8.05/3.61	YES
10.13/4.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.13/4.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.13/4.14	
10.13/4.14	
10.13/4.14	H-Termination with start terms of the given HASKELL could be proven:
10.13/4.14	
10.13/4.14	(0) HASKELL
10.13/4.14	(1) BR [EQUIVALENT, 0 ms]
10.13/4.14	(2) HASKELL
10.13/4.14	(3) COR [EQUIVALENT, 0 ms]
10.13/4.14	(4) HASKELL
10.13/4.14	(5) NumRed [SOUND, 0 ms]
10.13/4.14	(6) HASKELL
10.13/4.14	(7) Narrow [SOUND, 0 ms]
10.13/4.14	(8) AND
10.13/4.14	    (9) QDP
10.13/4.14	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.13/4.14	        (11) YES
10.13/4.14	    (12) QDP
10.13/4.14	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.13/4.14	        (14) YES
10.13/4.14	    (15) QDP
10.13/4.14	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.13/4.14	        (17) YES
10.13/4.14	
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(0)
10.13/4.14	Obligation:
10.13/4.14	mainModule Main 
10.13/4.14	module Main where {
10.13/4.14	import qualified Prelude;
10.13/4.14	}
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(1) BR (EQUIVALENT)
10.13/4.14	Replaced joker patterns by fresh variables and removed binding patterns.
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(2)
10.13/4.14	Obligation:
10.13/4.14	mainModule Main 
10.13/4.14	module Main where {
10.13/4.14	import qualified Prelude;
10.13/4.14	}
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(3) COR (EQUIVALENT)
10.13/4.14	Cond Reductions:
10.13/4.14	The following Function with conditions
10.13/4.14	"undefined |Falseundefined;
10.13/4.14	"
10.13/4.14	is transformed to
10.13/4.14	"undefined  = undefined1;
10.13/4.14	"
10.13/4.14	"undefined0 True = undefined;
10.13/4.14	"
10.13/4.14	"undefined1  = undefined0 False;
10.13/4.14	"
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(4)
10.13/4.14	Obligation:
10.13/4.14	mainModule Main 
10.13/4.14	module Main where {
10.13/4.14	import qualified Prelude;
10.13/4.14	}
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(5) NumRed (SOUND)
10.13/4.14	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(6)
10.13/4.14	Obligation:
10.13/4.14	mainModule Main 
10.13/4.14	module Main where {
10.13/4.14	import qualified Prelude;
10.13/4.14	}
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(7) Narrow (SOUND)
10.13/4.14	Haskell To QDPs
10.13/4.14	
10.13/4.14	digraph dp_graph {
10.13/4.14	node [outthreshold=100, inthreshold=100];1[label="product",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.13/4.14	3[label="product vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.13/4.14	4[label="foldl' (*) (fromInt (Pos (Succ Zero))) vx3",fontsize=16,color="burlywood",shape="box"];94[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 94[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	94 -> 5[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	95[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 95[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	95 -> 6[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	5[label="foldl' (*) (fromInt (Pos (Succ Zero))) (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.13/4.14	6[label="foldl' (*) (fromInt (Pos (Succ Zero))) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.13/4.14	7[label="(foldl' (*) $! (*) fromInt (Pos (Succ Zero)) vx30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.13/4.14	8[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
10.13/4.14	9 -> 11[label="",style="dashed", color="red", weight=0];
10.13/4.14	9[label="((*) fromInt (Pos (Succ Zero)) vx30 `seq` foldl' (*) ((*) fromInt (Pos (Succ Zero)) vx30))",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	9 -> 13[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	10[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];12 -> 8[label="",style="dashed", color="red", weight=0];
10.13/4.14	12[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
10.13/4.14	13[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];11[label="((*) vx4 vx30 `seq` foldl' (*) ((*) vx5 vx30))",fontsize=16,color="black",shape="triangle"];11 -> 14[label="",style="solid", color="black", weight=3];
10.13/4.14	14[label="enforceWHNF (WHNF ((*) vx4 vx30)) (foldl' (*) ((*) vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
10.13/4.14	15[label="enforceWHNF (WHNF (primMulInt vx4 vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="triangle"];96[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 96[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	96 -> 16[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	97[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 97[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	97 -> 17[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	16[label="enforceWHNF (WHNF (primMulInt (Pos vx40) vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];98[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 98[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	98 -> 18[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	99[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 99[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	99 -> 19[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	17[label="enforceWHNF (WHNF (primMulInt (Neg vx40) vx30)) (foldl' primMulInt (primMulInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];100[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 100[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	100 -> 20[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	101[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 101[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	101 -> 21[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	18[label="enforceWHNF (WHNF (primMulInt (Pos vx40) (Pos vx300))) (foldl' primMulInt (primMulInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.13/4.14	19[label="enforceWHNF (WHNF (primMulInt (Pos vx40) (Neg vx300))) (foldl' primMulInt (primMulInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.13/4.14	20[label="enforceWHNF (WHNF (primMulInt (Neg vx40) (Pos vx300))) (foldl' primMulInt (primMulInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.13/4.14	21[label="enforceWHNF (WHNF (primMulInt (Neg vx40) (Neg vx300))) (foldl' primMulInt (primMulInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.13/4.14	22[label="enforceWHNF (WHNF (Pos (primMulNat vx40 vx300))) (foldl' primMulInt (Pos (primMulNat vx40 vx300))) vx31",fontsize=16,color="black",shape="triangle"];22 -> 26[label="",style="solid", color="black", weight=3];
10.13/4.14	23[label="enforceWHNF (WHNF (Neg (primMulNat vx40 vx300))) (foldl' primMulInt (Neg (primMulNat vx40 vx300))) vx31",fontsize=16,color="black",shape="triangle"];23 -> 27[label="",style="solid", color="black", weight=3];
10.13/4.14	24 -> 23[label="",style="dashed", color="red", weight=0];
10.13/4.14	24[label="enforceWHNF (WHNF (Neg (primMulNat vx40 vx300))) (foldl' primMulInt (Neg (primMulNat vx40 vx300))) vx31",fontsize=16,color="magenta"];24 -> 28[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	24 -> 29[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	25 -> 22[label="",style="dashed", color="red", weight=0];
10.13/4.14	25[label="enforceWHNF (WHNF (Pos (primMulNat vx40 vx300))) (foldl' primMulInt (Pos (primMulNat vx40 vx300))) vx31",fontsize=16,color="magenta"];25 -> 30[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	25 -> 31[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	26[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];102[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];26 -> 102[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	102 -> 32[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	103[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 103[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	103 -> 33[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	27[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];104[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];27 -> 104[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	104 -> 34[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	105[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];27 -> 105[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	105 -> 35[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	28[label="vx40",fontsize=16,color="green",shape="box"];29[label="vx300",fontsize=16,color="green",shape="box"];30[label="vx300",fontsize=16,color="green",shape="box"];31[label="vx40",fontsize=16,color="green",shape="box"];32[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
10.13/4.14	33[label="foldl' primMulInt (Pos (primMulNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];33 -> 37[label="",style="solid", color="black", weight=3];
10.13/4.14	34[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];34 -> 38[label="",style="solid", color="black", weight=3];
10.13/4.14	35[label="foldl' primMulInt (Neg (primMulNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];35 -> 39[label="",style="solid", color="black", weight=3];
10.13/4.14	36[label="(foldl' primMulInt $! primMulInt (Pos (primMulNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3];
10.13/4.14	37[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];37 -> 41[label="",style="dashed", color="green", weight=3];
10.13/4.14	38[label="(foldl' primMulInt $! primMulInt (Neg (primMulNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3];
10.13/4.14	39[label="Neg (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];39 -> 43[label="",style="dashed", color="green", weight=3];
10.13/4.14	40[label="(primMulInt (Pos (primMulNat vx40 vx300)) vx310 `seq` foldl' primMulInt (primMulInt (Pos (primMulNat vx40 vx300)) vx310))",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3];
10.13/4.14	41[label="primMulNat vx40 vx300",fontsize=16,color="burlywood",shape="triangle"];106[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];41 -> 106[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	106 -> 45[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	107[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 107[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	107 -> 46[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	42 -> 47[label="",style="dashed", color="red", weight=0];
10.13/4.14	42[label="(primMulInt (Neg (primMulNat vx40 vx300)) vx310 `seq` foldl' primMulInt (primMulInt (Neg (primMulNat vx40 vx300)) vx310))",fontsize=16,color="magenta"];42 -> 48[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	42 -> 49[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	43 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	43[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];43 -> 50[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	44 -> 15[label="",style="dashed", color="red", weight=0];
10.13/4.14	44[label="enforceWHNF (WHNF (primMulInt (Pos (primMulNat vx40 vx300)) vx310)) (foldl' primMulInt (primMulInt (Pos (primMulNat vx40 vx300)) vx310)) vx311",fontsize=16,color="magenta"];44 -> 51[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	44 -> 52[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	44 -> 53[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	44 -> 54[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	45[label="primMulNat (Succ vx400) vx300",fontsize=16,color="burlywood",shape="box"];108[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];45 -> 108[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	108 -> 55[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	109[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 109[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	109 -> 56[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	46[label="primMulNat Zero vx300",fontsize=16,color="burlywood",shape="box"];110[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];46 -> 110[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	110 -> 57[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	111[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];46 -> 111[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	111 -> 58[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	48 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	48[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];48 -> 59[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	49 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	49[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];49 -> 60[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	47[label="(primMulInt (Neg vx6) vx310 `seq` foldl' primMulInt (primMulInt (Neg vx7) vx310))",fontsize=16,color="black",shape="triangle"];47 -> 61[label="",style="solid", color="black", weight=3];
10.13/4.14	50[label="vx300",fontsize=16,color="green",shape="box"];51[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];51 -> 62[label="",style="dashed", color="green", weight=3];
10.13/4.14	52[label="vx311",fontsize=16,color="green",shape="box"];53[label="Pos (primMulNat vx40 vx300)",fontsize=16,color="green",shape="box"];53 -> 63[label="",style="dashed", color="green", weight=3];
10.13/4.14	54[label="vx310",fontsize=16,color="green",shape="box"];55[label="primMulNat (Succ vx400) (Succ vx3000)",fontsize=16,color="black",shape="box"];55 -> 64[label="",style="solid", color="black", weight=3];
10.13/4.14	56[label="primMulNat (Succ vx400) Zero",fontsize=16,color="black",shape="box"];56 -> 65[label="",style="solid", color="black", weight=3];
10.13/4.14	57[label="primMulNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];57 -> 66[label="",style="solid", color="black", weight=3];
10.13/4.14	58[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3];
10.13/4.14	59[label="vx300",fontsize=16,color="green",shape="box"];60[label="vx300",fontsize=16,color="green",shape="box"];61 -> 15[label="",style="dashed", color="red", weight=0];
10.13/4.14	61[label="enforceWHNF (WHNF (primMulInt (Neg vx6) vx310)) (foldl' primMulInt (primMulInt (Neg vx7) vx310)) vx311",fontsize=16,color="magenta"];61 -> 68[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	61 -> 69[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	61 -> 70[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	61 -> 71[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	62 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	62[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];63 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	63[label="primMulNat vx40 vx300",fontsize=16,color="magenta"];64 -> 72[label="",style="dashed", color="red", weight=0];
10.13/4.14	64[label="primPlusNat (primMulNat vx400 (Succ vx3000)) (Succ vx3000)",fontsize=16,color="magenta"];64 -> 73[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	65[label="Zero",fontsize=16,color="green",shape="box"];66[label="Zero",fontsize=16,color="green",shape="box"];67[label="Zero",fontsize=16,color="green",shape="box"];68[label="Neg vx6",fontsize=16,color="green",shape="box"];69[label="vx311",fontsize=16,color="green",shape="box"];70[label="Neg vx7",fontsize=16,color="green",shape="box"];71[label="vx310",fontsize=16,color="green",shape="box"];73 -> 41[label="",style="dashed", color="red", weight=0];
10.13/4.14	73[label="primMulNat vx400 (Succ vx3000)",fontsize=16,color="magenta"];73 -> 74[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	73 -> 75[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	72[label="primPlusNat vx8 (Succ vx3000)",fontsize=16,color="burlywood",shape="triangle"];112[label="vx8/Succ vx80",fontsize=10,color="white",style="solid",shape="box"];72 -> 112[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	112 -> 76[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	113[label="vx8/Zero",fontsize=10,color="white",style="solid",shape="box"];72 -> 113[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	113 -> 77[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	74[label="Succ vx3000",fontsize=16,color="green",shape="box"];75[label="vx400",fontsize=16,color="green",shape="box"];76[label="primPlusNat (Succ vx80) (Succ vx3000)",fontsize=16,color="black",shape="box"];76 -> 78[label="",style="solid", color="black", weight=3];
10.13/4.14	77[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];77 -> 79[label="",style="solid", color="black", weight=3];
10.13/4.14	78[label="Succ (Succ (primPlusNat vx80 vx3000))",fontsize=16,color="green",shape="box"];78 -> 80[label="",style="dashed", color="green", weight=3];
10.13/4.14	79[label="Succ vx3000",fontsize=16,color="green",shape="box"];80[label="primPlusNat vx80 vx3000",fontsize=16,color="burlywood",shape="triangle"];114[label="vx80/Succ vx800",fontsize=10,color="white",style="solid",shape="box"];80 -> 114[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	114 -> 81[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	115[label="vx80/Zero",fontsize=10,color="white",style="solid",shape="box"];80 -> 115[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	115 -> 82[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	81[label="primPlusNat (Succ vx800) vx3000",fontsize=16,color="burlywood",shape="box"];116[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];81 -> 116[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	116 -> 83[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	117[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 117[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	117 -> 84[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	82[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];118[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];82 -> 118[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	118 -> 85[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	119[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 119[label="",style="solid", color="burlywood", weight=9];
10.13/4.14	119 -> 86[label="",style="solid", color="burlywood", weight=3];
10.13/4.14	83[label="primPlusNat (Succ vx800) (Succ vx30000)",fontsize=16,color="black",shape="box"];83 -> 87[label="",style="solid", color="black", weight=3];
10.13/4.14	84[label="primPlusNat (Succ vx800) Zero",fontsize=16,color="black",shape="box"];84 -> 88[label="",style="solid", color="black", weight=3];
10.13/4.14	85[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];85 -> 89[label="",style="solid", color="black", weight=3];
10.13/4.14	86[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];86 -> 90[label="",style="solid", color="black", weight=3];
10.13/4.14	87[label="Succ (Succ (primPlusNat vx800 vx30000))",fontsize=16,color="green",shape="box"];87 -> 91[label="",style="dashed", color="green", weight=3];
10.13/4.14	88[label="Succ vx800",fontsize=16,color="green",shape="box"];89[label="Succ vx30000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];91 -> 80[label="",style="dashed", color="red", weight=0];
10.13/4.14	91[label="primPlusNat vx800 vx30000",fontsize=16,color="magenta"];91 -> 92[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	91 -> 93[label="",style="dashed", color="magenta", weight=3];
10.13/4.14	92[label="vx800",fontsize=16,color="green",shape="box"];93[label="vx30000",fontsize=16,color="green",shape="box"];}
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(8)
10.13/4.14	Complex Obligation (AND)
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(9)
10.13/4.14	Obligation:
10.13/4.14	Q DP problem:
10.13/4.14	The TRS P consists of the following rules:
10.13/4.14	
10.13/4.14	   new_enforceWHNF0(Pos(vx40), Neg(vx300), vx5, :(vx310, vx311)) -> new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
10.13/4.14	   new_enforceWHNF0(Neg(vx40), Pos(vx300), vx5, vx31) -> new_enforceWHNF1(vx40, vx300, vx31)
10.13/4.14	   new_enforceWHNF1(vx40, vx300, :(vx310, vx311)) -> new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
10.13/4.14	   new_seq(vx6, vx310, vx7, vx311) -> new_enforceWHNF0(Neg(vx6), vx310, Neg(vx7), vx311)
10.13/4.14	   new_enforceWHNF(vx40, vx300, :(vx310, vx311)) -> new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
10.13/4.14	   new_enforceWHNF0(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) -> new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
10.13/4.14	   new_enforceWHNF0(Neg(vx40), Neg(vx300), vx5, vx31) -> new_enforceWHNF(vx40, vx300, vx31)
10.13/4.14	
10.13/4.14	The TRS R consists of the following rules:
10.13/4.14	
10.13/4.14	   new_primPlusNat0(Succ(vx800), Zero) -> Succ(vx800)
10.13/4.14	   new_primPlusNat0(Zero, Succ(vx30000)) -> Succ(vx30000)
10.13/4.14	   new_primMulNat0(Zero, Zero) -> Zero
10.13/4.14	   new_primPlusNat0(Succ(vx800), Succ(vx30000)) -> Succ(Succ(new_primPlusNat0(vx800, vx30000)))
10.13/4.14	   new_primPlusNat0(Zero, Zero) -> Zero
10.13/4.14	   new_primMulNat0(Succ(vx400), Succ(vx3000)) -> new_primPlusNat1(new_primMulNat0(vx400, Succ(vx3000)), vx3000)
10.13/4.14	   new_primPlusNat1(Succ(vx80), vx3000) -> Succ(Succ(new_primPlusNat0(vx80, vx3000)))
10.13/4.14	   new_primMulNat0(Succ(vx400), Zero) -> Zero
10.13/4.14	   new_primMulNat0(Zero, Succ(vx3000)) -> Zero
10.13/4.14	   new_primPlusNat1(Zero, vx3000) -> Succ(vx3000)
10.13/4.14	
10.13/4.14	The set Q consists of the following terms:
10.13/4.14	
10.13/4.14	   new_primMulNat0(Succ(x0), Zero)
10.13/4.14	   new_primMulNat0(Zero, Succ(x0))
10.13/4.14	   new_primMulNat0(Zero, Zero)
10.13/4.14	   new_primPlusNat0(Succ(x0), Zero)
10.13/4.14	   new_primPlusNat0(Succ(x0), Succ(x1))
10.13/4.14	   new_primPlusNat0(Zero, Succ(x0))
10.13/4.14	   new_primPlusNat0(Zero, Zero)
10.13/4.14	   new_primPlusNat1(Zero, x0)
10.13/4.14	   new_primMulNat0(Succ(x0), Succ(x1))
10.13/4.14	   new_primPlusNat1(Succ(x0), x1)
10.13/4.14	
10.13/4.14	We have to consider all minimal (P,Q,R)-chains.
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(10) QDPSizeChangeProof (EQUIVALENT)
10.13/4.14	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. 
10.13/4.14	
10.13/4.14	From the DPs we obtained the following set of size-change graphs:
10.13/4.14	*new_seq(vx6, vx310, vx7, vx311) -> new_enforceWHNF0(Neg(vx6), vx310, Neg(vx7), vx311)
10.13/4.14	The graph contains the following edges 2 >= 2, 4 >= 4
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF1(vx40, vx300, :(vx310, vx311)) -> new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
10.13/4.14	The graph contains the following edges 3 > 2, 3 > 4
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF0(Neg(vx40), Pos(vx300), vx5, vx31) -> new_enforceWHNF1(vx40, vx300, vx31)
10.13/4.14	The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF0(Neg(vx40), Neg(vx300), vx5, vx31) -> new_enforceWHNF(vx40, vx300, vx31)
10.13/4.14	The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF0(Pos(vx40), Neg(vx300), vx5, :(vx310, vx311)) -> new_seq(new_primMulNat0(vx40, vx300), vx310, new_primMulNat0(vx40, vx300), vx311)
10.13/4.14	The graph contains the following edges 4 > 2, 4 > 4
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF0(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) -> new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
10.13/4.14	The graph contains the following edges 4 > 2, 4 > 4
10.13/4.14	
10.13/4.14	
10.13/4.14	*new_enforceWHNF(vx40, vx300, :(vx310, vx311)) -> new_enforceWHNF0(Pos(new_primMulNat0(vx40, vx300)), vx310, Pos(new_primMulNat0(vx40, vx300)), vx311)
10.13/4.14	The graph contains the following edges 3 > 2, 3 > 4
10.13/4.14	
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(11)
10.13/4.14	YES
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(12)
10.13/4.14	Obligation:
10.13/4.14	Q DP problem:
10.13/4.14	The TRS P consists of the following rules:
10.13/4.14	
10.13/4.14	   new_primMulNat(Succ(vx400), Succ(vx3000)) -> new_primMulNat(vx400, Succ(vx3000))
10.13/4.14	
10.13/4.14	R is empty.
10.13/4.14	Q is empty.
10.13/4.14	We have to consider all minimal (P,Q,R)-chains.
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(13) QDPSizeChangeProof (EQUIVALENT)
10.13/4.14	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. 
10.13/4.14	
10.13/4.14	From the DPs we obtained the following set of size-change graphs:
10.13/4.14	*new_primMulNat(Succ(vx400), Succ(vx3000)) -> new_primMulNat(vx400, Succ(vx3000))
10.13/4.14	The graph contains the following edges 1 > 1, 2 >= 2
10.13/4.14	
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(14)
10.13/4.14	YES
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(15)
10.13/4.14	Obligation:
10.13/4.14	Q DP problem:
10.13/4.14	The TRS P consists of the following rules:
10.13/4.14	
10.13/4.14	   new_primPlusNat(Succ(vx800), Succ(vx30000)) -> new_primPlusNat(vx800, vx30000)
10.13/4.14	
10.13/4.14	R is empty.
10.13/4.14	Q is empty.
10.13/4.14	We have to consider all minimal (P,Q,R)-chains.
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(16) QDPSizeChangeProof (EQUIVALENT)
10.13/4.14	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. 
10.13/4.14	
10.13/4.14	From the DPs we obtained the following set of size-change graphs:
10.13/4.14	*new_primPlusNat(Succ(vx800), Succ(vx30000)) -> new_primPlusNat(vx800, vx30000)
10.13/4.14	The graph contains the following edges 1 > 1, 2 > 2
10.13/4.14	
10.13/4.14	
10.13/4.14	----------------------------------------
10.13/4.14	
10.13/4.14	(17)
10.13/4.14	YES
10.31/4.30	EOF