7.99/3.63	YES
9.48/4.07	proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs
9.48/4.07	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.48/4.07	
9.48/4.07	
9.48/4.07	H-Termination with start terms of the given HASKELL could be proven:
9.48/4.07	
9.48/4.07	(0) HASKELL
9.48/4.07	(1) BR [EQUIVALENT, 0 ms]
9.48/4.07	(2) HASKELL
9.48/4.07	(3) COR [EQUIVALENT, 0 ms]
9.48/4.07	(4) HASKELL
9.48/4.07	(5) Narrow [SOUND, 0 ms]
9.48/4.07	(6) AND
9.48/4.07	    (7) QDP
9.48/4.07	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.48/4.07	        (9) YES
9.48/4.07	    (10) QDP
9.48/4.07	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.48/4.07	        (12) YES
9.48/4.07	
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(0)
9.48/4.07	Obligation:
9.48/4.07	mainModule Main 
9.48/4.07	module Main where {
9.48/4.07	import qualified Prelude;
9.48/4.07	}
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(1) BR (EQUIVALENT)
9.48/4.07	Replaced joker patterns by fresh variables and removed binding patterns.
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(2)
9.48/4.07	Obligation:
9.48/4.07	mainModule Main 
9.48/4.07	module Main where {
9.48/4.07	import qualified Prelude;
9.48/4.07	}
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(3) COR (EQUIVALENT)
9.48/4.07	Cond Reductions:
9.48/4.07	The following Function with conditions
9.48/4.07	"undefined |Falseundefined;
9.48/4.07	"
9.48/4.07	is transformed to
9.48/4.07	"undefined  = undefined1;
9.48/4.07	"
9.48/4.07	"undefined0 True = undefined;
9.48/4.07	"
9.48/4.07	"undefined1  = undefined0 False;
9.48/4.07	"
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(4)
9.48/4.07	Obligation:
9.48/4.07	mainModule Main 
9.48/4.07	module Main where {
9.48/4.07	import qualified Prelude;
9.48/4.07	}
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(5) Narrow (SOUND)
9.48/4.07	Haskell To QDPs
9.48/4.07	
9.48/4.07	digraph dp_graph {
9.48/4.07	node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.48/4.07	3[label="notElem vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.48/4.07	4[label="notElem vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.48/4.07	5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.48/4.07	6[label="all ((/=) vx3) vx4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
9.48/4.07	7[label="and . map ((/=) vx3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
9.48/4.07	8[label="and (map ((/=) vx3) vx4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9.48/4.07	9[label="foldr (&&) True (map ((/=) vx3) vx4)",fontsize=16,color="burlywood",shape="triangle"];40[label="vx4/vx40 : vx41",fontsize=10,color="white",style="solid",shape="box"];9 -> 40[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	40 -> 10[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	41[label="vx4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 41[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	41 -> 11[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	10[label="foldr (&&) True (map ((/=) vx3) (vx40 : vx41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
9.48/4.07	11[label="foldr (&&) True (map ((/=) vx3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
9.48/4.07	12[label="foldr (&&) True (((/=) vx3 vx40) : map ((/=) vx3) vx41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
9.48/4.07	13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
9.48/4.07	14 -> 16[label="",style="dashed", color="red", weight=0];
9.48/4.07	14[label="(&&) (/=) vx3 vx40 foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
9.48/4.07	15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
9.48/4.07	17[label="foldr (&&) True (map ((/=) vx3) vx41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
9.48/4.07	16[label="(&&) (/=) vx3 vx40 vx5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
9.48/4.07	18[label="vx41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vx3 == vx40) vx5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
9.48/4.07	20[label="(&&) not (primEqChar vx3 vx40) vx5",fontsize=16,color="burlywood",shape="box"];42[label="vx3/Char vx30",fontsize=10,color="white",style="solid",shape="box"];20 -> 42[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	42 -> 21[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	21[label="(&&) not (primEqChar (Char vx30) vx40) vx5",fontsize=16,color="burlywood",shape="box"];43[label="vx40/Char vx400",fontsize=10,color="white",style="solid",shape="box"];21 -> 43[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	43 -> 22[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	22[label="(&&) not (primEqChar (Char vx30) (Char vx400)) vx5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3];
9.48/4.07	23[label="(&&) not (primEqNat vx30 vx400) vx5",fontsize=16,color="burlywood",shape="triangle"];44[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];23 -> 44[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	44 -> 24[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	45[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 45[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	45 -> 25[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	24[label="(&&) not (primEqNat (Succ vx300) vx400) vx5",fontsize=16,color="burlywood",shape="box"];46[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];24 -> 46[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	46 -> 26[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	47[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 47[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	47 -> 27[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	25[label="(&&) not (primEqNat Zero vx400) vx5",fontsize=16,color="burlywood",shape="box"];48[label="vx400/Succ vx4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 48[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	48 -> 28[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	49[label="vx400/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 49[label="",style="solid", color="burlywood", weight=9];
9.48/4.07	49 -> 29[label="",style="solid", color="burlywood", weight=3];
9.48/4.07	26[label="(&&) not (primEqNat (Succ vx300) (Succ vx4000)) vx5",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
9.48/4.07	27[label="(&&) not (primEqNat (Succ vx300) Zero) vx5",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
9.48/4.07	28[label="(&&) not (primEqNat Zero (Succ vx4000)) vx5",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
9.48/4.07	29[label="(&&) not (primEqNat Zero Zero) vx5",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
9.48/4.07	30 -> 23[label="",style="dashed", color="red", weight=0];
9.48/4.07	30[label="(&&) not (primEqNat vx300 vx4000) vx5",fontsize=16,color="magenta"];30 -> 34[label="",style="dashed", color="magenta", weight=3];
9.48/4.07	30 -> 35[label="",style="dashed", color="magenta", weight=3];
9.48/4.07	31[label="(&&) not False vx5",fontsize=16,color="black",shape="triangle"];31 -> 36[label="",style="solid", color="black", weight=3];
9.48/4.07	32 -> 31[label="",style="dashed", color="red", weight=0];
9.48/4.07	32[label="(&&) not False vx5",fontsize=16,color="magenta"];33[label="(&&) not True vx5",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
9.48/4.07	34[label="vx4000",fontsize=16,color="green",shape="box"];35[label="vx300",fontsize=16,color="green",shape="box"];36[label="(&&) True vx5",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3];
9.48/4.07	37[label="(&&) False vx5",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
9.48/4.07	38[label="vx5",fontsize=16,color="green",shape="box"];39[label="False",fontsize=16,color="green",shape="box"];}
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(6)
9.48/4.07	Complex Obligation (AND)
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(7)
9.48/4.07	Obligation:
9.48/4.07	Q DP problem:
9.48/4.07	The TRS P consists of the following rules:
9.48/4.07	
9.48/4.07	   new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.48/4.07	
9.48/4.07	R is empty.
9.48/4.07	Q is empty.
9.48/4.07	We have to consider all minimal (P,Q,R)-chains.
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(8) QDPSizeChangeProof (EQUIVALENT)
9.48/4.07	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. 
9.48/4.07	
9.48/4.07	From the DPs we obtained the following set of size-change graphs:
9.48/4.07	*new_foldr(vx3, :(vx40, vx41)) -> new_foldr(vx3, vx41)
9.48/4.07	The graph contains the following edges 1 >= 1, 2 > 2
9.48/4.07	
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(9)
9.48/4.07	YES
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(10)
9.48/4.07	Obligation:
9.48/4.07	Q DP problem:
9.48/4.07	The TRS P consists of the following rules:
9.48/4.07	
9.48/4.07	   new_asAs(Succ(vx300), Succ(vx4000), vx5) -> new_asAs(vx300, vx4000, vx5)
9.48/4.07	
9.48/4.07	R is empty.
9.48/4.07	Q is empty.
9.48/4.07	We have to consider all minimal (P,Q,R)-chains.
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(11) QDPSizeChangeProof (EQUIVALENT)
9.48/4.07	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. 
9.48/4.07	
9.48/4.07	From the DPs we obtained the following set of size-change graphs:
9.48/4.07	*new_asAs(Succ(vx300), Succ(vx4000), vx5) -> new_asAs(vx300, vx4000, vx5)
9.48/4.07	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
9.48/4.07	
9.48/4.07	
9.48/4.07	----------------------------------------
9.48/4.07	
9.48/4.07	(12)
9.48/4.07	YES
9.70/4.11	EOF