8.41/3.68	YES
10.42/4.21	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.42/4.21	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.42/4.21	
10.42/4.21	
10.42/4.21	H-Termination with start terms of the given HASKELL could be proven:
10.42/4.21	
10.42/4.21	(0) HASKELL
10.42/4.21	(1) BR [EQUIVALENT, 0 ms]
10.42/4.21	(2) HASKELL
10.42/4.21	(3) COR [EQUIVALENT, 0 ms]
10.42/4.21	(4) HASKELL
10.42/4.21	(5) NumRed [SOUND, 0 ms]
10.42/4.21	(6) HASKELL
10.42/4.21	(7) Narrow [SOUND, 0 ms]
10.42/4.21	(8) AND
10.42/4.21	    (9) QDP
10.42/4.21	        (10) TransformationProof [EQUIVALENT, 0 ms]
10.42/4.21	        (11) QDP
10.42/4.21	        (12) TransformationProof [EQUIVALENT, 0 ms]
10.42/4.21	        (13) QDP
10.42/4.21	        (14) UsableRulesProof [EQUIVALENT, 0 ms]
10.42/4.21	        (15) QDP
10.42/4.21	        (16) QReductionProof [EQUIVALENT, 0 ms]
10.42/4.21	        (17) QDP
10.42/4.21	        (18) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.42/4.21	        (19) YES
10.42/4.21	    (20) QDP
10.42/4.21	        (21) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.42/4.21	        (22) YES
10.42/4.21	    (23) QDP
10.42/4.21	        (24) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.42/4.21	        (25) YES
10.42/4.21	    (26) QDP
10.42/4.21	        (27) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.42/4.21	        (28) YES
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(0)
10.42/4.21	Obligation:
10.42/4.21	mainModule Main 
10.42/4.21	module Main where {
10.42/4.21	import qualified Prelude;
10.42/4.21	}
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(1) BR (EQUIVALENT)
10.42/4.21	Replaced joker patterns by fresh variables and removed binding patterns.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(2)
10.42/4.21	Obligation:
10.42/4.21	mainModule Main 
10.42/4.21	module Main where {
10.42/4.21	import qualified Prelude;
10.42/4.21	}
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(3) COR (EQUIVALENT)
10.42/4.21	Cond Reductions:
10.42/4.21	The following Function with conditions
10.42/4.21	"undefined |Falseundefined;
10.42/4.21	"
10.42/4.21	is transformed to
10.42/4.21	"undefined  = undefined1;
10.42/4.21	"
10.42/4.21	"undefined0 True = undefined;
10.42/4.21	"
10.42/4.21	"undefined1  = undefined0 False;
10.42/4.21	"
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(4)
10.42/4.21	Obligation:
10.42/4.21	mainModule Main 
10.42/4.21	module Main where {
10.42/4.21	import qualified Prelude;
10.42/4.21	}
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(5) NumRed (SOUND)
10.42/4.21	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(6)
10.42/4.21	Obligation:
10.42/4.21	mainModule Main 
10.42/4.21	module Main where {
10.42/4.21	import qualified Prelude;
10.42/4.21	}
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(7) Narrow (SOUND)
10.42/4.21	Haskell To QDPs
10.42/4.21	
10.42/4.21	digraph dp_graph {
10.42/4.21	node [outthreshold=100, inthreshold=100];1[label="sum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.42/4.21	3[label="sum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
10.42/4.21	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];
10.42/4.21	118 -> 5[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	119[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 119[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	119 -> 6[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	5[label="foldl' (+) (fromInt (Pos Zero)) (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
10.42/4.21	6[label="foldl' (+) (fromInt (Pos Zero)) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
10.42/4.21	7[label="(foldl' (+) $! (+) fromInt (Pos Zero) vx30)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
10.42/4.21	8[label="fromInt (Pos Zero)",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
10.42/4.21	9 -> 11[label="",style="dashed", color="red", weight=0];
10.42/4.21	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];
10.42/4.21	9 -> 13[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	10[label="primIntToFloat (Pos Zero)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
10.42/4.21	12 -> 8[label="",style="dashed", color="red", weight=0];
10.42/4.21	12[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
10.42/4.21	13[label="fromInt (Pos Zero)",fontsize=16,color="magenta"];11[label="((+) vx4 vx30 `seq` foldl' (+) ((+) vx5 vx30))",fontsize=16,color="black",shape="triangle"];11 -> 15[label="",style="solid", color="black", weight=3];
10.42/4.21	14[label="Float (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];15[label="enforceWHNF (WHNF ((+) vx4 vx30)) (foldl' (+) ((+) vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
10.42/4.21	16[label="enforceWHNF (WHNF (primPlusFloat vx4 vx30)) (foldl' primPlusFloat (primPlusFloat vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="triangle"];120[label="vx4/Float vx40 vx41",fontsize=10,color="white",style="solid",shape="box"];16 -> 120[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	120 -> 17[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	17[label="enforceWHNF (WHNF (primPlusFloat (Float vx40 vx41) vx30)) (foldl' primPlusFloat (primPlusFloat vx5 vx30)) vx31",fontsize=16,color="burlywood",shape="box"];121[label="vx30/Float vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];17 -> 121[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	121 -> 18[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	18[label="enforceWHNF (WHNF (primPlusFloat (Float vx40 vx41) (Float vx300 vx301))) (foldl' primPlusFloat (primPlusFloat vx5 (Float vx300 vx301))) vx31",fontsize=16,color="black",shape="box"];18 -> 19[label="",style="solid", color="black", weight=3];
10.42/4.21	19[label="enforceWHNF (WHNF (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301))) (foldl' primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301))) vx31",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
10.42/4.21	20[label="foldl' primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx31",fontsize=16,color="burlywood",shape="box"];122[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];20 -> 122[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	122 -> 21[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	123[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];20 -> 123[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	123 -> 22[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	21[label="foldl' primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) (vx310 : vx311)",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
10.42/4.21	22[label="foldl' primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) []",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
10.42/4.21	23[label="(foldl' primPlusFloat $! primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx310)",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
10.42/4.21	24[label="Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)",fontsize=16,color="green",shape="box"];24 -> 26[label="",style="dashed", color="green", weight=3];
10.42/4.21	24 -> 27[label="",style="dashed", color="green", weight=3];
10.42/4.21	25[label="(primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx310 `seq` foldl' primPlusFloat (primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx310))",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
10.42/4.21	26[label="vx40 * vx301 + vx300 * vx41",fontsize=16,color="black",shape="triangle"];26 -> 29[label="",style="solid", color="black", weight=3];
10.42/4.21	27[label="vx41 * vx301",fontsize=16,color="black",shape="triangle"];27 -> 30[label="",style="solid", color="black", weight=3];
10.42/4.21	28 -> 16[label="",style="dashed", color="red", weight=0];
10.42/4.21	28[label="enforceWHNF (WHNF (primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx310)) (foldl' primPlusFloat (primPlusFloat (Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)) vx310)) vx311",fontsize=16,color="magenta"];28 -> 31[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	28 -> 32[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	28 -> 33[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	28 -> 34[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	29 -> 35[label="",style="dashed", color="red", weight=0];
10.42/4.21	29[label="primPlusInt (vx40 * vx301) (vx300 * vx41)",fontsize=16,color="magenta"];29 -> 36[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	29 -> 37[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	30[label="primMulInt vx41 vx301",fontsize=16,color="burlywood",shape="box"];124[label="vx41/Pos vx410",fontsize=10,color="white",style="solid",shape="box"];30 -> 124[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	124 -> 38[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	125[label="vx41/Neg vx410",fontsize=10,color="white",style="solid",shape="box"];30 -> 125[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	125 -> 39[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	31[label="Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)",fontsize=16,color="green",shape="box"];31 -> 40[label="",style="dashed", color="green", weight=3];
10.42/4.21	31 -> 41[label="",style="dashed", color="green", weight=3];
10.42/4.21	32[label="vx310",fontsize=16,color="green",shape="box"];33[label="Float (vx40 * vx301 + vx300 * vx41) (vx41 * vx301)",fontsize=16,color="green",shape="box"];33 -> 42[label="",style="dashed", color="green", weight=3];
10.42/4.21	33 -> 43[label="",style="dashed", color="green", weight=3];
10.42/4.21	34[label="vx311",fontsize=16,color="green",shape="box"];36 -> 27[label="",style="dashed", color="red", weight=0];
10.42/4.21	36[label="vx40 * vx301",fontsize=16,color="magenta"];36 -> 44[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	37 -> 27[label="",style="dashed", color="red", weight=0];
10.42/4.21	37[label="vx300 * vx41",fontsize=16,color="magenta"];37 -> 45[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	37 -> 46[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	35[label="primPlusInt vx7 vx6",fontsize=16,color="burlywood",shape="triangle"];126[label="vx7/Pos vx70",fontsize=10,color="white",style="solid",shape="box"];35 -> 126[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	126 -> 47[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	127[label="vx7/Neg vx70",fontsize=10,color="white",style="solid",shape="box"];35 -> 127[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	127 -> 48[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	38[label="primMulInt (Pos vx410) vx301",fontsize=16,color="burlywood",shape="box"];128[label="vx301/Pos vx3010",fontsize=10,color="white",style="solid",shape="box"];38 -> 128[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	128 -> 49[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	129[label="vx301/Neg vx3010",fontsize=10,color="white",style="solid",shape="box"];38 -> 129[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	129 -> 50[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	39[label="primMulInt (Neg vx410) vx301",fontsize=16,color="burlywood",shape="box"];130[label="vx301/Pos vx3010",fontsize=10,color="white",style="solid",shape="box"];39 -> 130[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	130 -> 51[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	131[label="vx301/Neg vx3010",fontsize=10,color="white",style="solid",shape="box"];39 -> 131[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	131 -> 52[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	40 -> 26[label="",style="dashed", color="red", weight=0];
10.42/4.21	40[label="vx40 * vx301 + vx300 * vx41",fontsize=16,color="magenta"];41 -> 27[label="",style="dashed", color="red", weight=0];
10.42/4.21	41[label="vx41 * vx301",fontsize=16,color="magenta"];42 -> 26[label="",style="dashed", color="red", weight=0];
10.42/4.21	42[label="vx40 * vx301 + vx300 * vx41",fontsize=16,color="magenta"];43 -> 27[label="",style="dashed", color="red", weight=0];
10.42/4.21	43[label="vx41 * vx301",fontsize=16,color="magenta"];44[label="vx40",fontsize=16,color="green",shape="box"];45[label="vx41",fontsize=16,color="green",shape="box"];46[label="vx300",fontsize=16,color="green",shape="box"];47[label="primPlusInt (Pos vx70) vx6",fontsize=16,color="burlywood",shape="box"];132[label="vx6/Pos vx60",fontsize=10,color="white",style="solid",shape="box"];47 -> 132[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	132 -> 53[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	133[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];47 -> 133[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	133 -> 54[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	48[label="primPlusInt (Neg vx70) vx6",fontsize=16,color="burlywood",shape="box"];134[label="vx6/Pos vx60",fontsize=10,color="white",style="solid",shape="box"];48 -> 134[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	134 -> 55[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	135[label="vx6/Neg vx60",fontsize=10,color="white",style="solid",shape="box"];48 -> 135[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	135 -> 56[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	49[label="primMulInt (Pos vx410) (Pos vx3010)",fontsize=16,color="black",shape="box"];49 -> 57[label="",style="solid", color="black", weight=3];
10.42/4.21	50[label="primMulInt (Pos vx410) (Neg vx3010)",fontsize=16,color="black",shape="box"];50 -> 58[label="",style="solid", color="black", weight=3];
10.42/4.21	51[label="primMulInt (Neg vx410) (Pos vx3010)",fontsize=16,color="black",shape="box"];51 -> 59[label="",style="solid", color="black", weight=3];
10.42/4.21	52[label="primMulInt (Neg vx410) (Neg vx3010)",fontsize=16,color="black",shape="box"];52 -> 60[label="",style="solid", color="black", weight=3];
10.42/4.21	53[label="primPlusInt (Pos vx70) (Pos vx60)",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
10.42/4.21	54[label="primPlusInt (Pos vx70) (Neg vx60)",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
10.42/4.21	55[label="primPlusInt (Neg vx70) (Pos vx60)",fontsize=16,color="black",shape="box"];55 -> 63[label="",style="solid", color="black", weight=3];
10.42/4.21	56[label="primPlusInt (Neg vx70) (Neg vx60)",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3];
10.42/4.21	57[label="Pos (primMulNat vx410 vx3010)",fontsize=16,color="green",shape="box"];57 -> 65[label="",style="dashed", color="green", weight=3];
10.42/4.21	58[label="Neg (primMulNat vx410 vx3010)",fontsize=16,color="green",shape="box"];58 -> 66[label="",style="dashed", color="green", weight=3];
10.42/4.21	59[label="Neg (primMulNat vx410 vx3010)",fontsize=16,color="green",shape="box"];59 -> 67[label="",style="dashed", color="green", weight=3];
10.42/4.21	60[label="Pos (primMulNat vx410 vx3010)",fontsize=16,color="green",shape="box"];60 -> 68[label="",style="dashed", color="green", weight=3];
10.42/4.21	61[label="Pos (primPlusNat vx70 vx60)",fontsize=16,color="green",shape="box"];61 -> 69[label="",style="dashed", color="green", weight=3];
10.42/4.21	62[label="primMinusNat vx70 vx60",fontsize=16,color="burlywood",shape="triangle"];136[label="vx70/Succ vx700",fontsize=10,color="white",style="solid",shape="box"];62 -> 136[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	136 -> 70[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	137[label="vx70/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 137[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	137 -> 71[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	63 -> 62[label="",style="dashed", color="red", weight=0];
10.42/4.21	63[label="primMinusNat vx60 vx70",fontsize=16,color="magenta"];63 -> 72[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	63 -> 73[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	64[label="Neg (primPlusNat vx70 vx60)",fontsize=16,color="green",shape="box"];64 -> 74[label="",style="dashed", color="green", weight=3];
10.42/4.21	65[label="primMulNat vx410 vx3010",fontsize=16,color="burlywood",shape="triangle"];138[label="vx410/Succ vx4100",fontsize=10,color="white",style="solid",shape="box"];65 -> 138[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	138 -> 75[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	139[label="vx410/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 139[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	139 -> 76[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	66 -> 65[label="",style="dashed", color="red", weight=0];
10.42/4.21	66[label="primMulNat vx410 vx3010",fontsize=16,color="magenta"];66 -> 77[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	67 -> 65[label="",style="dashed", color="red", weight=0];
10.42/4.21	67[label="primMulNat vx410 vx3010",fontsize=16,color="magenta"];67 -> 78[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	68 -> 65[label="",style="dashed", color="red", weight=0];
10.42/4.21	68[label="primMulNat vx410 vx3010",fontsize=16,color="magenta"];68 -> 79[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	68 -> 80[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	69[label="primPlusNat vx70 vx60",fontsize=16,color="burlywood",shape="triangle"];140[label="vx70/Succ vx700",fontsize=10,color="white",style="solid",shape="box"];69 -> 140[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	140 -> 81[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	141[label="vx70/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 141[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	141 -> 82[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	70[label="primMinusNat (Succ vx700) vx60",fontsize=16,color="burlywood",shape="box"];142[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];70 -> 142[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	142 -> 83[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	143[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 143[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	143 -> 84[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	71[label="primMinusNat Zero vx60",fontsize=16,color="burlywood",shape="box"];144[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];71 -> 144[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	144 -> 85[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	145[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];71 -> 145[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	145 -> 86[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	72[label="vx60",fontsize=16,color="green",shape="box"];73[label="vx70",fontsize=16,color="green",shape="box"];74 -> 69[label="",style="dashed", color="red", weight=0];
10.42/4.21	74[label="primPlusNat vx70 vx60",fontsize=16,color="magenta"];74 -> 87[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	74 -> 88[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	75[label="primMulNat (Succ vx4100) vx3010",fontsize=16,color="burlywood",shape="box"];146[label="vx3010/Succ vx30100",fontsize=10,color="white",style="solid",shape="box"];75 -> 146[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	146 -> 89[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	147[label="vx3010/Zero",fontsize=10,color="white",style="solid",shape="box"];75 -> 147[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	147 -> 90[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	76[label="primMulNat Zero vx3010",fontsize=16,color="burlywood",shape="box"];148[label="vx3010/Succ vx30100",fontsize=10,color="white",style="solid",shape="box"];76 -> 148[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	148 -> 91[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	149[label="vx3010/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 149[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	149 -> 92[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	77[label="vx3010",fontsize=16,color="green",shape="box"];78[label="vx410",fontsize=16,color="green",shape="box"];79[label="vx3010",fontsize=16,color="green",shape="box"];80[label="vx410",fontsize=16,color="green",shape="box"];81[label="primPlusNat (Succ vx700) vx60",fontsize=16,color="burlywood",shape="box"];150[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];81 -> 150[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	150 -> 93[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	151[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 151[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	151 -> 94[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	82[label="primPlusNat Zero vx60",fontsize=16,color="burlywood",shape="box"];152[label="vx60/Succ vx600",fontsize=10,color="white",style="solid",shape="box"];82 -> 152[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	152 -> 95[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	153[label="vx60/Zero",fontsize=10,color="white",style="solid",shape="box"];82 -> 153[label="",style="solid", color="burlywood", weight=9];
10.42/4.21	153 -> 96[label="",style="solid", color="burlywood", weight=3];
10.42/4.21	83[label="primMinusNat (Succ vx700) (Succ vx600)",fontsize=16,color="black",shape="box"];83 -> 97[label="",style="solid", color="black", weight=3];
10.42/4.21	84[label="primMinusNat (Succ vx700) Zero",fontsize=16,color="black",shape="box"];84 -> 98[label="",style="solid", color="black", weight=3];
10.42/4.21	85[label="primMinusNat Zero (Succ vx600)",fontsize=16,color="black",shape="box"];85 -> 99[label="",style="solid", color="black", weight=3];
10.42/4.21	86[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];86 -> 100[label="",style="solid", color="black", weight=3];
10.42/4.21	87[label="vx70",fontsize=16,color="green",shape="box"];88[label="vx60",fontsize=16,color="green",shape="box"];89[label="primMulNat (Succ vx4100) (Succ vx30100)",fontsize=16,color="black",shape="box"];89 -> 101[label="",style="solid", color="black", weight=3];
10.42/4.21	90[label="primMulNat (Succ vx4100) Zero",fontsize=16,color="black",shape="box"];90 -> 102[label="",style="solid", color="black", weight=3];
10.42/4.21	91[label="primMulNat Zero (Succ vx30100)",fontsize=16,color="black",shape="box"];91 -> 103[label="",style="solid", color="black", weight=3];
10.42/4.21	92[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];92 -> 104[label="",style="solid", color="black", weight=3];
10.42/4.21	93[label="primPlusNat (Succ vx700) (Succ vx600)",fontsize=16,color="black",shape="box"];93 -> 105[label="",style="solid", color="black", weight=3];
10.42/4.21	94[label="primPlusNat (Succ vx700) Zero",fontsize=16,color="black",shape="box"];94 -> 106[label="",style="solid", color="black", weight=3];
10.42/4.21	95[label="primPlusNat Zero (Succ vx600)",fontsize=16,color="black",shape="box"];95 -> 107[label="",style="solid", color="black", weight=3];
10.42/4.21	96[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];96 -> 108[label="",style="solid", color="black", weight=3];
10.42/4.21	97 -> 62[label="",style="dashed", color="red", weight=0];
10.42/4.21	97[label="primMinusNat vx700 vx600",fontsize=16,color="magenta"];97 -> 109[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	97 -> 110[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	98[label="Pos (Succ vx700)",fontsize=16,color="green",shape="box"];99[label="Neg (Succ vx600)",fontsize=16,color="green",shape="box"];100[label="Pos Zero",fontsize=16,color="green",shape="box"];101 -> 69[label="",style="dashed", color="red", weight=0];
10.42/4.21	101[label="primPlusNat (primMulNat vx4100 (Succ vx30100)) (Succ vx30100)",fontsize=16,color="magenta"];101 -> 111[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	101 -> 112[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	102[label="Zero",fontsize=16,color="green",shape="box"];103[label="Zero",fontsize=16,color="green",shape="box"];104[label="Zero",fontsize=16,color="green",shape="box"];105[label="Succ (Succ (primPlusNat vx700 vx600))",fontsize=16,color="green",shape="box"];105 -> 113[label="",style="dashed", color="green", weight=3];
10.42/4.21	106[label="Succ vx700",fontsize=16,color="green",shape="box"];107[label="Succ vx600",fontsize=16,color="green",shape="box"];108[label="Zero",fontsize=16,color="green",shape="box"];109[label="vx700",fontsize=16,color="green",shape="box"];110[label="vx600",fontsize=16,color="green",shape="box"];111 -> 65[label="",style="dashed", color="red", weight=0];
10.42/4.21	111[label="primMulNat vx4100 (Succ vx30100)",fontsize=16,color="magenta"];111 -> 114[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	111 -> 115[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	112[label="Succ vx30100",fontsize=16,color="green",shape="box"];113 -> 69[label="",style="dashed", color="red", weight=0];
10.42/4.21	113[label="primPlusNat vx700 vx600",fontsize=16,color="magenta"];113 -> 116[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	113 -> 117[label="",style="dashed", color="magenta", weight=3];
10.42/4.21	114[label="Succ vx30100",fontsize=16,color="green",shape="box"];115[label="vx4100",fontsize=16,color="green",shape="box"];116[label="vx700",fontsize=16,color="green",shape="box"];117[label="vx600",fontsize=16,color="green",shape="box"];}
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(8)
10.42/4.21	Complex Obligation (AND)
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(9)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311)
10.42/4.21	
10.42/4.21	The TRS R consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.42/4.21	   new_primPlusNat0(Succ(vx700), Zero) -> Succ(vx700)
10.42/4.21	   new_primPlusNat0(Zero, Succ(vx600)) -> Succ(vx600)
10.42/4.21	   new_sr(Pos(vx410), Neg(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Pos(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Zero, Zero) -> Zero
10.42/4.21	   new_ps(vx40, vx301, vx300, vx41) -> new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41))
10.42/4.21	   new_primPlusNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primPlusInt(Neg(vx70), Neg(vx60)) -> Neg(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Pos(vx60)) -> Pos(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primMinusNat0(Zero, Succ(vx600)) -> Neg(Succ(vx600))
10.42/4.21	   new_primPlusInt(Pos(vx70), Neg(vx60)) -> new_primMinusNat0(vx70, vx60)
10.42/4.21	   new_primPlusInt(Neg(vx70), Pos(vx60)) -> new_primMinusNat0(vx60, vx70)
10.42/4.21	   new_primMulNat0(Succ(vx4100), Zero) -> Zero
10.42/4.21	   new_primMulNat0(Zero, Succ(vx30100)) -> Zero
10.42/4.21	   new_sr(Pos(vx410), Pos(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Succ(vx4100), Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
10.42/4.21	   new_primPlusNat0(Succ(vx700), Succ(vx600)) -> Succ(Succ(new_primPlusNat0(vx700, vx600)))
10.42/4.21	   new_sr(Neg(vx410), Neg(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMinusNat0(Succ(vx700), Succ(vx600)) -> new_primMinusNat0(vx700, vx600)
10.42/4.21	   new_primMinusNat0(Succ(vx700), Zero) -> Pos(Succ(vx700))
10.42/4.21	
10.42/4.21	The set Q consists of the following terms:
10.42/4.21	
10.42/4.21	   new_primMulNat0(Succ(x0), Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(x0))
10.42/4.21	   new_primPlusInt(Neg(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Neg(x1))
10.42/4.21	   new_primMulNat0(Zero, Zero)
10.42/4.21	   new_primMinusNat0(Zero, Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primMulNat0(Zero, Succ(x0))
10.42/4.21	   new_primMinusNat0(Succ(x0), Zero)
10.42/4.21	   new_sr(Pos(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Pos(x1))
10.42/4.21	   new_primPlusInt(Pos(x0), Neg(x1))
10.42/4.21	   new_primPlusInt(Neg(x0), Pos(x1))
10.42/4.21	   new_ps(x0, x1, x2, x3)
10.42/4.21	   new_primMinusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Zero)
10.42/4.21	   new_primPlusInt(Pos(x0), Pos(x1))
10.42/4.21	   new_sr(Pos(x0), Pos(x1))
10.42/4.21	   new_primMulNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Succ(x0))
10.42/4.21	
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(10) TransformationProof (EQUIVALENT)
10.42/4.21	By rewriting [LPAR04] the rule new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311) at position [0,0] we obtained the following new rules [LPAR04]:
10.42/4.21	
10.42/4.21	   (new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311),new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311))
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(11)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311)
10.42/4.21	
10.42/4.21	The TRS R consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.42/4.21	   new_primPlusNat0(Succ(vx700), Zero) -> Succ(vx700)
10.42/4.21	   new_primPlusNat0(Zero, Succ(vx600)) -> Succ(vx600)
10.42/4.21	   new_sr(Pos(vx410), Neg(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Pos(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Zero, Zero) -> Zero
10.42/4.21	   new_ps(vx40, vx301, vx300, vx41) -> new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41))
10.42/4.21	   new_primPlusNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primPlusInt(Neg(vx70), Neg(vx60)) -> Neg(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Pos(vx60)) -> Pos(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primMinusNat0(Zero, Succ(vx600)) -> Neg(Succ(vx600))
10.42/4.21	   new_primPlusInt(Pos(vx70), Neg(vx60)) -> new_primMinusNat0(vx70, vx60)
10.42/4.21	   new_primPlusInt(Neg(vx70), Pos(vx60)) -> new_primMinusNat0(vx60, vx70)
10.42/4.21	   new_primMulNat0(Succ(vx4100), Zero) -> Zero
10.42/4.21	   new_primMulNat0(Zero, Succ(vx30100)) -> Zero
10.42/4.21	   new_sr(Pos(vx410), Pos(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Succ(vx4100), Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
10.42/4.21	   new_primPlusNat0(Succ(vx700), Succ(vx600)) -> Succ(Succ(new_primPlusNat0(vx700, vx600)))
10.42/4.21	   new_sr(Neg(vx410), Neg(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMinusNat0(Succ(vx700), Succ(vx600)) -> new_primMinusNat0(vx700, vx600)
10.42/4.21	   new_primMinusNat0(Succ(vx700), Zero) -> Pos(Succ(vx700))
10.42/4.21	
10.42/4.21	The set Q consists of the following terms:
10.42/4.21	
10.42/4.21	   new_primMulNat0(Succ(x0), Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(x0))
10.42/4.21	   new_primPlusInt(Neg(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Neg(x1))
10.42/4.21	   new_primMulNat0(Zero, Zero)
10.42/4.21	   new_primMinusNat0(Zero, Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primMulNat0(Zero, Succ(x0))
10.42/4.21	   new_primMinusNat0(Succ(x0), Zero)
10.42/4.21	   new_sr(Pos(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Pos(x1))
10.42/4.21	   new_primPlusInt(Pos(x0), Neg(x1))
10.42/4.21	   new_primPlusInt(Neg(x0), Pos(x1))
10.42/4.21	   new_ps(x0, x1, x2, x3)
10.42/4.21	   new_primMinusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Zero)
10.42/4.21	   new_primPlusInt(Pos(x0), Pos(x1))
10.42/4.21	   new_sr(Pos(x0), Pos(x1))
10.42/4.21	   new_primMulNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Succ(x0))
10.42/4.21	
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(12) TransformationProof (EQUIVALENT)
10.42/4.21	By rewriting [LPAR04] the rule new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_ps(vx40, vx301, vx300, vx41), new_sr(vx41, vx301)), vx311) at position [2,0] we obtained the following new rules [LPAR04]:
10.42/4.21	
10.42/4.21	   (new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311),new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311))
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(13)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311)
10.42/4.21	
10.42/4.21	The TRS R consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.42/4.21	   new_primPlusNat0(Succ(vx700), Zero) -> Succ(vx700)
10.42/4.21	   new_primPlusNat0(Zero, Succ(vx600)) -> Succ(vx600)
10.42/4.21	   new_sr(Pos(vx410), Neg(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Pos(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Zero, Zero) -> Zero
10.42/4.21	   new_ps(vx40, vx301, vx300, vx41) -> new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41))
10.42/4.21	   new_primPlusNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primPlusInt(Neg(vx70), Neg(vx60)) -> Neg(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Pos(vx60)) -> Pos(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primMinusNat0(Zero, Succ(vx600)) -> Neg(Succ(vx600))
10.42/4.21	   new_primPlusInt(Pos(vx70), Neg(vx60)) -> new_primMinusNat0(vx70, vx60)
10.42/4.21	   new_primPlusInt(Neg(vx70), Pos(vx60)) -> new_primMinusNat0(vx60, vx70)
10.42/4.21	   new_primMulNat0(Succ(vx4100), Zero) -> Zero
10.42/4.21	   new_primMulNat0(Zero, Succ(vx30100)) -> Zero
10.42/4.21	   new_sr(Pos(vx410), Pos(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMulNat0(Succ(vx4100), Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
10.42/4.21	   new_primPlusNat0(Succ(vx700), Succ(vx600)) -> Succ(Succ(new_primPlusNat0(vx700, vx600)))
10.42/4.21	   new_sr(Neg(vx410), Neg(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primMinusNat0(Succ(vx700), Succ(vx600)) -> new_primMinusNat0(vx700, vx600)
10.42/4.21	   new_primMinusNat0(Succ(vx700), Zero) -> Pos(Succ(vx700))
10.42/4.21	
10.42/4.21	The set Q consists of the following terms:
10.42/4.21	
10.42/4.21	   new_primMulNat0(Succ(x0), Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(x0))
10.42/4.21	   new_primPlusInt(Neg(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Neg(x1))
10.42/4.21	   new_primMulNat0(Zero, Zero)
10.42/4.21	   new_primMinusNat0(Zero, Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primMulNat0(Zero, Succ(x0))
10.42/4.21	   new_primMinusNat0(Succ(x0), Zero)
10.42/4.21	   new_sr(Pos(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Pos(x1))
10.42/4.21	   new_primPlusInt(Pos(x0), Neg(x1))
10.42/4.21	   new_primPlusInt(Neg(x0), Pos(x1))
10.42/4.21	   new_ps(x0, x1, x2, x3)
10.42/4.21	   new_primMinusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Zero)
10.42/4.21	   new_primPlusInt(Pos(x0), Pos(x1))
10.42/4.21	   new_sr(Pos(x0), Pos(x1))
10.42/4.21	   new_primMulNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Succ(x0))
10.42/4.21	
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(14) UsableRulesProof (EQUIVALENT)
10.42/4.21	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(15)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311)
10.42/4.21	
10.42/4.21	The TRS R consists of the following rules:
10.42/4.21	
10.42/4.21	   new_sr(Pos(vx410), Neg(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Pos(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Pos(vx410), Pos(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Neg(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primPlusInt(Neg(vx70), Neg(vx60)) -> Neg(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Pos(vx60)) -> Pos(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Neg(vx60)) -> new_primMinusNat0(vx70, vx60)
10.42/4.21	   new_primPlusInt(Neg(vx70), Pos(vx60)) -> new_primMinusNat0(vx60, vx70)
10.42/4.21	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(vx600)) -> Neg(Succ(vx600))
10.42/4.21	   new_primMinusNat0(Succ(vx700), Succ(vx600)) -> new_primMinusNat0(vx700, vx600)
10.42/4.21	   new_primMinusNat0(Succ(vx700), Zero) -> Pos(Succ(vx700))
10.42/4.21	   new_primPlusNat0(Succ(vx700), Zero) -> Succ(vx700)
10.42/4.21	   new_primPlusNat0(Zero, Succ(vx600)) -> Succ(vx600)
10.42/4.21	   new_primPlusNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primPlusNat0(Succ(vx700), Succ(vx600)) -> Succ(Succ(new_primPlusNat0(vx700, vx600)))
10.42/4.21	   new_primMulNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primMulNat0(Succ(vx4100), Zero) -> Zero
10.42/4.21	   new_primMulNat0(Zero, Succ(vx30100)) -> Zero
10.42/4.21	   new_primMulNat0(Succ(vx4100), Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
10.42/4.21	
10.42/4.21	The set Q consists of the following terms:
10.42/4.21	
10.42/4.21	   new_primMulNat0(Succ(x0), Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(x0))
10.42/4.21	   new_primPlusInt(Neg(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Neg(x1))
10.42/4.21	   new_primMulNat0(Zero, Zero)
10.42/4.21	   new_primMinusNat0(Zero, Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primMulNat0(Zero, Succ(x0))
10.42/4.21	   new_primMinusNat0(Succ(x0), Zero)
10.42/4.21	   new_sr(Pos(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Pos(x1))
10.42/4.21	   new_primPlusInt(Pos(x0), Neg(x1))
10.42/4.21	   new_primPlusInt(Neg(x0), Pos(x1))
10.42/4.21	   new_ps(x0, x1, x2, x3)
10.42/4.21	   new_primMinusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Zero)
10.42/4.21	   new_primPlusInt(Pos(x0), Pos(x1))
10.42/4.21	   new_sr(Pos(x0), Pos(x1))
10.42/4.21	   new_primMulNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Succ(x0))
10.42/4.21	
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(16) QReductionProof (EQUIVALENT)
10.42/4.21	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.42/4.21	
10.42/4.21	   new_ps(x0, x1, x2, x3)
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(17)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311)
10.42/4.21	
10.42/4.21	The TRS R consists of the following rules:
10.42/4.21	
10.42/4.21	   new_sr(Pos(vx410), Neg(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Pos(vx3010)) -> Neg(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Pos(vx410), Pos(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_sr(Neg(vx410), Neg(vx3010)) -> Pos(new_primMulNat0(vx410, vx3010))
10.42/4.21	   new_primPlusInt(Neg(vx70), Neg(vx60)) -> Neg(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Pos(vx60)) -> Pos(new_primPlusNat0(vx70, vx60))
10.42/4.21	   new_primPlusInt(Pos(vx70), Neg(vx60)) -> new_primMinusNat0(vx70, vx60)
10.42/4.21	   new_primPlusInt(Neg(vx70), Pos(vx60)) -> new_primMinusNat0(vx60, vx70)
10.42/4.21	   new_primMinusNat0(Zero, Zero) -> Pos(Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(vx600)) -> Neg(Succ(vx600))
10.42/4.21	   new_primMinusNat0(Succ(vx700), Succ(vx600)) -> new_primMinusNat0(vx700, vx600)
10.42/4.21	   new_primMinusNat0(Succ(vx700), Zero) -> Pos(Succ(vx700))
10.42/4.21	   new_primPlusNat0(Succ(vx700), Zero) -> Succ(vx700)
10.42/4.21	   new_primPlusNat0(Zero, Succ(vx600)) -> Succ(vx600)
10.42/4.21	   new_primPlusNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primPlusNat0(Succ(vx700), Succ(vx600)) -> Succ(Succ(new_primPlusNat0(vx700, vx600)))
10.42/4.21	   new_primMulNat0(Zero, Zero) -> Zero
10.42/4.21	   new_primMulNat0(Succ(vx4100), Zero) -> Zero
10.42/4.21	   new_primMulNat0(Zero, Succ(vx30100)) -> Zero
10.42/4.21	   new_primMulNat0(Succ(vx4100), Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx4100, Succ(vx30100)), Succ(vx30100))
10.42/4.21	
10.42/4.21	The set Q consists of the following terms:
10.42/4.21	
10.42/4.21	   new_primMulNat0(Succ(x0), Zero)
10.42/4.21	   new_primMinusNat0(Zero, Succ(x0))
10.42/4.21	   new_primPlusInt(Neg(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Neg(x1))
10.42/4.21	   new_primMulNat0(Zero, Zero)
10.42/4.21	   new_primMinusNat0(Zero, Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Zero)
10.42/4.21	   new_primPlusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primMulNat0(Zero, Succ(x0))
10.42/4.21	   new_primMinusNat0(Succ(x0), Zero)
10.42/4.21	   new_sr(Pos(x0), Neg(x1))
10.42/4.21	   new_sr(Neg(x0), Pos(x1))
10.42/4.21	   new_primPlusInt(Pos(x0), Neg(x1))
10.42/4.21	   new_primPlusInt(Neg(x0), Pos(x1))
10.42/4.21	   new_primMinusNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Zero)
10.42/4.21	   new_primPlusInt(Pos(x0), Pos(x1))
10.42/4.21	   new_sr(Pos(x0), Pos(x1))
10.42/4.21	   new_primMulNat0(Succ(x0), Succ(x1))
10.42/4.21	   new_primPlusNat0(Zero, Succ(x0))
10.42/4.21	
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(18) QDPSizeChangeProof (EQUIVALENT)
10.42/4.21	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. 
10.42/4.21	
10.42/4.21	From the DPs we obtained the following set of size-change graphs:
10.42/4.21	*new_enforceWHNF(Float(vx40, vx41), Float(vx300, vx301), vx5, :(vx310, vx311)) -> new_enforceWHNF(Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx310, Float(new_primPlusInt(new_sr(vx40, vx301), new_sr(vx300, vx41)), new_sr(vx41, vx301)), vx311)
10.42/4.21	The graph contains the following edges 4 > 2, 4 > 4
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(19)
10.42/4.21	YES
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(20)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primMulNat(Succ(vx4100), Succ(vx30100)) -> new_primMulNat(vx4100, Succ(vx30100))
10.42/4.21	
10.42/4.21	R is empty.
10.42/4.21	Q is empty.
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(21) QDPSizeChangeProof (EQUIVALENT)
10.42/4.21	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. 
10.42/4.21	
10.42/4.21	From the DPs we obtained the following set of size-change graphs:
10.42/4.21	*new_primMulNat(Succ(vx4100), Succ(vx30100)) -> new_primMulNat(vx4100, Succ(vx30100))
10.42/4.21	The graph contains the following edges 1 > 1, 2 >= 2
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(22)
10.42/4.21	YES
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(23)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primMinusNat(Succ(vx700), Succ(vx600)) -> new_primMinusNat(vx700, vx600)
10.42/4.21	
10.42/4.21	R is empty.
10.42/4.21	Q is empty.
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(24) QDPSizeChangeProof (EQUIVALENT)
10.42/4.21	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. 
10.42/4.21	
10.42/4.21	From the DPs we obtained the following set of size-change graphs:
10.42/4.21	*new_primMinusNat(Succ(vx700), Succ(vx600)) -> new_primMinusNat(vx700, vx600)
10.42/4.21	The graph contains the following edges 1 > 1, 2 > 2
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(25)
10.42/4.21	YES
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(26)
10.42/4.21	Obligation:
10.42/4.21	Q DP problem:
10.42/4.21	The TRS P consists of the following rules:
10.42/4.21	
10.42/4.21	   new_primPlusNat(Succ(vx700), Succ(vx600)) -> new_primPlusNat(vx700, vx600)
10.42/4.21	
10.42/4.21	R is empty.
10.42/4.21	Q is empty.
10.42/4.21	We have to consider all minimal (P,Q,R)-chains.
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(27) QDPSizeChangeProof (EQUIVALENT)
10.42/4.21	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. 
10.42/4.21	
10.42/4.21	From the DPs we obtained the following set of size-change graphs:
10.42/4.21	*new_primPlusNat(Succ(vx700), Succ(vx600)) -> new_primPlusNat(vx700, vx600)
10.42/4.21	The graph contains the following edges 1 > 1, 2 > 2
10.42/4.21	
10.42/4.21	
10.42/4.21	----------------------------------------
10.42/4.21	
10.42/4.21	(28)
10.42/4.21	YES
10.53/4.27	EOF