8.44/3.66	YES
10.34/4.14	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.34/4.14	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.34/4.14	
10.34/4.14	
10.34/4.14	H-Termination with start terms of the given HASKELL could be proven:
10.34/4.14	
10.34/4.14	(0) HASKELL
10.34/4.14	(1) BR [EQUIVALENT, 0 ms]
10.34/4.14	(2) HASKELL
10.34/4.14	(3) COR [EQUIVALENT, 0 ms]
10.34/4.14	(4) HASKELL
10.34/4.14	(5) NumRed [SOUND, 0 ms]
10.34/4.14	(6) HASKELL
10.34/4.14	(7) Narrow [SOUND, 0 ms]
10.34/4.14	(8) AND
10.34/4.14	    (9) QDP
10.34/4.14	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.34/4.14	        (11) YES
10.34/4.14	    (12) QDP
10.34/4.14	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.34/4.14	        (14) YES
10.34/4.14	
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(0)
10.34/4.14	Obligation:
10.34/4.14	mainModule Main 
10.34/4.14	module Main where {
10.34/4.14	import qualified Prelude;
10.34/4.14	}
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(1) BR (EQUIVALENT)
10.34/4.14	Replaced joker patterns by fresh variables and removed binding patterns.
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(2)
10.34/4.14	Obligation:
10.34/4.14	mainModule Main 
10.34/4.14	module Main where {
10.34/4.14	import qualified Prelude;
10.34/4.14	}
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(3) COR (EQUIVALENT)
10.34/4.14	Cond Reductions:
10.34/4.14	The following Function with conditions
10.34/4.14	"undefined |Falseundefined;
10.34/4.14	"
10.34/4.14	is transformed to
10.34/4.14	"undefined  = undefined1;
10.34/4.14	"
10.34/4.14	"undefined0 True = undefined;
10.34/4.14	"
10.34/4.14	"undefined1  = undefined0 False;
10.34/4.14	"
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(4)
10.34/4.14	Obligation:
10.34/4.14	mainModule Main 
10.34/4.14	module Main where {
10.34/4.14	import qualified Prelude;
10.34/4.14	}
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(5) NumRed (SOUND)
10.34/4.14	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(6)
10.34/4.14	Obligation:
10.34/4.14	mainModule Main 
10.34/4.14	module Main where {
10.34/4.14	import qualified Prelude;
10.34/4.14	}
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(7) Narrow (SOUND)
10.34/4.14	Haskell To QDPs
10.34/4.14	
10.34/4.14	digraph dp_graph {
10.34/4.14	node [outthreshold=100, inthreshold=100];1[label="sum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.34/4.14	3[label="sum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.34/4.14	4[label="foldl' (+) (fromInt (Pos Zero)) vx3",fontsize=16,color="burlywood",shape="box"];118[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 118[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	118 -> 5[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	119[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 119[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	119 -> 6[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	5[label="foldl' (+) (fromInt (Pos Zero)) (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.34/4.14	6[label="foldl' (+) (fromInt (Pos Zero)) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.34/4.14	7[label="(foldl' (+) $! (+) fromInt (Pos Zero) vx30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.34/4.14	8[label="fromInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
10.34/4.14	9 -> 11[label="",style="dashed", color="red", weight=0];
10.34/4.14	9[label="((+) fromInt (Pos Zero) vx30 `seq` foldl' (+) ((+) fromInt (Pos Zero) vx30))",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	9 -> 13[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	10[label="Pos Zero",fontsize=16,color="green",shape="box"];12 -> 8[label="",style="dashed", color="red", weight=0];
10.34/4.14	12[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
10.34/4.14	13[label="fromInt (Pos 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];
10.34/4.14	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];
10.34/4.14	15[label="enforceWHNF (WHNF (primPlusInt vx4 vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="triangle"];120[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 120[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	120 -> 16[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	121[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];15 -> 121[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	121 -> 17[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	16[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];122[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 122[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	122 -> 18[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	123[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];16 -> 123[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	123 -> 19[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	17[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) vx30)) (foldl' primPlusInt (primPlusInt vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];124[label="vx30/Pos vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 124[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	124 -> 20[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	125[label="vx30/Neg vx300",fontsize=10,color="white",style="solid",shape="box"];17 -> 125[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	125 -> 21[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	18[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) (Pos vx300))) (foldl' primPlusInt (primPlusInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.34/4.14	19[label="enforceWHNF (WHNF (primPlusInt (Pos vx40) (Neg vx300))) (foldl' primPlusInt (primPlusInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.34/4.14	20[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) (Pos vx300))) (foldl' primPlusInt (primPlusInt vx5 (Pos vx300))) vx31",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.34/4.14	21[label="enforceWHNF (WHNF (primPlusInt (Neg vx40) (Neg vx300))) (foldl' primPlusInt (primPlusInt vx5 (Neg vx300))) vx31",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.34/4.14	22[label="enforceWHNF (WHNF (Pos (primPlusNat vx40 vx300))) (foldl' primPlusInt (Pos (primPlusNat vx40 vx300))) vx31",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.34/4.14	23[label="enforceWHNF (WHNF (primMinusNat vx40 vx300)) (foldl' primPlusInt (primMinusNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="triangle"];126[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];23 -> 126[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	126 -> 27[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	127[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 127[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	127 -> 28[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	24 -> 23[label="",style="dashed", color="red", weight=0];
10.34/4.14	24[label="enforceWHNF (WHNF (primMinusNat vx300 vx40)) (foldl' primPlusInt (primMinusNat vx300 vx40)) vx31",fontsize=16,color="magenta"];24 -> 29[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	24 -> 30[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	25[label="enforceWHNF (WHNF (Neg (primPlusNat vx40 vx300))) (foldl' primPlusInt (Neg (primPlusNat vx40 vx300))) vx31",fontsize=16,color="black",shape="box"];25 -> 31[label="",style="solid", color="black", weight=3];
10.34/4.14	26[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];128[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];26 -> 128[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	128 -> 32[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	129[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];26 -> 129[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	129 -> 33[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	27[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) vx300)) (foldl' primPlusInt (primMinusNat (Succ vx400) vx300)) vx31",fontsize=16,color="burlywood",shape="box"];130[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];27 -> 130[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	130 -> 34[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	131[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 131[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	131 -> 35[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	28[label="enforceWHNF (WHNF (primMinusNat Zero vx300)) (foldl' primPlusInt (primMinusNat Zero vx300)) vx31",fontsize=16,color="burlywood",shape="box"];132[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];28 -> 132[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	132 -> 36[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	133[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 133[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	133 -> 37[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	29[label="vx300",fontsize=16,color="green",shape="box"];30[label="vx40",fontsize=16,color="green",shape="box"];31[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) vx31",fontsize=16,color="burlywood",shape="box"];134[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];31 -> 134[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	134 -> 38[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	135[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];31 -> 135[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	135 -> 39[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	32[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];32 -> 40[label="",style="solid", color="black", weight=3];
10.34/4.14	33[label="foldl' primPlusInt (Pos (primPlusNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3];
10.34/4.14	34[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) (Succ vx3000))) (foldl' primPlusInt (primMinusNat (Succ vx400) (Succ vx3000))) vx31",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3];
10.34/4.14	35[label="enforceWHNF (WHNF (primMinusNat (Succ vx400) Zero)) (foldl' primPlusInt (primMinusNat (Succ vx400) Zero)) vx31",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3];
10.34/4.14	36[label="enforceWHNF (WHNF (primMinusNat Zero (Succ vx3000))) (foldl' primPlusInt (primMinusNat Zero (Succ vx3000))) vx31",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3];
10.34/4.14	37[label="enforceWHNF (WHNF (primMinusNat Zero Zero)) (foldl' primPlusInt (primMinusNat Zero Zero)) vx31",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3];
10.34/4.14	38[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
10.34/4.14	39[label="foldl' primPlusInt (Neg (primPlusNat vx40 vx300)) []",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
10.34/4.14	40[label="(foldl' primPlusInt $! primPlusInt (Pos (primPlusNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
10.34/4.14	41[label="Pos (primPlusNat vx40 vx300)",fontsize=16,color="green",shape="box"];41 -> 49[label="",style="dashed", color="green", weight=3];
10.34/4.14	42 -> 23[label="",style="dashed", color="red", weight=0];
10.34/4.14	42[label="enforceWHNF (WHNF (primMinusNat vx400 vx3000)) (foldl' primPlusInt (primMinusNat vx400 vx3000)) vx31",fontsize=16,color="magenta"];42 -> 50[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	42 -> 51[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	43[label="enforceWHNF (WHNF (Pos (Succ vx400))) (foldl' primPlusInt (Pos (Succ vx400))) vx31",fontsize=16,color="black",shape="box"];43 -> 52[label="",style="solid", color="black", weight=3];
10.34/4.14	44[label="enforceWHNF (WHNF (Neg (Succ vx3000))) (foldl' primPlusInt (Neg (Succ vx3000))) vx31",fontsize=16,color="black",shape="box"];44 -> 53[label="",style="solid", color="black", weight=3];
10.34/4.14	45[label="enforceWHNF (WHNF (Pos Zero)) (foldl' primPlusInt (Pos Zero)) vx31",fontsize=16,color="black",shape="box"];45 -> 54[label="",style="solid", color="black", weight=3];
10.34/4.14	46[label="(foldl' primPlusInt $! primPlusInt (Neg (primPlusNat vx40 vx300)) vx310)",fontsize=16,color="black",shape="box"];46 -> 55[label="",style="solid", color="black", weight=3];
10.34/4.14	47[label="Neg (primPlusNat vx40 vx300)",fontsize=16,color="green",shape="box"];47 -> 56[label="",style="dashed", color="green", weight=3];
10.34/4.14	48[label="(primPlusInt (Pos (primPlusNat vx40 vx300)) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310))",fontsize=16,color="black",shape="box"];48 -> 57[label="",style="solid", color="black", weight=3];
10.34/4.14	49[label="primPlusNat vx40 vx300",fontsize=16,color="burlywood",shape="triangle"];136[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];49 -> 136[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	136 -> 58[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	137[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 137[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	137 -> 59[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	50[label="vx400",fontsize=16,color="green",shape="box"];51[label="vx3000",fontsize=16,color="green",shape="box"];52[label="foldl' primPlusInt (Pos (Succ vx400)) vx31",fontsize=16,color="burlywood",shape="box"];138[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];52 -> 138[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	138 -> 60[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	139[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];52 -> 139[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	139 -> 61[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	53[label="foldl' primPlusInt (Neg (Succ vx3000)) vx31",fontsize=16,color="burlywood",shape="box"];140[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];53 -> 140[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	140 -> 62[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	141[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];53 -> 141[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	141 -> 63[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	54[label="foldl' primPlusInt (Pos Zero) vx31",fontsize=16,color="burlywood",shape="box"];142[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];54 -> 142[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	142 -> 64[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	143[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];54 -> 143[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	143 -> 65[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	55 -> 66[label="",style="dashed", color="red", weight=0];
10.34/4.14	55[label="(primPlusInt (Neg (primPlusNat vx40 vx300)) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg (primPlusNat vx40 vx300)) vx310))",fontsize=16,color="magenta"];55 -> 67[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	55 -> 68[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	56 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	56[label="primPlusNat vx40 vx300",fontsize=16,color="magenta"];56 -> 69[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	56 -> 70[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	57 -> 15[label="",style="dashed", color="red", weight=0];
10.34/4.14	57[label="enforceWHNF (WHNF (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310)) (foldl' primPlusInt (primPlusInt (Pos (primPlusNat vx40 vx300)) vx310)) vx311",fontsize=16,color="magenta"];57 -> 71[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	57 -> 72[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	57 -> 73[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	57 -> 74[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	58[label="primPlusNat (Succ vx400) vx300",fontsize=16,color="burlywood",shape="box"];144[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];58 -> 144[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	144 -> 75[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	145[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 145[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	145 -> 76[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	59[label="primPlusNat Zero vx300",fontsize=16,color="burlywood",shape="box"];146[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];59 -> 146[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	146 -> 77[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	147[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 147[label="",style="solid", color="burlywood", weight=9];
10.34/4.14	147 -> 78[label="",style="solid", color="burlywood", weight=3];
10.34/4.14	60[label="foldl' primPlusInt (Pos (Succ vx400)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];60 -> 79[label="",style="solid", color="black", weight=3];
10.34/4.14	61[label="foldl' primPlusInt (Pos (Succ vx400)) []",fontsize=16,color="black",shape="box"];61 -> 80[label="",style="solid", color="black", weight=3];
10.34/4.14	62[label="foldl' primPlusInt (Neg (Succ vx3000)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];62 -> 81[label="",style="solid", color="black", weight=3];
10.34/4.14	63[label="foldl' primPlusInt (Neg (Succ vx3000)) []",fontsize=16,color="black",shape="box"];63 -> 82[label="",style="solid", color="black", weight=3];
10.34/4.14	64[label="foldl' primPlusInt (Pos Zero) (vx310 : vx311)",fontsize=16,color="black",shape="box"];64 -> 83[label="",style="solid", color="black", weight=3];
10.34/4.14	65[label="foldl' primPlusInt (Pos Zero) []",fontsize=16,color="black",shape="box"];65 -> 84[label="",style="solid", color="black", weight=3];
10.34/4.14	67 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	67[label="primPlusNat vx40 vx300",fontsize=16,color="magenta"];67 -> 85[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	67 -> 86[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	68 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	68[label="primPlusNat vx40 vx300",fontsize=16,color="magenta"];68 -> 87[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	68 -> 88[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	66[label="(primPlusInt (Neg vx6) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg vx7) vx310))",fontsize=16,color="black",shape="triangle"];66 -> 89[label="",style="solid", color="black", weight=3];
10.34/4.14	69[label="vx300",fontsize=16,color="green",shape="box"];70[label="vx40",fontsize=16,color="green",shape="box"];71[label="Pos (primPlusNat vx40 vx300)",fontsize=16,color="green",shape="box"];71 -> 90[label="",style="dashed", color="green", weight=3];
10.34/4.14	72[label="vx311",fontsize=16,color="green",shape="box"];73[label="Pos (primPlusNat vx40 vx300)",fontsize=16,color="green",shape="box"];73 -> 91[label="",style="dashed", color="green", weight=3];
10.34/4.14	74[label="vx310",fontsize=16,color="green",shape="box"];75[label="primPlusNat (Succ vx400) (Succ vx3000)",fontsize=16,color="black",shape="box"];75 -> 92[label="",style="solid", color="black", weight=3];
10.34/4.14	76[label="primPlusNat (Succ vx400) Zero",fontsize=16,color="black",shape="box"];76 -> 93[label="",style="solid", color="black", weight=3];
10.34/4.14	77[label="primPlusNat Zero (Succ vx3000)",fontsize=16,color="black",shape="box"];77 -> 94[label="",style="solid", color="black", weight=3];
10.34/4.14	78[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
10.34/4.14	79[label="(foldl' primPlusInt $! primPlusInt (Pos (Succ vx400)) vx310)",fontsize=16,color="black",shape="box"];79 -> 96[label="",style="solid", color="black", weight=3];
10.34/4.14	80[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];81[label="(foldl' primPlusInt $! primPlusInt (Neg (Succ vx3000)) vx310)",fontsize=16,color="black",shape="box"];81 -> 97[label="",style="solid", color="black", weight=3];
10.34/4.14	82[label="Neg (Succ vx3000)",fontsize=16,color="green",shape="box"];83[label="(foldl' primPlusInt $! primPlusInt (Pos Zero) vx310)",fontsize=16,color="black",shape="box"];83 -> 98[label="",style="solid", color="black", weight=3];
10.34/4.14	84[label="Pos Zero",fontsize=16,color="green",shape="box"];85[label="vx300",fontsize=16,color="green",shape="box"];86[label="vx40",fontsize=16,color="green",shape="box"];87[label="vx300",fontsize=16,color="green",shape="box"];88[label="vx40",fontsize=16,color="green",shape="box"];89 -> 15[label="",style="dashed", color="red", weight=0];
10.34/4.14	89[label="enforceWHNF (WHNF (primPlusInt (Neg vx6) vx310)) (foldl' primPlusInt (primPlusInt (Neg vx7) vx310)) vx311",fontsize=16,color="magenta"];89 -> 99[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	89 -> 100[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	89 -> 101[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	89 -> 102[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	90 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	90[label="primPlusNat vx40 vx300",fontsize=16,color="magenta"];91 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	91[label="primPlusNat vx40 vx300",fontsize=16,color="magenta"];92[label="Succ (Succ (primPlusNat vx400 vx3000))",fontsize=16,color="green",shape="box"];92 -> 103[label="",style="dashed", color="green", weight=3];
10.34/4.14	93[label="Succ vx400",fontsize=16,color="green",shape="box"];94[label="Succ vx3000",fontsize=16,color="green",shape="box"];95[label="Zero",fontsize=16,color="green",shape="box"];96[label="(primPlusInt (Pos (Succ vx400)) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos (Succ vx400)) vx310))",fontsize=16,color="black",shape="box"];96 -> 104[label="",style="solid", color="black", weight=3];
10.34/4.14	97 -> 66[label="",style="dashed", color="red", weight=0];
10.34/4.14	97[label="(primPlusInt (Neg (Succ vx3000)) vx310 `seq` foldl' primPlusInt (primPlusInt (Neg (Succ vx3000)) vx310))",fontsize=16,color="magenta"];97 -> 105[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	97 -> 106[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	98[label="(primPlusInt (Pos Zero) vx310 `seq` foldl' primPlusInt (primPlusInt (Pos Zero) vx310))",fontsize=16,color="black",shape="box"];98 -> 107[label="",style="solid", color="black", weight=3];
10.34/4.14	99[label="Neg vx6",fontsize=16,color="green",shape="box"];100[label="vx311",fontsize=16,color="green",shape="box"];101[label="Neg vx7",fontsize=16,color="green",shape="box"];102[label="vx310",fontsize=16,color="green",shape="box"];103 -> 49[label="",style="dashed", color="red", weight=0];
10.34/4.14	103[label="primPlusNat vx400 vx3000",fontsize=16,color="magenta"];103 -> 108[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	103 -> 109[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	104 -> 15[label="",style="dashed", color="red", weight=0];
10.34/4.14	104[label="enforceWHNF (WHNF (primPlusInt (Pos (Succ vx400)) vx310)) (foldl' primPlusInt (primPlusInt (Pos (Succ vx400)) vx310)) vx311",fontsize=16,color="magenta"];104 -> 110[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	104 -> 111[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	104 -> 112[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	104 -> 113[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	105[label="Succ vx3000",fontsize=16,color="green",shape="box"];106[label="Succ vx3000",fontsize=16,color="green",shape="box"];107 -> 15[label="",style="dashed", color="red", weight=0];
10.34/4.14	107[label="enforceWHNF (WHNF (primPlusInt (Pos Zero) vx310)) (foldl' primPlusInt (primPlusInt (Pos Zero) vx310)) vx311",fontsize=16,color="magenta"];107 -> 114[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	107 -> 115[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	107 -> 116[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	107 -> 117[label="",style="dashed", color="magenta", weight=3];
10.34/4.14	108[label="vx3000",fontsize=16,color="green",shape="box"];109[label="vx400",fontsize=16,color="green",shape="box"];110[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];111[label="vx311",fontsize=16,color="green",shape="box"];112[label="Pos (Succ vx400)",fontsize=16,color="green",shape="box"];113[label="vx310",fontsize=16,color="green",shape="box"];114[label="Pos Zero",fontsize=16,color="green",shape="box"];115[label="vx311",fontsize=16,color="green",shape="box"];116[label="Pos Zero",fontsize=16,color="green",shape="box"];117[label="vx310",fontsize=16,color="green",shape="box"];}
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(8)
10.34/4.14	Complex Obligation (AND)
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(9)
10.34/4.14	Obligation:
10.34/4.14	Q DP problem:
10.34/4.14	The TRS P consists of the following rules:
10.34/4.14	
10.34/4.14	   new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) -> new_enforceWHNF0(vx400, vx3000, vx31)
10.34/4.14	   new_seq(vx6, vx310, vx7, vx311) -> new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
10.34/4.14	   new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) -> new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
10.34/4.14	   new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) -> new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
10.34/4.14	   new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) -> new_enforceWHNF0(vx300, vx40, vx31)
10.34/4.14	   new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) -> new_enforceWHNF0(vx400, vx3000, vx31)
10.34/4.14	   new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) -> new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
10.34/4.14	   new_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
10.34/4.14	   new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
10.34/4.14	   new_enforceWHNF(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) -> new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
10.34/4.14	   new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) -> new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
10.34/4.14	   new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
10.34/4.14	
10.34/4.14	The TRS R consists of the following rules:
10.34/4.14	
10.34/4.14	   new_primPlusNat0(Succ(vx400), Zero) -> Succ(vx400)
10.34/4.14	   new_primPlusNat0(Zero, Succ(vx3000)) -> Succ(vx3000)
10.34/4.14	   new_primPlusNat0(Succ(vx400), Succ(vx3000)) -> Succ(Succ(new_primPlusNat0(vx400, vx3000)))
10.34/4.14	   new_primPlusNat0(Zero, Zero) -> Zero
10.34/4.14	
10.34/4.14	The set Q consists of the following terms:
10.34/4.14	
10.34/4.14	   new_primPlusNat0(Succ(x0), Zero)
10.34/4.14	   new_primPlusNat0(Succ(x0), Succ(x1))
10.34/4.14	   new_primPlusNat0(Zero, Succ(x0))
10.34/4.14	   new_primPlusNat0(Zero, Zero)
10.34/4.14	
10.34/4.14	We have to consider all minimal (P,Q,R)-chains.
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(10) QDPSizeChangeProof (EQUIVALENT)
10.34/4.14	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. 
10.34/4.14	
10.34/4.14	From the DPs we obtained the following set of size-change graphs:
10.34/4.14	*new_enforceWHNF0(Succ(vx400), Succ(vx3000), vx31) -> new_enforceWHNF0(vx400, vx3000, vx31)
10.34/4.14	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF0(Zero, Succ(vx3000), :(vx310, vx311)) -> new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
10.34/4.14	The graph contains the following edges 2 >= 1, 3 > 2, 2 >= 3, 3 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Neg(vx40), Pos(vx300), vx5, vx31) -> new_enforceWHNF0(vx300, vx40, vx31)
10.34/4.14	The graph contains the following edges 2 > 1, 1 > 2, 4 >= 3
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Pos(Succ(vx400)), Neg(Succ(vx3000)), vx5, vx31) -> new_enforceWHNF0(vx400, vx3000, vx31)
10.34/4.14	The graph contains the following edges 1 > 1, 2 > 2, 4 >= 3
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Neg(vx40), Neg(vx300), vx5, :(vx310, vx311)) -> new_seq(new_primPlusNat0(vx40, vx300), vx310, new_primPlusNat0(vx40, vx300), vx311)
10.34/4.14	The graph contains the following edges 4 > 2, 4 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Pos(Zero), Neg(Succ(vx3000)), vx5, :(vx310, vx311)) -> new_seq(Succ(vx3000), vx310, Succ(vx3000), vx311)
10.34/4.14	The graph contains the following edges 2 > 1, 4 > 2, 2 > 3, 4 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_seq(vx6, vx310, vx7, vx311) -> new_enforceWHNF(Neg(vx6), vx310, Neg(vx7), vx311)
10.34/4.14	The graph contains the following edges 2 >= 2, 4 >= 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Pos(Zero), Neg(Zero), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
10.34/4.14	The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF0(Succ(vx400), Zero, :(vx310, vx311)) -> new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
10.34/4.14	The graph contains the following edges 3 > 2, 3 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF0(Zero, Zero, :(vx310, vx311)) -> new_enforceWHNF(Pos(Zero), vx310, Pos(Zero), vx311)
10.34/4.14	The graph contains the following edges 3 > 2, 3 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Pos(vx40), Pos(vx300), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(new_primPlusNat0(vx40, vx300)), vx310, Pos(new_primPlusNat0(vx40, vx300)), vx311)
10.34/4.14	The graph contains the following edges 4 > 2, 4 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	*new_enforceWHNF(Pos(Succ(vx400)), Neg(Zero), vx5, :(vx310, vx311)) -> new_enforceWHNF(Pos(Succ(vx400)), vx310, Pos(Succ(vx400)), vx311)
10.34/4.14	The graph contains the following edges 1 >= 1, 4 > 2, 1 >= 3, 4 > 4
10.34/4.14	
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(11)
10.34/4.14	YES
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(12)
10.34/4.14	Obligation:
10.34/4.14	Q DP problem:
10.34/4.14	The TRS P consists of the following rules:
10.34/4.14	
10.34/4.14	   new_primPlusNat(Succ(vx400), Succ(vx3000)) -> new_primPlusNat(vx400, vx3000)
10.34/4.14	
10.34/4.14	R is empty.
10.34/4.14	Q is empty.
10.34/4.14	We have to consider all minimal (P,Q,R)-chains.
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(13) QDPSizeChangeProof (EQUIVALENT)
10.34/4.14	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. 
10.34/4.14	
10.34/4.14	From the DPs we obtained the following set of size-change graphs:
10.34/4.14	*new_primPlusNat(Succ(vx400), Succ(vx3000)) -> new_primPlusNat(vx400, vx3000)
10.34/4.14	The graph contains the following edges 1 > 1, 2 > 2
10.34/4.14	
10.34/4.14	
10.34/4.14	----------------------------------------
10.34/4.14	
10.34/4.14	(14)
10.34/4.14	YES
10.39/4.18	EOF