="solid", color="burlywood", weight=9];
9.48/4.03	33 -> 23[label="",style="solid", color="burlywood", weight=3];
9.48/4.03	21[label="vx310 : vx311",fontsize=16,color="green",shape="box"];22[label="(vx50 : vx51) ++ Char (Succ vx6) : vx9",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
9.48/4.03	23[label="[] ++ Char (Succ vx6) : vx9",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
9.48/4.03	24[label="vx50 : vx51 ++ Char (Succ vx6) : vx9",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
9.48/4.03	25[label="Char (Succ vx6) : vx9",fontsize=16,color="green",shape="box"];26 -> 16[label="",style="dashed", color="red", weight=0];
9.48/4.03	26[label="vx51 ++ Char (Succ vx6) : vx9",fontsize=16,color="magenta"];26 -> 27[label="",style="dashed", color="magenta", weight=3];
9.48/4.03	27[label="vx51",fontsize=16,color="green",shape="box"];}
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(8)
9.48/4.03	Complex Obligation (AND)
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(9)
9.48/4.03	Obligation:
9.48/4.03	Q DP problem:
9.48/4.03	The TRS P consists of the following rules:
9.48/4.03	
9.48/4.03	   new_unwords(:(vx30, :(vx310, vx311))) -> new_unwords(:(vx310, vx311))
9.48/4.03	
9.48/4.03	R is empty.
9.48/4.03	Q is empty.
9.48/4.03	We have to consider all minimal (P,Q,R)-chains.
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(10) QDPSizeChangeProof (EQUIVALENT)
9.48/4.03	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. 
9.48/4.03	
9.48/4.03	From the DPs we obtained the following set of size-change graphs:
9.48/4.03	*new_unwords(:(vx30, :(vx310, vx311))) -> new_unwords(:(vx310, vx311))
9.48/4.03	The graph contains the following edges 1 > 1
9.48/4.03	
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(11)
9.48/4.03	YES
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(12)
9.48/4.03	Obligation:
9.48/4.03	Q DP problem:
9.48/4.03	The TRS P consists of the following rules:
9.48/4.03	
9.48/4.03	   new_psPs(:(vx50, vx51), vx6, vx9) -> new_psPs(vx51, vx6, vx9)
9.48/4.03	
9.48/4.03	R is empty.
9.48/4.03	Q is empty.
9.48/4.03	We have to consider all minimal (P,Q,R)-chains.
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(13) QDPSizeChangeProof (EQUIVALENT)
9.48/4.03	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. 
9.48/4.03	
9.48/4.03	From the DPs we obtained the following set of size-change graphs:
9.48/4.03	*new_psPs(:(vx50, vx51), vx6, vx9) -> new_psPs(vx51, vx6, vx9)
9.48/4.03	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.48/4.03	
9.48/4.03	
9.48/4.03	----------------------------------------
9.48/4.03	
9.48/4.03	(14)
9.48/4.03	YES
9.85/4.07	EOF