40.85/21.36	YES
42.75/21.89	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
42.75/21.89	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
42.75/21.89	
42.75/21.89	
42.75/21.89	H-Termination with start terms of the given HASKELL could be proven:
42.75/21.89	
42.75/21.89	(0) HASKELL
42.75/21.89	(1) BR [EQUIVALENT, 0 ms]
42.75/21.89	(2) HASKELL
42.75/21.89	(3) COR [EQUIVALENT, 0 ms]
42.75/21.89	(4) HASKELL
42.75/21.89	(5) NumRed [SOUND, 0 ms]
42.75/21.89	(6) HASKELL
42.75/21.89	(7) Narrow [SOUND, 0 ms]
42.75/21.89	(8) QDP
42.75/21.89	(9) QDPPairToRuleProof [EQUIVALENT, 0 ms]
42.75/21.89	(10) AND
42.75/21.89	    (11) QDP
42.75/21.89	        (12) DependencyGraphProof [EQUIVALENT, 0 ms]
42.75/21.89	        (13) QDP
42.75/21.89	        (14) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (15) QDP
42.75/21.89	        (16) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (17) QDP
42.75/21.89	        (18) UsableRulesProof [EQUIVALENT, 0 ms]
42.75/21.89	        (19) QDP
42.75/21.89	        (20) QReductionProof [EQUIVALENT, 0 ms]
42.75/21.89	        (21) QDP
42.75/21.89	        (22) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (23) QDP
42.75/21.89	        (24) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (25) QDP
42.75/21.89	        (26) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (27) QDP
42.75/21.89	        (28) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (29) QDP
42.75/21.89	        (30) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (31) QDP
42.75/21.89	        (32) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (33) QDP
42.75/21.89	        (34) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (35) QDP
42.75/21.89	        (36) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (37) QDP
42.75/21.89	        (38) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (39) QDP
42.75/21.89	        (40) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (41) QDP
42.75/21.89	        (42) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (43) QDP
42.75/21.89	        (44) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (45) QDP
42.75/21.89	        (46) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (47) QDP
42.75/21.89	        (48) UsableRulesProof [EQUIVALENT, 0 ms]
42.75/21.89	        (49) QDP
42.75/21.89	        (50) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (51) QDP
42.75/21.89	        (52) UsableRulesProof [EQUIVALENT, 0 ms]
42.75/21.89	        (53) QDP
42.75/21.89	        (54) QReductionProof [EQUIVALENT, 0 ms]
42.75/21.89	        (55) QDP
42.75/21.89	        (56) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (57) QDP
42.75/21.89	        (58) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (59) QDP
42.75/21.89	        (60) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (61) QDP
42.75/21.89	        (62) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (63) QDP
42.75/21.89	        (64) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (65) QDP
42.75/21.89	        (66) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (67) QDP
42.75/21.89	        (68) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (69) QDP
42.75/21.89	        (70) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (71) QDP
42.75/21.89	        (72) QDPOrderProof [EQUIVALENT, 0 ms]
42.75/21.89	        (73) QDP
42.75/21.89	        (74) TransformationProof [EQUIVALENT, 0 ms]
42.75/21.89	        (75) QDP
42.75/21.89	        (76) DependencyGraphProof [EQUIVALENT, 0 ms]
42.75/21.89	        (77) QDP
42.75/21.89	        (78) InductionCalculusProof [EQUIVALENT, 0 ms]
42.75/21.89	        (79) QDP
42.75/21.89	        (80) NonInfProof [EQUIVALENT, 130 ms]
42.75/21.89	        (81) QDP
42.75/21.89	        (82) InductionCalculusProof [EQUIVALENT, 0 ms]
42.75/21.89	        (83) QDP
42.75/21.89	        (84) NonInfProof [EQUIVALENT, 40 ms]
42.75/21.89	        (85) AND
42.75/21.89	            (86) QDP
42.75/21.89	                (87) DependencyGraphProof [EQUIVALENT, 0 ms]
42.75/21.89	                (88) TRUE
42.75/21.89	            (89) QDP
42.75/21.89	                (90) InductionCalculusProof [EQUIVALENT, 0 ms]
42.75/21.89	                (91) QDP
42.75/21.89	                (92) NonInfProof [EQUIVALENT, 38 ms]
42.75/21.89	                (93) QDP
42.75/21.89	                (94) DependencyGraphProof [EQUIVALENT, 0 ms]
42.75/21.89	                (95) TRUE
42.75/21.89	    (96) QDP
42.75/21.89	        (97) QDPSizeChangeProof [EQUIVALENT, 0 ms]
42.75/21.89	        (98) YES
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(0)
42.75/21.89	Obligation:
42.75/21.89	mainModule Main 
42.75/21.89	module Main where {
42.75/21.89	import qualified Prelude;
42.75/21.89	}
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(1) BR (EQUIVALENT)
42.75/21.89	Replaced joker patterns by fresh variables and removed binding patterns.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(2)
42.75/21.89	Obligation:
42.75/21.89	mainModule Main 
42.75/21.89	module Main where {
42.75/21.89	import qualified Prelude;
42.75/21.89	}
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(3) COR (EQUIVALENT)
42.75/21.89	Cond Reductions:
42.75/21.89	The following Function with conditions
42.75/21.89	"takeWhile p [] = [];
42.75/21.89	takeWhile p (x : xs)|p xx : takeWhile p xs|otherwise[];
42.75/21.89	"
42.75/21.89	is transformed to
42.75/21.89	"takeWhile p [] = takeWhile3 p [];
42.75/21.89	takeWhile p (x : xs) = takeWhile2 p (x : xs);
42.75/21.89	"
42.75/21.89	"takeWhile1 p x xs True = x : takeWhile p xs;
42.75/21.89	takeWhile1 p x xs False = takeWhile0 p x xs otherwise;
42.75/21.89	"
42.75/21.89	"takeWhile0 p x xs True = [];
42.75/21.89	"
42.75/21.89	"takeWhile2 p (x : xs) = takeWhile1 p x xs (p x);
42.75/21.89	"
42.75/21.89	"takeWhile3 p [] = [];
42.75/21.89	takeWhile3 vz wu = takeWhile2 vz wu;
42.75/21.89	"
42.75/21.89	The following Function with conditions
42.75/21.89	"undefined |Falseundefined;
42.75/21.89	"
42.75/21.89	is transformed to
42.75/21.89	"undefined  = undefined1;
42.75/21.89	"
42.75/21.89	"undefined0 True = undefined;
42.75/21.89	"
42.75/21.89	"undefined1  = undefined0 False;
42.75/21.89	"
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(4)
42.75/21.89	Obligation:
42.75/21.89	mainModule Main 
42.75/21.89	module Main where {
42.75/21.89	import qualified Prelude;
42.75/21.89	}
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(5) NumRed (SOUND)
42.75/21.89	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(6)
42.75/21.89	Obligation:
42.75/21.89	mainModule Main 
42.75/21.89	module Main where {
42.75/21.89	import qualified Prelude;
42.75/21.89	}
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(7) Narrow (SOUND)
42.75/21.89	Haskell To QDPs
42.75/21.89	
42.75/21.89	digraph dp_graph {
42.75/21.89	node [outthreshold=100, inthreshold=100];1[label="enumFromTo",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
42.75/21.89	3[label="enumFromTo wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
42.75/21.89	4[label="enumFromTo wv3 wv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
42.75/21.89	5[label="map toEnum (enumFromTo (fromEnum wv3) (fromEnum wv4))",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
42.75/21.89	6[label="map toEnum (numericEnumFromTo (fromEnum wv3) (fromEnum wv4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
42.75/21.89	7[label="map toEnum (takeWhile (flip (<=) (fromEnum wv4)) (numericEnumFrom (fromEnum wv3)))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
42.75/21.89	8[label="map toEnum (takeWhile (flip (<=) (fromEnum wv4)) (fromEnum wv3 : (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
42.75/21.89	9[label="map toEnum (takeWhile2 (flip (<=) (fromEnum wv4)) (fromEnum wv3 : (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3];
42.75/21.89	10[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (fromEnum wv3) (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero))) (flip (<=) (fromEnum wv4) (fromEnum wv3)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3];
42.75/21.89	11[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (fromEnum wv3) (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero))) ((<=) fromEnum wv3 fromEnum wv4))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
42.75/21.89	12[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (fromEnum wv3) (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero))) (compare (fromEnum wv3) (fromEnum wv4) /= GT))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
42.75/21.89	13[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (fromEnum wv3) (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero))) (not (compare (fromEnum wv3) (fromEnum wv4) == GT)))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
42.75/21.89	14[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (fromEnum wv3) (numericEnumFrom $! fromEnum wv3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromEnum wv3) (fromEnum wv4) == GT)))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
42.75/21.89	15[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (primCharToInt wv3) (numericEnumFrom $! primCharToInt wv3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primCharToInt wv3) (fromEnum wv4) == GT)))",fontsize=16,color="burlywood",shape="box"];559[label="wv3/Char wv30",fontsize=10,color="white",style="solid",shape="box"];15 -> 559[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	559 -> 16[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	16[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (primCharToInt (Char wv30)) (numericEnumFrom $! primCharToInt (Char wv30) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primCharToInt (Char wv30)) (fromEnum wv4) == GT)))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3];
42.75/21.89	17[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (Pos wv30) (numericEnumFrom $! Pos wv30 + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos wv30) (fromEnum wv4) == GT)))",fontsize=16,color="burlywood",shape="box"];560[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];17 -> 560[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	560 -> 18[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	561[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 561[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	561 -> 19[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	18[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ wv300)) (fromEnum wv4) == GT)))",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
42.75/21.89	19[label="map toEnum (takeWhile1 (flip (<=) (fromEnum wv4)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (fromEnum wv4) == GT)))",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
42.75/21.89	20[label="map toEnum (takeWhile1 (flip (<=) (primCharToInt wv4)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ wv300)) (primCharToInt wv4) == GT)))",fontsize=16,color="burlywood",shape="box"];562[label="wv4/Char wv40",fontsize=10,color="white",style="solid",shape="box"];20 -> 562[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	562 -> 22[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	21[label="map toEnum (takeWhile1 (flip (<=) (primCharToInt wv4)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (primCharToInt wv4) == GT)))",fontsize=16,color="burlywood",shape="box"];563[label="wv4/Char wv40",fontsize=10,color="white",style="solid",shape="box"];21 -> 563[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	563 -> 23[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	22[label="map toEnum (takeWhile1 (flip (<=) (primCharToInt (Char wv40))) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ wv300)) (primCharToInt (Char wv40)) == GT)))",fontsize=16,color="black",shape="box"];22 -> 24[label="",style="solid", color="black", weight=3];
42.75/21.89	23[label="map toEnum (takeWhile1 (flip (<=) (primCharToInt (Char wv40))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (primCharToInt (Char wv40)) == GT)))",fontsize=16,color="black",shape="box"];23 -> 25[label="",style="solid", color="black", weight=3];
42.75/21.89	24[label="map toEnum (takeWhile1 (flip (<=) (Pos wv40)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ wv300)) (Pos wv40) == GT)))",fontsize=16,color="black",shape="triangle"];24 -> 26[label="",style="solid", color="black", weight=3];
42.75/21.89	25[label="map toEnum (takeWhile1 (flip (<=) (Pos wv40)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (Pos wv40) == GT)))",fontsize=16,color="burlywood",shape="box"];564[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];25 -> 564[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	564 -> 27[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	565[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 565[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	565 -> 28[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	26[label="map toEnum (takeWhile1 (flip (<=) (Pos wv40)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv300) wv40 == GT)))",fontsize=16,color="burlywood",shape="box"];566[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];26 -> 566[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	566 -> 29[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	567[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 567[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	567 -> 30[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	27[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (Pos (Succ wv400)) == GT)))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
42.75/21.89	28[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
42.75/21.89	29[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv300) (Succ wv400) == GT)))",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
42.75/21.89	30[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv300) Zero == GT)))",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3];
42.75/21.89	31[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (primCmpNat Zero (Succ wv400) == GT)))",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3];
42.75/21.89	32[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3];
42.75/21.89	33 -> 450[label="",style="dashed", color="red", weight=0];
42.75/21.89	33[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (primCmpNat wv300 wv400 == GT)))",fontsize=16,color="magenta"];33 -> 451[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	33 -> 452[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	33 -> 453[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	33 -> 454[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	34[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not (GT == GT)))",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3];
42.75/21.89	35[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not (LT == GT)))",fontsize=16,color="black",shape="box"];35 -> 40[label="",style="solid", color="black", weight=3];
42.75/21.89	36[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not False))",fontsize=16,color="black",shape="box"];36 -> 41[label="",style="solid", color="black", weight=3];
42.75/21.89	451[label="wv300",fontsize=16,color="green",shape="box"];452[label="wv300",fontsize=16,color="green",shape="box"];453[label="wv400",fontsize=16,color="green",shape="box"];454[label="wv400",fontsize=16,color="green",shape="box"];450[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat wv24 wv25 == GT)))",fontsize=16,color="burlywood",shape="triangle"];568[label="wv24/Succ wv240",fontsize=10,color="white",style="solid",shape="box"];450 -> 568[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	568 -> 491[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	569[label="wv24/Zero",fontsize=10,color="white",style="solid",shape="box"];450 -> 569[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	569 -> 492[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	39[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) (not True))",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
42.75/21.89	40[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) (not False))",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
42.75/21.89	41[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) True)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
42.75/21.89	491[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv240) wv25 == GT)))",fontsize=16,color="burlywood",shape="box"];570[label="wv25/Succ wv250",fontsize=10,color="white",style="solid",shape="box"];491 -> 570[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	570 -> 493[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	571[label="wv25/Zero",fontsize=10,color="white",style="solid",shape="box"];491 -> 571[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	571 -> 494[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	492[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat Zero wv25 == GT)))",fontsize=16,color="burlywood",shape="box"];572[label="wv25/Succ wv250",fontsize=10,color="white",style="solid",shape="box"];492 -> 572[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	572 -> 495[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	573[label="wv25/Zero",fontsize=10,color="white",style="solid",shape="box"];492 -> 573[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	573 -> 496[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	46[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) False)",fontsize=16,color="black",shape="box"];46 -> 53[label="",style="solid", color="black", weight=3];
42.75/21.89	47[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos Zero) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))) True)",fontsize=16,color="black",shape="box"];47 -> 54[label="",style="solid", color="black", weight=3];
42.75/21.89	48[label="map toEnum (Pos Zero : takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3];
42.75/21.89	493[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv240) (Succ wv250) == GT)))",fontsize=16,color="black",shape="box"];493 -> 497[label="",style="solid", color="black", weight=3];
42.75/21.89	494[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ wv240) Zero == GT)))",fontsize=16,color="black",shape="box"];494 -> 498[label="",style="solid", color="black", weight=3];
42.75/21.89	495[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat Zero (Succ wv250) == GT)))",fontsize=16,color="black",shape="box"];495 -> 499[label="",style="solid", color="black", weight=3];
42.75/21.89	496[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat Zero Zero == GT)))",fontsize=16,color="black",shape="box"];496 -> 500[label="",style="solid", color="black", weight=3];
42.75/21.89	53[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) otherwise)",fontsize=16,color="black",shape="box"];53 -> 61[label="",style="solid", color="black", weight=3];
42.75/21.89	54[label="map toEnum (Pos Zero : takeWhile (flip (<=) (Pos (Succ wv400))) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];54 -> 62[label="",style="solid", color="black", weight=3];
42.75/21.89	55[label="toEnum (Pos Zero) : map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="green",shape="box"];55 -> 63[label="",style="dashed", color="green", weight=3];
42.75/21.89	55 -> 64[label="",style="dashed", color="green", weight=3];
42.75/21.89	497 -> 450[label="",style="dashed", color="red", weight=0];
42.75/21.89	497[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (primCmpNat wv240 wv250 == GT)))",fontsize=16,color="magenta"];497 -> 501[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	497 -> 502[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	498[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (GT == GT)))",fontsize=16,color="black",shape="box"];498 -> 503[label="",style="solid", color="black", weight=3];
42.75/21.89	499[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (LT == GT)))",fontsize=16,color="black",shape="box"];499 -> 504[label="",style="solid", color="black", weight=3];
42.75/21.89	500[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];500 -> 505[label="",style="solid", color="black", weight=3];
42.75/21.89	61[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos (Succ wv300)) (numericEnumFrom $! Pos (Succ wv300) + fromInt (Pos (Succ Zero))) True)",fontsize=16,color="black",shape="box"];61 -> 72[label="",style="solid", color="black", weight=3];
42.75/21.89	62[label="toEnum (Pos Zero) : map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="green",shape="box"];62 -> 73[label="",style="dashed", color="green", weight=3];
42.75/21.89	62 -> 74[label="",style="dashed", color="green", weight=3];
42.75/21.89	63[label="toEnum (Pos Zero)",fontsize=16,color="black",shape="triangle"];63 -> 75[label="",style="solid", color="black", weight=3];
42.75/21.89	64[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];64 -> 76[label="",style="solid", color="black", weight=3];
42.75/21.89	501[label="wv240",fontsize=16,color="green",shape="box"];502[label="wv250",fontsize=16,color="green",shape="box"];503[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not True))",fontsize=16,color="black",shape="box"];503 -> 506[label="",style="solid", color="black", weight=3];
42.75/21.89	504[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not False))",fontsize=16,color="black",shape="triangle"];504 -> 507[label="",style="solid", color="black", weight=3];
42.75/21.89	505 -> 504[label="",style="dashed", color="red", weight=0];
42.75/21.89	505[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) (not False))",fontsize=16,color="magenta"];72[label="map toEnum []",fontsize=16,color="black",shape="triangle"];72 -> 84[label="",style="solid", color="black", weight=3];
42.75/21.89	73 -> 63[label="",style="dashed", color="red", weight=0];
42.75/21.89	73[label="toEnum (Pos Zero)",fontsize=16,color="magenta"];74[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];74 -> 85[label="",style="solid", color="black", weight=3];
42.75/21.89	75[label="primIntToChar (Pos Zero)",fontsize=16,color="black",shape="box"];75 -> 86[label="",style="solid", color="black", weight=3];
42.75/21.89	76[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];76 -> 87[label="",style="solid", color="black", weight=3];
42.75/21.89	506[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) False)",fontsize=16,color="black",shape="box"];506 -> 508[label="",style="solid", color="black", weight=3];
42.75/21.89	507[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) True)",fontsize=16,color="black",shape="box"];507 -> 509[label="",style="solid", color="black", weight=3];
42.75/21.89	84[label="[]",fontsize=16,color="green",shape="box"];85[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (Pos Zero + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];85 -> 96[label="",style="solid", color="black", weight=3];
42.75/21.89	86[label="Char Zero",fontsize=16,color="green",shape="box"];87[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos Zero + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];87 -> 97[label="",style="solid", color="black", weight=3];
42.75/21.89	508[label="map toEnum (takeWhile0 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) otherwise)",fontsize=16,color="black",shape="box"];508 -> 510[label="",style="solid", color="black", weight=3];
42.75/21.89	509[label="map toEnum (Pos (Succ wv23) : takeWhile (flip (<=) (Pos (Succ wv22))) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];509 -> 511[label="",style="solid", color="black", weight=3];
42.75/21.89	96[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (enforceWHNF (WHNF (Pos Zero + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos Zero + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];96 -> 110[label="",style="solid", color="black", weight=3];
42.75/21.89	97[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];97 -> 111[label="",style="solid", color="black", weight=3];
42.75/21.89	510[label="map toEnum (takeWhile0 (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23)) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))) True)",fontsize=16,color="black",shape="box"];510 -> 512[label="",style="solid", color="black", weight=3];
42.75/21.89	511[label="toEnum (Pos (Succ wv23)) : map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))))",fontsize=16,color="green",shape="box"];511 -> 513[label="",style="dashed", color="green", weight=3];
42.75/21.89	511 -> 514[label="",style="dashed", color="green", weight=3];
42.75/21.89	110[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos Zero) (fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];110 -> 122[label="",style="solid", color="black", weight=3];
42.75/21.89	111[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos Zero) (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];111 -> 123[label="",style="solid", color="black", weight=3];
42.75/21.89	512[label="map toEnum []",fontsize=16,color="black",shape="box"];512 -> 515[label="",style="solid", color="black", weight=3];
42.75/21.89	513[label="toEnum (Pos (Succ wv23))",fontsize=16,color="blue",shape="box"];574[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];513 -> 574[label="",style="solid", color="blue", weight=9];
42.75/21.89	574 -> 516[label="",style="solid", color="blue", weight=3];
42.75/21.89	575[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];513 -> 575[label="",style="solid", color="blue", weight=9];
42.75/21.89	575 -> 517[label="",style="solid", color="blue", weight=3];
42.75/21.89	576[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];513 -> 576[label="",style="solid", color="blue", weight=9];
42.75/21.89	576 -> 518[label="",style="solid", color="blue", weight=3];
42.75/21.89	577[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];513 -> 577[label="",style="solid", color="blue", weight=9];
42.75/21.89	577 -> 519[label="",style="solid", color="blue", weight=3];
42.75/21.89	578[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];513 -> 578[label="",style="solid", color="blue", weight=9];
42.75/21.89	578 -> 520[label="",style="solid", color="blue", weight=3];
42.75/21.89	579[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];513 -> 579[label="",style="solid", color="blue", weight=9];
42.75/21.89	579 -> 521[label="",style="solid", color="blue", weight=3];
42.75/21.89	580[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];513 -> 580[label="",style="solid", color="blue", weight=9];
42.75/21.89	580 -> 522[label="",style="solid", color="blue", weight=3];
42.75/21.89	581[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];513 -> 581[label="",style="solid", color="blue", weight=9];
42.75/21.89	581 -> 523[label="",style="solid", color="blue", weight=3];
42.75/21.89	582[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];513 -> 582[label="",style="solid", color="blue", weight=9];
42.75/21.89	582 -> 524[label="",style="solid", color="blue", weight=3];
42.75/21.89	514[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (numericEnumFrom $! Pos (Succ wv23) + fromInt (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];514 -> 525[label="",style="solid", color="black", weight=3];
42.75/21.89	122[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (enforceWHNF (WHNF (primPlusInt (Pos Zero) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos Zero) (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];122 -> 135[label="",style="solid", color="black", weight=3];
42.75/21.89	123[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos (primPlusNat Zero (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero))))))",fontsize=16,color="black",shape="box"];123 -> 136[label="",style="solid", color="black", weight=3];
42.75/21.89	515[label="[]",fontsize=16,color="green",shape="box"];516[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];516 -> 526[label="",style="solid", color="black", weight=3];
42.75/21.89	517[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];517 -> 527[label="",style="solid", color="black", weight=3];
42.75/21.89	518[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];518 -> 528[label="",style="solid", color="black", weight=3];
42.75/21.89	519[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];519 -> 529[label="",style="solid", color="black", weight=3];
42.75/21.89	520[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];520 -> 530[label="",style="solid", color="black", weight=3];
42.75/21.89	521[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];521 -> 531[label="",style="solid", color="black", weight=3];
42.75/21.89	522[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];522 -> 532[label="",style="solid", color="black", weight=3];
42.75/21.89	523[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];523 -> 533[label="",style="solid", color="black", weight=3];
42.75/21.89	524[label="toEnum (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];524 -> 534[label="",style="solid", color="black", weight=3];
42.75/21.89	525[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (Pos (Succ wv23) + fromInt (Pos (Succ Zero)) `seq` numericEnumFrom (Pos (Succ wv23) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];525 -> 535[label="",style="solid", color="black", weight=3];
42.75/21.89	135[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (enforceWHNF (WHNF (Pos (primPlusNat Zero (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero))))))",fontsize=16,color="black",shape="box"];135 -> 151[label="",style="solid", color="black", weight=3];
42.75/21.89	136[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero)))))",fontsize=16,color="black",shape="box"];136 -> 152[label="",style="solid", color="black", weight=3];
42.75/21.89	526[label="primIntToChar (Pos (Succ wv23))",fontsize=16,color="black",shape="box"];526 -> 536[label="",style="solid", color="black", weight=3];
42.75/21.89	527[label="error []",fontsize=16,color="red",shape="box"];528[label="error []",fontsize=16,color="red",shape="box"];529[label="error []",fontsize=16,color="red",shape="box"];530[label="error []",fontsize=16,color="red",shape="box"];531[label="error []",fontsize=16,color="red",shape="box"];532[label="error []",fontsize=16,color="red",shape="box"];533[label="error []",fontsize=16,color="red",shape="box"];534[label="error []",fontsize=16,color="red",shape="box"];535[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (enforceWHNF (WHNF (Pos (Succ wv23) + fromInt (Pos (Succ Zero)))) (numericEnumFrom (Pos (Succ wv23) + fromInt (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];535 -> 537[label="",style="solid", color="black", weight=3];
42.75/21.89	151[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (numericEnumFrom (Pos (primPlusNat Zero (Succ Zero)))))",fontsize=16,color="black",shape="box"];151 -> 165[label="",style="solid", color="black", weight=3];
42.75/21.89	152[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];152 -> 166[label="",style="solid", color="black", weight=3];
42.75/21.89	536[label="Char (Succ wv23)",fontsize=16,color="green",shape="box"];537[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (enforceWHNF (WHNF (primPlusInt (Pos (Succ wv23)) (fromInt (Pos (Succ Zero))))) (numericEnumFrom (primPlusInt (Pos (Succ wv23)) (fromInt (Pos (Succ Zero)))))))",fontsize=16,color="black",shape="box"];537 -> 538[label="",style="solid", color="black", weight=3];
42.75/21.89	165[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];165 -> 180[label="",style="solid", color="black", weight=3];
42.75/21.89	166[label="map toEnum (takeWhile2 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];166 -> 181[label="",style="solid", color="black", weight=3];
42.75/21.89	538[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (enforceWHNF (WHNF (primPlusInt (Pos (Succ wv23)) (Pos (Succ Zero)))) (numericEnumFrom (primPlusInt (Pos (Succ wv23)) (Pos (Succ Zero))))))",fontsize=16,color="black",shape="box"];538 -> 539[label="",style="solid", color="black", weight=3];
42.75/21.89	180[label="map toEnum (takeWhile2 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];180 -> 198[label="",style="solid", color="black", weight=3];
42.75/21.89	181[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (flip (<=) (Pos Zero) (Pos (primPlusNat Zero (Succ Zero)))))",fontsize=16,color="black",shape="box"];181 -> 199[label="",style="solid", color="black", weight=3];
42.75/21.89	539[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (enforceWHNF (WHNF (Pos (primPlusNat (Succ wv23) (Succ Zero)))) (numericEnumFrom (Pos (primPlusNat (Succ wv23) (Succ Zero))))))",fontsize=16,color="black",shape="box"];539 -> 540[label="",style="solid", color="black", weight=3];
42.75/21.89	198[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (flip (<=) (Pos (Succ wv400)) (Pos (primPlusNat Zero (Succ Zero)))))",fontsize=16,color="black",shape="box"];198 -> 214[label="",style="solid", color="black", weight=3];
42.75/21.89	199[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) ((<=) Pos (primPlusNat Zero (Succ Zero)) Pos Zero))",fontsize=16,color="black",shape="box"];199 -> 215[label="",style="solid", color="black", weight=3];
42.75/21.89	540[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (numericEnumFrom (Pos (primPlusNat (Succ wv23) (Succ Zero)))))",fontsize=16,color="black",shape="box"];540 -> 541[label="",style="solid", color="black", weight=3];
42.75/21.89	214[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) ((<=) Pos (primPlusNat Zero (Succ Zero)) Pos (Succ wv400)))",fontsize=16,color="black",shape="box"];214 -> 231[label="",style="solid", color="black", weight=3];
42.75/21.89	215[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (compare (Pos (primPlusNat Zero (Succ Zero))) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];215 -> 232[label="",style="solid", color="black", weight=3];
42.75/21.89	541[label="map toEnum (takeWhile (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];541 -> 542[label="",style="solid", color="black", weight=3];
42.75/21.89	231[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (compare (Pos (primPlusNat Zero (Succ Zero))) (Pos (Succ wv400)) /= GT))",fontsize=16,color="black",shape="box"];231 -> 253[label="",style="solid", color="black", weight=3];
42.75/21.89	232[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (compare (Pos (primPlusNat Zero (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];232 -> 254[label="",style="solid", color="black", weight=3];
42.75/21.89	542[label="map toEnum (takeWhile2 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero)) : (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];542 -> 543[label="",style="solid", color="black", weight=3];
42.75/21.89	253[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (compare (Pos (primPlusNat Zero (Succ Zero))) (Pos (Succ wv400)) == GT)))",fontsize=16,color="black",shape="box"];253 -> 270[label="",style="solid", color="black", weight=3];
42.75/21.89	254[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primPlusNat Zero (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];254 -> 271[label="",style="solid", color="black", weight=3];
42.75/21.89	543[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero))) (flip (<=) (Pos (Succ wv22)) (Pos (primPlusNat (Succ wv23) (Succ Zero)))))",fontsize=16,color="black",shape="box"];543 -> 544[label="",style="solid", color="black", weight=3];
42.75/21.89	270[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (primPlusNat Zero (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat Zero (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primPlusNat Zero (Succ Zero))) (Pos (Succ wv400)) == GT)))",fontsize=16,color="black",shape="box"];270 -> 288[label="",style="solid", color="black", weight=3];
42.75/21.89	271 -> 24[label="",style="dashed", color="red", weight=0];
42.75/21.89	271[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos (Succ Zero)) (numericEnumFrom $! Pos (Succ Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="magenta"];271 -> 289[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	271 -> 290[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	544[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero))) ((<=) Pos (primPlusNat (Succ wv23) (Succ Zero)) Pos (Succ wv22)))",fontsize=16,color="black",shape="box"];544 -> 545[label="",style="solid", color="black", weight=3];
42.75/21.89	288 -> 24[label="",style="dashed", color="red", weight=0];
42.75/21.89	288[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv400))) (Pos (Succ Zero)) (numericEnumFrom $! Pos (Succ Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ Zero)) (Pos (Succ wv400)) == GT)))",fontsize=16,color="magenta"];288 -> 310[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	288 -> 311[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	289[label="Zero",fontsize=16,color="green",shape="box"];290[label="Zero",fontsize=16,color="green",shape="box"];545[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero))) (compare (Pos (primPlusNat (Succ wv23) (Succ Zero))) (Pos (Succ wv22)) /= GT))",fontsize=16,color="black",shape="box"];545 -> 546[label="",style="solid", color="black", weight=3];
42.75/21.89	310[label="Succ wv400",fontsize=16,color="green",shape="box"];311[label="Zero",fontsize=16,color="green",shape="box"];546[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (compare (Pos (primPlusNat (Succ wv23) (Succ Zero))) (Pos (Succ wv22)) == GT)))",fontsize=16,color="black",shape="box"];546 -> 547[label="",style="solid", color="black", weight=3];
42.75/21.89	547[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (primPlusNat (Succ wv23) (Succ Zero))) (numericEnumFrom $! Pos (primPlusNat (Succ wv23) (Succ Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primPlusNat (Succ wv23) (Succ Zero))) (Pos (Succ wv22)) == GT)))",fontsize=16,color="black",shape="box"];547 -> 548[label="",style="solid", color="black", weight=3];
42.75/21.89	548[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ (Succ (primPlusNat wv23 Zero)))) (numericEnumFrom $! Pos (Succ (Succ (primPlusNat wv23 Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (Succ (Succ (primPlusNat wv23 Zero)))) (Pos (Succ wv22)) == GT)))",fontsize=16,color="black",shape="box"];548 -> 549[label="",style="solid", color="black", weight=3];
42.75/21.89	549 -> 450[label="",style="dashed", color="red", weight=0];
42.75/21.89	549[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ wv22))) (Pos (Succ (Succ (primPlusNat wv23 Zero)))) (numericEnumFrom $! Pos (Succ (Succ (primPlusNat wv23 Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpNat (Succ (Succ (primPlusNat wv23 Zero))) (Succ wv22) == GT)))",fontsize=16,color="magenta"];549 -> 550[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	549 -> 551[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	549 -> 552[label="",style="dashed", color="magenta", weight=3];
42.75/21.89	550[label="Succ (primPlusNat wv23 Zero)",fontsize=16,color="green",shape="box"];550 -> 553[label="",style="dashed", color="green", weight=3];
42.75/21.89	551[label="Succ (Succ (primPlusNat wv23 Zero))",fontsize=16,color="green",shape="box"];551 -> 554[label="",style="dashed", color="green", weight=3];
42.75/21.89	552[label="Succ wv22",fontsize=16,color="green",shape="box"];553[label="primPlusNat wv23 Zero",fontsize=16,color="burlywood",shape="triangle"];583[label="wv23/Succ wv230",fontsize=10,color="white",style="solid",shape="box"];553 -> 583[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	583 -> 555[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	584[label="wv23/Zero",fontsize=10,color="white",style="solid",shape="box"];553 -> 584[label="",style="solid", color="burlywood", weight=9];
42.75/21.89	584 -> 556[label="",style="solid", color="burlywood", weight=3];
42.75/21.89	554 -> 553[label="",style="dashed", color="red", weight=0];
42.75/21.89	554[label="primPlusNat wv23 Zero",fontsize=16,color="magenta"];555[label="primPlusNat (Succ wv230) Zero",fontsize=16,color="black",shape="box"];555 -> 557[label="",style="solid", color="black", weight=3];
42.75/21.89	556[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];556 -> 558[label="",style="solid", color="black", weight=3];
42.75/21.89	557[label="Succ wv230",fontsize=16,color="green",shape="box"];558[label="Zero",fontsize=16,color="green",shape="box"];}
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(8)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   new_map0(wv22, wv23, h) -> new_map(wv22, Succ(new_primPlusNat(wv23)), Succ(Succ(new_primPlusNat(wv23))), Succ(wv22), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> new_map(wv22, Succ(new_primPlusNat(wv23)), Succ(Succ(new_primPlusNat(wv23))), Succ(wv22), h)
42.75/21.89	   new_map(wv22, wv23, Succ(wv240), Succ(wv250), h) -> new_map(wv22, wv23, wv240, wv250, h)
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(9) QDPPairToRuleProof (EQUIVALENT)
42.75/21.89	The dependency pair new_map(wv22, wv23, Succ(wv240), Succ(wv250), h) -> new_map(wv22, wv23, wv240, wv250, h) was transformed to the following new rules:
42.75/21.89	   anew_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	the following new pairs maintain the fan-in:
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	
42.75/21.89	the following new pairs maintain the fan-out:
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(10)
42.75/21.89	Complex Obligation (AND)
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(11)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   new_map0(wv22, wv23, h) -> new_map(wv22, Succ(new_primPlusNat(wv23)), Succ(Succ(new_primPlusNat(wv23))), Succ(wv22), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> new_map(wv22, Succ(new_primPlusNat(wv23)), Succ(Succ(new_primPlusNat(wv23))), Succ(wv22), h)
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   anew_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(12) DependencyGraphProof (EQUIVALENT)
42.75/21.89	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(13)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   anew_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(14) TransformationProof (EQUIVALENT)
42.75/21.89	By rewriting [LPAR04] the rule new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22))) at position [3] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)),new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(15)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22)))
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   anew_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(16) TransformationProof (EQUIVALENT)
42.75/21.89	By rewriting [LPAR04] the rule new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, anew_new_map(Succ(Succ(new_primPlusNat(wv23))), Succ(wv22))) at position [3] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)),new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(17)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   anew_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(18) UsableRulesProof (EQUIVALENT)
42.75/21.89	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.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(19)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(20) QReductionProof (EQUIVALENT)
42.75/21.89	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
42.75/21.89	
42.75/21.89	   anew_new_map(Succ(x0), Succ(x1))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(21)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(22) TransformationProof (EQUIVALENT)
42.75/21.89	By narrowing [LPAR04] the rule new_map0(wv22, wv23, h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)) at position [3] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1)),new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1)))
42.75/21.89	   (new_map0(y0, Zero, y2) -> H(y0, Succ(new_primPlusNat(Zero)), y2, new_new_map(Succ(Zero), y0)),new_map0(y0, Zero, y2) -> H(y0, Succ(new_primPlusNat(Zero)), y2, new_new_map(Succ(Zero), y0)))
42.75/21.89	   (new_map0(y0, Succ(x0), y2) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(Succ(x0)), y0)),new_map0(y0, Succ(x0), y2) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(Succ(x0)), y0)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(23)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
42.75/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(new_primPlusNat(Zero)), y2, new_new_map(Succ(Zero), y0))
42.75/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(Succ(x0)), y0))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(24) TransformationProof (EQUIVALENT)
42.75/21.89	By rewriting [LPAR04] the rule new_map0(y0, Zero, y2) -> H(y0, Succ(new_primPlusNat(Zero)), y2, new_new_map(Succ(Zero), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0)),new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(25)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
42.75/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(Succ(x0)), y0))
42.75/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(26) TransformationProof (EQUIVALENT)
42.75/21.89	By rewriting [LPAR04] the rule new_map0(y0, Succ(x0), y2) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(Succ(x0)), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0)),new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(27)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22))
42.75/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
42.75/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
42.75/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(x0))
42.75/21.89	
42.75/21.89	We have to consider all minimal (P,Q,R)-chains.
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(28) TransformationProof (EQUIVALENT)
42.75/21.89	By narrowing [LPAR04] the rule new_map(wv22, wv23, Zero, Succ(wv250), h) -> H(wv22, Succ(new_primPlusNat(wv23)), h, new_new_map(Succ(new_primPlusNat(wv23)), wv22)) at position [3] we obtained the following new rules [LPAR04]:
42.75/21.89	
42.75/21.89	   (new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1)),new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1)))
42.75/21.89	   (new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Zero)), y3, new_new_map(Succ(Zero), y0)),new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Zero)), y3, new_new_map(Succ(Zero), y0)))
42.75/21.89	   (new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(Succ(x0)), y0)),new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(Succ(x0)), y0)))
42.75/21.89	
42.75/21.89	
42.75/21.89	----------------------------------------
42.75/21.89	
42.75/21.89	(29)
42.75/21.89	Obligation:
42.75/21.89	Q DP problem:
42.75/21.89	The TRS P consists of the following rules:
42.75/21.89	
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
42.75/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
42.75/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
42.75/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
42.75/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
42.75/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
42.75/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
42.75/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Zero)), y3, new_new_map(Succ(Zero), y0))
42.75/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(Succ(x0)), y0))
42.75/21.89	
42.75/21.89	The TRS R consists of the following rules:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Zero) -> Zero
42.75/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
42.75/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
42.75/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
42.75/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
42.75/21.89	
42.75/21.89	The set Q consists of the following terms:
42.75/21.89	
42.75/21.89	   new_primPlusNat(Succ(x0))
42.75/21.89	   new_primPlusNat(Zero)
42.75/21.89	   new_new_map(Succ(x0), Succ(x1))
42.75/21.89	   new_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(x0))
43.03/21.89	
43.03/21.89	We have to consider all minimal (P,Q,R)-chains.
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(30) TransformationProof (EQUIVALENT)
43.03/21.89	By rewriting [LPAR04] the rule new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Zero)), y3, new_new_map(Succ(Zero), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.89	
43.03/21.89	   (new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0)),new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0)))
43.03/21.89	
43.03/21.89	
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(31)
43.03/21.89	Obligation:
43.03/21.89	Q DP problem:
43.03/21.89	The TRS P consists of the following rules:
43.03/21.89	
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.89	
43.03/21.89	The TRS R consists of the following rules:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Zero) -> Zero
43.03/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.89	
43.03/21.89	The set Q consists of the following terms:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Succ(x0))
43.03/21.89	   new_primPlusNat(Zero)
43.03/21.89	   new_new_map(Succ(x0), Succ(x1))
43.03/21.89	   new_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(x0))
43.03/21.89	
43.03/21.89	We have to consider all minimal (P,Q,R)-chains.
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(32) TransformationProof (EQUIVALENT)
43.03/21.89	By rewriting [LPAR04] the rule new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(Succ(x0)), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.89	
43.03/21.89	   (new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0)),new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0)))
43.03/21.89	
43.03/21.89	
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(33)
43.03/21.89	Obligation:
43.03/21.89	Q DP problem:
43.03/21.89	The TRS P consists of the following rules:
43.03/21.89	
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.89	   new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	
43.03/21.89	The TRS R consists of the following rules:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Zero) -> Zero
43.03/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.89	
43.03/21.89	The set Q consists of the following terms:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Succ(x0))
43.03/21.89	   new_primPlusNat(Zero)
43.03/21.89	   new_new_map(Succ(x0), Succ(x1))
43.03/21.89	   new_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(x0))
43.03/21.89	
43.03/21.89	We have to consider all minimal (P,Q,R)-chains.
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(34) TransformationProof (EQUIVALENT)
43.03/21.89	By narrowing [LPAR04] the rule new_map0(Succ(x1), y1, y2) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y2, new_new_map(new_primPlusNat(y1), x1)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.89	
43.03/21.89	   (new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y2, new_new_map(Zero, y0)),new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y2, new_new_map(Zero, y0)))
43.03/21.89	   (new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(x0), y0)),new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(x0), y0)))
43.03/21.89	
43.03/21.89	
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(35)
43.03/21.89	Obligation:
43.03/21.89	Q DP problem:
43.03/21.89	The TRS P consists of the following rules:
43.03/21.89	
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y2, new_new_map(Zero, y0))
43.03/21.89	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(x0), y0))
43.03/21.89	
43.03/21.89	The TRS R consists of the following rules:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Zero) -> Zero
43.03/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.89	
43.03/21.89	The set Q consists of the following terms:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Succ(x0))
43.03/21.89	   new_primPlusNat(Zero)
43.03/21.89	   new_new_map(Succ(x0), Succ(x1))
43.03/21.89	   new_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(x0))
43.03/21.89	
43.03/21.89	We have to consider all minimal (P,Q,R)-chains.
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(36) TransformationProof (EQUIVALENT)
43.03/21.89	By rewriting [LPAR04] the rule new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y2, new_new_map(Zero, y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.89	
43.03/21.89	   (new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0)),new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0)))
43.03/21.89	
43.03/21.89	
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(37)
43.03/21.89	Obligation:
43.03/21.89	Q DP problem:
43.03/21.89	The TRS P consists of the following rules:
43.03/21.89	
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(x0), y0))
43.03/21.89	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.89	
43.03/21.89	The TRS R consists of the following rules:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Zero) -> Zero
43.03/21.89	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.89	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.89	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.89	
43.03/21.89	The set Q consists of the following terms:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Succ(x0))
43.03/21.89	   new_primPlusNat(Zero)
43.03/21.89	   new_new_map(Succ(x0), Succ(x1))
43.03/21.89	   new_new_map(Zero, Zero)
43.03/21.89	   new_new_map(Zero, Succ(x0))
43.03/21.89	
43.03/21.89	We have to consider all minimal (P,Q,R)-chains.
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(38) TransformationProof (EQUIVALENT)
43.03/21.89	By rewriting [LPAR04] the rule new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y2, new_new_map(Succ(x0), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.89	
43.03/21.89	   (new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0)),new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0)))
43.03/21.89	
43.03/21.89	
43.03/21.89	----------------------------------------
43.03/21.89	
43.03/21.89	(39)
43.03/21.89	Obligation:
43.03/21.89	Q DP problem:
43.03/21.89	The TRS P consists of the following rules:
43.03/21.89	
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.89	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.89	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.89	   new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.89	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.89	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.89	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.89	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.89	
43.03/21.89	The TRS R consists of the following rules:
43.03/21.89	
43.03/21.89	   new_primPlusNat(Zero) -> Zero
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(40) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map0(y0, Zero, y2) -> H(y0, Succ(Zero), y2, new_new_map(Succ(Zero), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map0(Succ(x1), Zero, y1) -> H(Succ(x1), Succ(Zero), y1, new_new_map(Zero, x1)),new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(41)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Zero) -> Zero
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(42) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map0(y0, Succ(x0), y2) -> H(y0, Succ(Succ(x0)), y2, new_new_map(Succ(Succ(x0)), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map0(Succ(x1), Succ(y1), y2) -> H(Succ(x1), Succ(Succ(y1)), y2, new_new_map(Succ(y1), x1)),new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(43)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1))
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Zero) -> Zero
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(44) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(Succ(x1), y1, Zero, Succ(y2), y3) -> H(Succ(x1), Succ(new_primPlusNat(y1)), y3, new_new_map(new_primPlusNat(y1), x1)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y3, new_new_map(Zero, y0)),new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y3, new_new_map(Zero, y0)))
43.03/21.90	   (new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0)),new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(45)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Zero) -> Zero
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(46) TransformationProof (EQUIVALENT)
43.03/21.90	By rewriting [LPAR04] the rule new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Zero)), y3, new_new_map(Zero, y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0)),new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(47)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Zero) -> Zero
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(48) UsableRulesProof (EQUIVALENT)
43.03/21.90	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.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(49)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(50) TransformationProof (EQUIVALENT)
43.03/21.90	By rewriting [LPAR04] the rule new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(new_primPlusNat(Succ(x0))), y3, new_new_map(Succ(x0), y0)) at position [1,0] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0)),new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(51)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	   new_primPlusNat(Succ(wv230)) -> Succ(wv230)
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(52) UsableRulesProof (EQUIVALENT)
43.03/21.90	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.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(53)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(54) QReductionProof (EQUIVALENT)
43.03/21.90	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
43.03/21.90	
43.03/21.90	   new_primPlusNat(Succ(x0))
43.03/21.90	   new_primPlusNat(Zero)
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(55)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0))
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(56) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(y0, Zero, Zero, Succ(y2), y3) -> H(y0, Succ(Zero), y3, new_new_map(Succ(Zero), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(x1), Zero, Zero, Succ(y1), y2) -> H(Succ(x1), Succ(Zero), y2, new_new_map(Zero, x1)),new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(57)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0))
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(58) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(y0, Succ(x0), Zero, Succ(y2), y3) -> H(y0, Succ(Succ(x0)), y3, new_new_map(Succ(Succ(x0)), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(x1), Succ(y1), Zero, Succ(y2), y3) -> H(Succ(x1), Succ(Succ(y1)), y3, new_new_map(Succ(y1), x1)),new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(59)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0))
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(60) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map0(Succ(y0), Zero, y2) -> H(Succ(y0), Succ(Zero), y2, new_new_map(Zero, y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero)),new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero)))
43.03/21.90	   (new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0))),new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0))))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(61)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(62) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map0(Succ(y0), Succ(x0), y2) -> H(Succ(y0), Succ(Succ(x0)), y2, new_new_map(Succ(x0), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1)),new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(63)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0))
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(64) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(Succ(y0), Zero, Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Zero), y3, new_new_map(Zero, y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero)),new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero)))
43.03/21.90	   (new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0))),new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0))))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(65)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0))
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(66) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(Succ(y0), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(y0), Succ(Succ(x0)), y3, new_new_map(Succ(x0), y0)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(Succ(x1)), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y3, new_new_map(x0, x1)),new_map(Succ(Succ(x1)), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y3, new_new_map(x0, x1)))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(67)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Succ(x1)), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y3, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(68) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map0(Succ(Succ(x1)), Succ(x0), y2) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y2, new_new_map(x0, x1)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)),new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)))
43.03/21.90	   (new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero)),new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero)))
43.03/21.90	   (new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))),new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(69)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Succ(x1)), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(70) TransformationProof (EQUIVALENT)
43.03/21.90	By narrowing [LPAR04] the rule new_map(Succ(Succ(x1)), Succ(x0), Zero, Succ(y2), y3) -> H(Succ(Succ(x1)), Succ(Succ(x0)), y3, new_new_map(x0, x1)) at position [3] we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)),new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)))
43.03/21.90	   (new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y3, cons_new_map(Zero, Zero)),new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y3, cons_new_map(Zero, Zero)))
43.03/21.90	   (new_map(Succ(Succ(Succ(x0))), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y3, cons_new_map(Zero, Succ(x0))),new_map(Succ(Succ(Succ(x0))), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y3, cons_new_map(Zero, Succ(x0))))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(71)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y3, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(Succ(x0))), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y3, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(72) QDPOrderProof (EQUIVALENT)
43.03/21.90	We use the reduction pair processor [LPAR04,JAR06].
43.03/21.90	
43.03/21.90	
43.03/21.90	The following pairs can be oriented strictly and are deleted.
43.03/21.90	
43.03/21.90	   new_map0(Succ(Zero), Zero, y1) -> H(Succ(Zero), Succ(Zero), y1, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(x0)), Zero, y1) -> H(Succ(Succ(x0)), Succ(Zero), y1, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Zero), Zero, Zero, Succ(y1), y2) -> H(Succ(Zero), Succ(Zero), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(x0)), Zero, Zero, Succ(y1), y2) -> H(Succ(Succ(x0)), Succ(Zero), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	The remaining pairs can at least be oriented weakly.
43.03/21.90	Used ordering:  Polynomial interpretation [POLO]:
43.03/21.90	
43.03/21.90	   POL(H(x_1, x_2, x_3, x_4)) = x_2
43.03/21.90	   POL(Succ(x_1)) = 0
43.03/21.90	   POL(Zero) = 1
43.03/21.90	   POL(cons_new_map(x_1, x_2)) = 0
43.03/21.90	   POL(new_map(x_1, x_2, x_3, x_4, x_5)) = x_2
43.03/21.90	   POL(new_map0(x_1, x_2, x_3)) = x_2
43.03/21.90	   POL(new_new_map(x_1, x_2)) = 0
43.03/21.90	
43.03/21.90	The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented:
43.03/21.90	none
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(73)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y3, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(Succ(x0))), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y3, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(74) TransformationProof (EQUIVALENT)
43.03/21.90	By instantiating [LPAR04] the rule H(wv22, wv23, h, cons_new_map(Zero, Succ(wv250))) -> new_map(wv22, wv23, Zero, Succ(wv250), h) we obtained the following new rules [LPAR04]:
43.03/21.90	
43.03/21.90	   (H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2),H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2))
43.03/21.90	   (H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1),H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1))
43.03/21.90	
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(75)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y3, cons_new_map(Zero, Zero))
43.03/21.90	   new_map(Succ(Succ(Succ(x0))), Succ(Zero), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y3, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(76) DependencyGraphProof (EQUIVALENT)
43.03/21.90	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(77)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(78) InductionCalculusProof (EQUIVALENT)
43.03/21.90	Note that final constraints are written in bold face.
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.90	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x18, x19, Zero, Zero, x20) -> new_map0(x18, x19, x20), new_map0(Succ(Succ(Zero)), Succ(Zero), x21) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), x21, cons_new_map(Zero, Zero)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x18, x19, x20)=new_map0(Succ(Succ(Zero)), Succ(Zero), x21)  ==>  new_map(x18, x19, Zero, Zero, x20)_>=_new_map0(x18, x19, x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Zero, x20)_>=_new_map0(Succ(Succ(Zero)), Succ(Zero), x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x22, x23, Zero, Zero, x24) -> new_map0(x22, x23, x24), new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26) -> H(Succ(Succ(Succ(x25))), Succ(Succ(Zero)), x26, cons_new_map(Zero, Succ(x25))) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x22, x23, x24)=new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26)  ==>  new_map(x22, x23, Zero, Zero, x24)_>=_new_map0(x22, x23, x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.90	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Zero)), Succ(Zero), x165) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)), H(x166, x167, x168, cons_new_map(Zero, Zero)) -> new_map(x166, x167, Zero, Zero, x168) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero))=H(x166, x167, x168, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189) -> H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))), H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190))) -> new_map(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), Zero, Succ(x190), x191) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188)))=H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190)))  ==>  new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200))) -> new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201), new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205) -> H(Succ(Succ(Succ(x202))), Succ(Succ(Succ(x203))), x205, new_new_map(x203, x202)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201)=new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205)  ==>  H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.90	
43.03/21.90	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Zero, x20)_>=_new_map0(Succ(Succ(Zero)), Succ(Zero), x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	
43.03/21.90	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(79)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(80) NonInfProof (EQUIVALENT)
43.03/21.90	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
43.03/21.90	
43.03/21.90	Note that final constraints are written in bold face.
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.90	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x18, x19, Zero, Zero, x20) -> new_map0(x18, x19, x20), new_map0(Succ(Succ(Zero)), Succ(Zero), x21) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), x21, cons_new_map(Zero, Zero)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x18, x19, x20)=new_map0(Succ(Succ(Zero)), Succ(Zero), x21)  ==>  new_map(x18, x19, Zero, Zero, x20)_>=_new_map0(x18, x19, x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Zero, x20)_>=_new_map0(Succ(Succ(Zero)), Succ(Zero), x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x22, x23, Zero, Zero, x24) -> new_map0(x22, x23, x24), new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26) -> H(Succ(Succ(Succ(x25))), Succ(Succ(Zero)), x26, cons_new_map(Zero, Succ(x25))) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x22, x23, x24)=new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26)  ==>  new_map(x22, x23, Zero, Zero, x24)_>=_new_map0(x22, x23, x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.90	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Zero)), Succ(Zero), x165) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)), H(x166, x167, x168, cons_new_map(Zero, Zero)) -> new_map(x166, x167, Zero, Zero, x168) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero))=H(x166, x167, x168, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189) -> H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))), H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190))) -> new_map(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), Zero, Succ(x190), x191) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188)))=H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190)))  ==>  new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200))) -> new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201), new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205) -> H(Succ(Succ(Succ(x202))), Succ(Succ(Succ(x203))), x205, new_new_map(x203, x202)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201)=new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205)  ==>  H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.90	
43.03/21.90	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Zero)), Succ(Zero), Zero, Zero, x20)_>=_new_map0(Succ(Succ(Zero)), Succ(Zero), x20))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	
43.03/21.90	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Zero)), Succ(Zero), x165)_>=_H(Succ(Succ(Zero)), Succ(Succ(Zero)), x165, cons_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.90	
43.03/21.90	Using the following integer polynomial ordering the  resulting constraints can be solved 
43.03/21.90	
43.03/21.90	Polynomial interpretation [NONINF]:
43.03/21.90	
43.03/21.90	   POL(H(x_1, x_2, x_3, x_4)) = -1 - x_1 - x_2 + x_3 - x_4
43.03/21.90	   POL(Succ(x_1)) = 1 + x_1
43.03/21.90	   POL(Zero) = 0
43.03/21.90	   POL(c) = -7
43.03/21.90	   POL(cons_new_map(x_1, x_2)) = 0
43.03/21.90	   POL(new_map(x_1, x_2, x_3, x_4, x_5)) = -1 - x_1 - x_2 + x_3 + x_5
43.03/21.90	   POL(new_map0(x_1, x_2, x_3)) = -1 - x_1 - x_2 + x_3
43.03/21.90	   POL(new_new_map(x_1, x_2)) = 0
43.03/21.90	
43.03/21.90	
43.03/21.90	The following pairs  are in P_>:
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	The following pairs are in P_bound:
43.03/21.90	   new_map0(Succ(Succ(Zero)), Succ(Zero), y2) -> H(Succ(Succ(Zero)), Succ(Succ(Zero)), y2, cons_new_map(Zero, Zero))
43.03/21.90	The following rules are usable:
43.03/21.90	   new_new_map(wv240, wv250) -> new_new_map(Succ(wv240), Succ(wv250))
43.03/21.90	   cons_new_map(Zero, Zero) -> new_new_map(Zero, Zero)
43.03/21.90	   cons_new_map(Zero, Succ(wv250)) -> new_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(81)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(82) InductionCalculusProof (EQUIVALENT)
43.03/21.90	Note that final constraints are written in bold face.
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.90	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x22, x23, Zero, Zero, x24) -> new_map0(x22, x23, x24), new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26) -> H(Succ(Succ(Succ(x25))), Succ(Succ(Zero)), x26, cons_new_map(Zero, Succ(x25))) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x22, x23, x24)=new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26)  ==>  new_map(x22, x23, Zero, Zero, x24)_>=_new_map0(x22, x23, x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.90	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (5) using rules (I), (II).
43.03/21.90	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.90	
43.03/21.90	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189) -> H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))), H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190))) -> new_map(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), Zero, Succ(x190), x191) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188)))=H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190)))  ==>  new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1) the following chains were created:
43.03/21.90	*We consider the chain H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200))) -> new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201), new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205) -> H(Succ(Succ(Succ(x202))), Succ(Succ(Succ(x203))), x205, new_new_map(x203, x202)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201)=new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205)  ==>  H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.90	
43.03/21.90	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	
43.03/21.90	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.90	
43.03/21.90	
43.03/21.90	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	
43.03/21.90	*(new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	*(H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(83)
43.03/21.90	Obligation:
43.03/21.90	Q DP problem:
43.03/21.90	The TRS P consists of the following rules:
43.03/21.90	
43.03/21.90	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.90	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.90	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.90	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.90	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.90	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.90	
43.03/21.90	The TRS R consists of the following rules:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.90	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.90	
43.03/21.90	The set Q consists of the following terms:
43.03/21.90	
43.03/21.90	   new_new_map(Succ(x0), Succ(x1))
43.03/21.90	   new_new_map(Zero, Zero)
43.03/21.90	   new_new_map(Zero, Succ(x0))
43.03/21.90	
43.03/21.90	We have to consider all minimal (P,Q,R)-chains.
43.03/21.90	----------------------------------------
43.03/21.90	
43.03/21.90	(84) NonInfProof (EQUIVALENT)
43.03/21.90	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
43.03/21.90	
43.03/21.90	Note that final constraints are written in bold face.
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.90	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	*We consider the chain new_map(x22, x23, Zero, Zero, x24) -> new_map0(x22, x23, x24), new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26) -> H(Succ(Succ(Succ(x25))), Succ(Succ(Zero)), x26, cons_new_map(Zero, Succ(x25))) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (new_map0(x22, x23, x24)=new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x26)  ==>  new_map(x22, x23, Zero, Zero, x24)_>=_new_map0(x22, x23, x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.90	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.90	
43.03/21.90	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.90	
43.03/21.90	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.90	
43.03/21.90	
43.03/21.90	
43.03/21.90	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.90	
43.03/21.90	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.91	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.91	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0))) the following chains were created:
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189) -> H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))), H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190))) -> new_map(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), Zero, Succ(x190), x191) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188)))=H(Succ(Succ(Succ(x190))), Succ(Succ(Zero)), x191, cons_new_map(Zero, Succ(x190)))  ==>  new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1) the following chains were created:
43.03/21.91	*We consider the chain H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200))) -> new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201), new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205) -> H(Succ(Succ(Succ(x202))), Succ(Succ(Succ(x203))), x205, new_new_map(x203, x202)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201)=new_map(Succ(Succ(Succ(x202))), Succ(Succ(x203)), Zero, Succ(x204), x205)  ==>  H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.91	
43.03/21.91	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x25))), Succ(Zero), Zero, Zero, x24)_>=_new_map0(Succ(Succ(Succ(x25))), Succ(Zero), x24))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	
43.03/21.91	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	
43.03/21.91	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x188))), Succ(Zero), x189)_>=_H(Succ(Succ(Succ(x188))), Succ(Succ(Zero)), x189, cons_new_map(Zero, Succ(x188))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.91	
43.03/21.91	*(H(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), x201, cons_new_map(Zero, Succ(x200)))_>=_new_map(Succ(Succ(Succ(x200))), Succ(Succ(Zero)), Zero, Succ(x200), x201))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.91	
43.03/21.91	Using the following integer polynomial ordering the  resulting constraints can be solved 
43.03/21.91	
43.03/21.91	Polynomial interpretation [NONINF]:
43.03/21.91	
43.03/21.91	   POL(H(x_1, x_2, x_3, x_4)) = -1 - x_2 + x_3 - x_4
43.03/21.91	   POL(Succ(x_1)) = 1 + x_1
43.03/21.91	   POL(Zero) = 0
43.03/21.91	   POL(c) = -4
43.03/21.91	   POL(cons_new_map(x_1, x_2)) = 0
43.03/21.91	   POL(new_map(x_1, x_2, x_3, x_4, x_5)) = -1 - x_2 + x_3 + x_5
43.03/21.91	   POL(new_map0(x_1, x_2, x_3)) = -1 - x_2 + x_3
43.03/21.91	   POL(new_new_map(x_1, x_2)) = 0
43.03/21.91	
43.03/21.91	
43.03/21.91	The following pairs  are in P_>:
43.03/21.91	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.91	The following pairs are in P_bound:
43.03/21.91	   new_map0(Succ(Succ(Succ(x0))), Succ(Zero), y2) -> H(Succ(Succ(Succ(x0))), Succ(Succ(Zero)), y2, cons_new_map(Zero, Succ(x0)))
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.91	The following rules are usable:
43.03/21.91	   new_new_map(wv240, wv250) -> new_new_map(Succ(wv240), Succ(wv250))
43.03/21.91	   cons_new_map(Zero, Zero) -> new_new_map(Zero, Zero)
43.03/21.91	   cons_new_map(Zero, Succ(wv250)) -> new_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(85)
43.03/21.91	Complex Obligation (AND)
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(86)
43.03/21.91	Obligation:
43.03/21.91	Q DP problem:
43.03/21.91	The TRS P consists of the following rules:
43.03/21.91	
43.03/21.91	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), z1, cons_new_map(Zero, Succ(z0))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Zero)), Zero, Succ(z0), z1)
43.03/21.91	
43.03/21.91	The TRS R consists of the following rules:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.91	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	The set Q consists of the following terms:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(x0), Succ(x1))
43.03/21.91	   new_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(x0))
43.03/21.91	
43.03/21.91	We have to consider all minimal (P,Q,R)-chains.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(87) DependencyGraphProof (EQUIVALENT)
43.03/21.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(88)
43.03/21.91	TRUE
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(89)
43.03/21.91	Obligation:
43.03/21.91	Q DP problem:
43.03/21.91	The TRS P consists of the following rules:
43.03/21.91	
43.03/21.91	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	The TRS R consists of the following rules:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.91	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	The set Q consists of the following terms:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(x0), Succ(x1))
43.03/21.91	   new_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(x0))
43.03/21.91	
43.03/21.91	We have to consider all minimal (P,Q,R)-chains.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(90) InductionCalculusProof (EQUIVALENT)
43.03/21.91	Note that final constraints are written in bold face.
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.91	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.91	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.91	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.91	
43.03/21.91	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	
43.03/21.91	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	
43.03/21.91	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(91)
43.03/21.91	Obligation:
43.03/21.91	Q DP problem:
43.03/21.91	The TRS P consists of the following rules:
43.03/21.91	
43.03/21.91	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	The TRS R consists of the following rules:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.91	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	The set Q consists of the following terms:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(x0), Succ(x1))
43.03/21.91	   new_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(x0))
43.03/21.91	
43.03/21.91	We have to consider all minimal (P,Q,R)-chains.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(92) NonInfProof (EQUIVALENT)
43.03/21.91	The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
43.03/21.91	
43.03/21.91	Note that final constraints are written in bold face.
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h) the following chains were created:
43.03/21.91	*We consider the chain new_map(x3, x4, Zero, Zero, x5) -> new_map0(x3, x4, x5), new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8) -> H(Succ(Succ(Succ(x6))), Succ(Succ(Succ(x7))), x8, new_new_map(x7, x6)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map0(x3, x4, x5)=new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x8)  ==>  new_map(x3, x4, Zero, Zero, x5)_>=_new_map0(x3, x4, x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1)) the following chains were created:
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38) -> H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)), H(x39, x40, x41, cons_new_map(Zero, Zero)) -> new_map(x39, x40, Zero, Zero, x41) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36))=H(x39, x40, x41, cons_new_map(Zero, Zero))  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x37, x36)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x36))), Succ(Succ(x37)), x38)_>=_H(Succ(Succ(Succ(x36))), Succ(Succ(Succ(x37))), x38, new_new_map(x37, x36)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x37, x36)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x213, x212)=cons_new_map(Zero, Zero) & (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212)))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x215))=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Succ(x215)))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Succ(x215))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x214:new_new_map(x213, x212)=cons_new_map(Zero, Zero)  ==>  new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x214)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x214, new_new_map(x213, x212))) with sigma = [x214 / x38] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44) -> H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)), H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48))) -> new_map(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), Zero, Succ(x48), x47) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42))=H(Succ(Succ(Succ(x45))), Succ(Succ(Succ(x46))), x47, cons_new_map(Zero, Succ(x48)))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(x42))), Succ(Succ(x43)), x44)_>=_H(Succ(Succ(Succ(x42))), Succ(Succ(Succ(x43))), x44, new_new_map(x43, x42)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x43, x42)=cons_new_map(Zero, Succ(x48)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x217, x216)=cons_new_map(Zero, Succ(x48)) & (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216)))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x220))=cons_new_map(Zero, Succ(x48))  ==>  new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x218,x219:new_new_map(x217, x216)=cons_new_map(Zero, Succ(x218))  ==>  new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x219)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x219, new_new_map(x217, x216))) with sigma = [x218 / x48, x219 / x44] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h) the following chains were created:
43.03/21.91	*We consider the chain H(x61, x62, x63, cons_new_map(Zero, Zero)) -> new_map(x61, x62, Zero, Zero, x63), new_map(x64, x65, Zero, Zero, x66) -> new_map0(x64, x65, x66) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(x61, x62, Zero, Zero, x63)=new_map(x64, x65, Zero, Zero, x66)  ==>  H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2) the following chains were created:
43.03/21.91	*We consider the chain H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107))) -> new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106), new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111) -> H(Succ(Succ(Succ(x108))), Succ(Succ(Succ(x109))), x111, new_new_map(x109, x108)) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106)=new_map(Succ(Succ(Succ(x108))), Succ(Succ(x109)), Zero, Succ(x110), x111)  ==>  H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	For Pair new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1)) the following chains were created:
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135) -> H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)), H(x136, x137, x138, cons_new_map(Zero, Zero)) -> new_map(x136, x137, Zero, Zero, x138) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132))=H(x136, x137, x138, cons_new_map(Zero, Zero))  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x133, x132)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x132))), Succ(Succ(x133)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x132))), Succ(Succ(Succ(x133))), x135, new_new_map(x133, x132)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x133, x132)=cons_new_map(Zero, Zero) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x222, x221)=cons_new_map(Zero, Zero) & (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221)))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x225))=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x225)))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Succ(x225))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x223,x224:new_new_map(x222, x221)=cons_new_map(Zero, Zero)  ==>  new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x223), x224)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x224, new_new_map(x222, x221))) with sigma = [x223 / x134, x224 / x135] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (4) using rules (I), (II) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (5) using rules (I), (II).
43.03/21.91	*We consider the chain new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142) -> H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)), H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146))) -> new_map(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), Zero, Succ(x146), x145) which results in the following constraint:
43.03/21.91	
43.03/21.91	(1)    (H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139))=H(Succ(Succ(Succ(x143))), Succ(Succ(Succ(x144))), x145, cons_new_map(Zero, Succ(x146)))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(2)    (new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(x139))), Succ(Succ(x140)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x139))), Succ(Succ(Succ(x140))), x142, new_new_map(x140, x139)))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x140, x139)=cons_new_map(Zero, Succ(x146)) which results in the following new constraints:
43.03/21.91	
43.03/21.91	(3)    (new_new_map(x227, x226)=cons_new_map(Zero, Succ(x146)) & (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226)))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	(4)    (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	(5)    (cons_new_map(Zero, Succ(x231))=cons_new_map(Zero, Succ(x146))  ==>  new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We simplified constraint (3) using rule (VI) where we applied the induction hypothesis (\/x228,x229,x230:new_new_map(x227, x226)=cons_new_map(Zero, Succ(x228))  ==>  new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x229), x230)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x230, new_new_map(x227, x226))) with sigma = [x228 / x146, x229 / x141, x230 / x142] which results in the following new constraint:
43.03/21.91	
43.03/21.91	(6)    (new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint:
43.03/21.91	
43.03/21.91	(7)    (new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	To summarize, we get the following constraints P__>=_ for the following pairs.
43.03/21.91	
43.03/21.91	*new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x6))), Succ(Succ(x7)), Zero, Zero, x5)_>=_new_map0(Succ(Succ(Succ(x6))), Succ(Succ(x7)), x5))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x212))), Succ(Succ(x213)), x38)_>=_H(Succ(Succ(Succ(x212))), Succ(Succ(Succ(x213))), x38, new_new_map(x213, x212))  ==>  new_map0(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(x213))), x38)_>=_H(Succ(Succ(Succ(Succ(x212)))), Succ(Succ(Succ(Succ(x213)))), x38, new_new_map(Succ(x213), Succ(x212))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), x38)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x38, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(x216))), Succ(Succ(x217)), x44)_>=_H(Succ(Succ(Succ(x216))), Succ(Succ(Succ(x217))), x44, new_new_map(x217, x216))  ==>  new_map0(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(x217))), x44)_>=_H(Succ(Succ(Succ(Succ(x216)))), Succ(Succ(Succ(Succ(x217)))), x44, new_new_map(Succ(x217), Succ(x216))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map0(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Zero)), x44)_>=_H(Succ(Succ(Succ(Succ(x220)))), Succ(Succ(Succ(Zero))), x44, new_new_map(Zero, Succ(x220))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	
43.03/21.91	*(H(x61, x62, x63, cons_new_map(Zero, Zero))_>=_new_map(x61, x62, Zero, Zero, x63))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	
43.03/21.91	*(H(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), x106, cons_new_map(Zero, Succ(x107)))_>=_new_map(Succ(Succ(Succ(x104))), Succ(Succ(Succ(x105))), Zero, Succ(x107), x106))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	*new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x221))), Succ(Succ(x222)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(x221))), Succ(Succ(Succ(x222))), x135, new_new_map(x222, x221))  ==>  new_map(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(x222))), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Succ(x221)))), Succ(Succ(Succ(Succ(x222)))), x135, new_new_map(Succ(x222), Succ(x221))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Zero))), Succ(Succ(Zero)), Zero, Succ(x134), x135)_>=_H(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), x135, new_new_map(Zero, Zero)))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(x226))), Succ(Succ(x227)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(x226))), Succ(Succ(Succ(x227))), x142, new_new_map(x227, x226))  ==>  new_map(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(x227))), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x226)))), Succ(Succ(Succ(Succ(x227)))), x142, new_new_map(Succ(x227), Succ(x226))))
43.03/21.91	
43.03/21.91	
43.03/21.91	*(new_map(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Zero)), Zero, Succ(x141), x142)_>=_H(Succ(Succ(Succ(Succ(x231)))), Succ(Succ(Succ(Zero))), x142, new_new_map(Zero, Succ(x231))))
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	
43.03/21.91	The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by  "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. 
43.03/21.91	
43.03/21.91	Using the following integer polynomial ordering the  resulting constraints can be solved 
43.03/21.91	
43.03/21.91	Polynomial interpretation [NONINF]:
43.03/21.91	
43.03/21.91	   POL(H(x_1, x_2, x_3, x_4)) = -1 + x_1 - x_2 + x_3 - x_4
43.03/21.91	   POL(Succ(x_1)) = 1 + x_1
43.03/21.91	   POL(Zero) = 0
43.03/21.91	   POL(c) = -1
43.03/21.91	   POL(cons_new_map(x_1, x_2)) = 0
43.03/21.91	   POL(new_map(x_1, x_2, x_3, x_4, x_5)) = -1 + x_1 - x_2 + x_3 + x_5
43.03/21.91	   POL(new_map0(x_1, x_2, x_3)) = -1 + x_1 - x_2 + x_3
43.03/21.91	   POL(new_new_map(x_1, x_2)) = 0
43.03/21.91	
43.03/21.91	
43.03/21.91	The following pairs  are in P_>:
43.03/21.91	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	The following pairs are in P_bound:
43.03/21.91	   new_map0(Succ(Succ(Succ(x1))), Succ(Succ(x0)), y2) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y2, new_new_map(x0, x1))
43.03/21.91	   new_map(Succ(Succ(Succ(x1))), Succ(Succ(x0)), Zero, Succ(y2), y3) -> H(Succ(Succ(Succ(x1))), Succ(Succ(Succ(x0))), y3, new_new_map(x0, x1))
43.03/21.91	The following rules are usable:
43.03/21.91	   new_new_map(wv240, wv250) -> new_new_map(Succ(wv240), Succ(wv250))
43.03/21.91	   cons_new_map(Zero, Zero) -> new_new_map(Zero, Zero)
43.03/21.91	   cons_new_map(Zero, Succ(wv250)) -> new_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(93)
43.03/21.91	Obligation:
43.03/21.91	Q DP problem:
43.03/21.91	The TRS P consists of the following rules:
43.03/21.91	
43.03/21.91	   new_map(wv22, wv23, Zero, Zero, h) -> new_map0(wv22, wv23, h)
43.03/21.91	   H(wv22, wv23, h, cons_new_map(Zero, Zero)) -> new_map(wv22, wv23, Zero, Zero, h)
43.03/21.91	   H(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), z2, cons_new_map(Zero, Succ(x3))) -> new_map(Succ(Succ(Succ(z0))), Succ(Succ(Succ(z1))), Zero, Succ(x3), z2)
43.03/21.91	
43.03/21.91	The TRS R consists of the following rules:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(wv240), Succ(wv250)) -> new_new_map(wv240, wv250)
43.03/21.91	   new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(wv250)) -> cons_new_map(Zero, Succ(wv250))
43.03/21.91	
43.03/21.91	The set Q consists of the following terms:
43.03/21.91	
43.03/21.91	   new_new_map(Succ(x0), Succ(x1))
43.03/21.91	   new_new_map(Zero, Zero)
43.03/21.91	   new_new_map(Zero, Succ(x0))
43.03/21.91	
43.03/21.91	We have to consider all minimal (P,Q,R)-chains.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(94) DependencyGraphProof (EQUIVALENT)
43.03/21.91	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(95)
43.03/21.91	TRUE
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(96)
43.03/21.91	Obligation:
43.03/21.91	Q DP problem:
43.03/21.91	The TRS P consists of the following rules:
43.03/21.91	
43.03/21.91	   new_map(wv22, wv23, Succ(wv240), Succ(wv250), h) -> new_map(wv22, wv23, wv240, wv250, h)
43.03/21.91	
43.03/21.91	R is empty.
43.03/21.91	Q is empty.
43.03/21.91	We have to consider all minimal (P,Q,R)-chains.
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(97) QDPSizeChangeProof (EQUIVALENT)
43.03/21.91	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. 
43.03/21.91	
43.03/21.91	From the DPs we obtained the following set of size-change graphs:
43.03/21.91	*new_map(wv22, wv23, Succ(wv240), Succ(wv250), h) -> new_map(wv22, wv23, wv240, wv250, h)
43.03/21.91	The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4, 5 >= 5
43.03/21.91	
43.03/21.91	
43.03/21.91	----------------------------------------
43.03/21.91	
43.03/21.91	(98)
43.03/21.91	YES
43.03/21.94	EOF