7.59/3.46	YES
9.31/3.93	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.31/3.93	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.31/3.93	
9.31/3.93	
9.31/3.93	H-Termination with start terms of the given HASKELL could be proven:
9.31/3.93	
9.31/3.93	(0) HASKELL
9.31/3.93	(1) BR [EQUIVALENT, 0 ms]
9.31/3.93	(2) HASKELL
9.31/3.93	(3) COR [EQUIVALENT, 0 ms]
9.31/3.93	(4) HASKELL
9.31/3.93	(5) Narrow [SOUND, 0 ms]
9.31/3.93	(6) AND
9.31/3.93	    (7) QDP
9.31/3.93	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.31/3.93	        (9) YES
9.31/3.93	    (10) QDP
9.31/3.93	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.31/3.93	        (12) YES
9.31/3.93	
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(0)
9.31/3.93	Obligation:
9.31/3.93	mainModule Main 
9.31/3.93	module Main where {
9.31/3.93	import qualified Prelude;
9.31/3.93	}
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(1) BR (EQUIVALENT)
9.31/3.93	Replaced joker patterns by fresh variables and removed binding patterns.
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(2)
9.31/3.93	Obligation:
9.31/3.93	mainModule Main 
9.31/3.93	module Main where {
9.31/3.93	import qualified Prelude;
9.31/3.93	}
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(3) COR (EQUIVALENT)
9.31/3.93	Cond Reductions:
9.31/3.93	The following Function with conditions
9.31/3.93	"undefined |Falseundefined;
9.31/3.93	"
9.31/3.93	is transformed to
9.31/3.93	"undefined  = undefined1;
9.31/3.93	"
9.31/3.93	"undefined0 True = undefined;
9.31/3.93	"
9.31/3.93	"undefined1  = undefined0 False;
9.31/3.93	"
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(4)
9.31/3.93	Obligation:
9.31/3.93	mainModule Main 
9.31/3.93	module Main where {
9.31/3.93	import qualified Prelude;
9.31/3.93	}
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(5) Narrow (SOUND)
9.31/3.93	Haskell To QDPs
9.31/3.93	
9.31/3.93	digraph dp_graph {
9.31/3.93	node [outthreshold=100, inthreshold=100];1[label="concatMap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.31/3.93	3[label="concatMap vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.31/3.93	4[label="concatMap vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.31/3.93	5[label="concat . map vx3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.31/3.93	6[label="concat (map vx3 vx4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.31/3.93	7[label="foldr (++) [] (map vx3 vx4)",fontsize=16,color="burlywood",shape="triangle"];30[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];7 -> 30[label="",style="solid", color="burlywood", weight=9];
9.31/3.93	30 -> 8[label="",style="solid", color="burlywood", weight=3];
9.31/3.93	31[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 31[label="",style="solid", color="burlywood", weight=9];
9.31/3.93	31 -> 9[label="",style="solid", color="burlywood", weight=3];
9.31/3.93	8[label="foldr (++) [] (map vx3 (vx40 : vx41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.31/3.93	9[label="foldr (++) [] (map vx3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
9.31/3.93	10[label="foldr (++) [] (vx3 vx40 : map vx3 vx41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.31/3.93	11[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.31/3.93	12 -> 17[label="",style="dashed", color="red", weight=0];
9.31/3.93	12[label="(++) vx3 vx40 foldr (++) [] (map vx3 vx41)",fontsize=16,color="magenta"];12 -> 18[label="",style="dashed", color="magenta", weight=3];
9.31/3.93	12 -> 19[label="",style="dashed", color="magenta", weight=3];
9.31/3.93	13[label="[]",fontsize=16,color="green",shape="box"];18 -> 7[label="",style="dashed", color="red", weight=0];
9.31/3.93	18[label="foldr (++) [] (map vx3 vx41)",fontsize=16,color="magenta"];18 -> 21[label="",style="dashed", color="magenta", weight=3];
9.31/3.93	19[label="vx3 vx40",fontsize=16,color="green",shape="box"];19 -> 22[label="",style="dashed", color="green", weight=3];
9.31/3.93	17[label="(++) vx6 vx5",fontsize=16,color="burlywood",shape="triangle"];32[label="vx6/vx60 : vx61",fontsize=10,color="white",style="solid",shape="box"];17 -> 32[label="",style="solid", color="burlywood", weight=9];
9.31/3.93	32 -> 23[label="",style="solid", color="burlywood", weight=3];
9.31/3.93	33[label="vx6/[]",fontsize=10,color="white",style="solid",shape="box"];17 -> 33[label="",style="solid", color="burlywood", weight=9];
9.31/3.93	33 -> 24[label="",style="solid", color="burlywood", weight=3];
9.31/3.93	21[label="vx41",fontsize=16,color="green",shape="box"];22[label="vx40",fontsize=16,color="green",shape="box"];23[label="(++) (vx60 : vx61) vx5",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
9.31/3.93	24[label="(++) [] vx5",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
9.31/3.93	26[label="vx60 : vx61 ++ vx5",fontsize=16,color="green",shape="box"];26 -> 28[label="",style="dashed", color="green", weight=3];
9.31/3.93	27[label="vx5",fontsize=16,color="green",shape="box"];28 -> 17[label="",style="dashed", color="red", weight=0];
9.31/3.93	28[label="vx61 ++ vx5",fontsize=16,color="magenta"];28 -> 29[label="",style="dashed", color="magenta", weight=3];
9.31/3.93	29[label="vx61",fontsize=16,color="green",shape="box"];}
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(6)
9.31/3.93	Complex Obligation (AND)
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(7)
9.31/3.93	Obligation:
9.31/3.93	Q DP problem:
9.31/3.93	The TRS P consists of the following rules:
9.31/3.93	
9.31/3.93	   new_foldr(vx3, :(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)
9.31/3.93	
9.31/3.93	R is empty.
9.31/3.93	Q is empty.
9.31/3.93	We have to consider all minimal (P,Q,R)-chains.
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(8) QDPSizeChangeProof (EQUIVALENT)
9.31/3.93	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. 
9.31/3.93	
9.31/3.93	From the DPs we obtained the following set of size-change graphs:
9.31/3.93	*new_foldr(vx3, :(vx40, vx41), h, ba) -> new_foldr(vx3, vx41, h, ba)
9.31/3.93	The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4
9.31/3.93	
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(9)
9.31/3.93	YES
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(10)
9.31/3.93	Obligation:
9.31/3.93	Q DP problem:
9.31/3.93	The TRS P consists of the following rules:
9.31/3.93	
9.31/3.93	   new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.31/3.93	
9.31/3.93	R is empty.
9.31/3.93	Q is empty.
9.31/3.93	We have to consider all minimal (P,Q,R)-chains.
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(11) QDPSizeChangeProof (EQUIVALENT)
9.31/3.93	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. 
9.31/3.93	
9.31/3.93	From the DPs we obtained the following set of size-change graphs:
9.31/3.93	*new_psPs(:(vx60, vx61), vx5, h) -> new_psPs(vx61, vx5, h)
9.31/3.93	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
9.31/3.93	
9.31/3.93	
9.31/3.93	----------------------------------------
9.31/3.93	
9.31/3.93	(12)
9.31/3.93	YES
9.31/3.98	EOF