/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.hs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) IFR [EQUIVALENT, 0 ms] (4) HASKELL (5) BR [EQUIVALENT, 0 ms] (6) HASKELL (7) COR [EQUIVALENT, 24 ms] (8) HASKELL (9) LetRed [EQUIVALENT, 0 ms] (10) HASKELL (11) NumRed [SOUND, 0 ms] (12) HASKELL (13) Narrow [SOUND, 0 ms] (14) AND (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPOrderProof [EQUIVALENT, 48 ms] (19) QDP (20) DependencyGraphProof [EQUIVALENT, 0 ms] (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) TransformationProof [EQUIVALENT, 0 ms] (26) QDP (27) TransformationProof [EQUIVALENT, 0 ms] (28) QDP (29) UsableRulesProof [EQUIVALENT, 0 ms] (30) QDP (31) QReductionProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) TransformationProof [EQUIVALENT, 0 ms] (38) QDP (39) DependencyGraphProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) TransformationProof [EQUIVALENT, 0 ms] (44) QDP (45) DependencyGraphProof [EQUIVALENT, 0 ms] (46) TRUE (47) QDP (48) QDPSizeChangeProof [EQUIVALENT, 0 ms] (49) YES (50) QDP (51) DependencyGraphProof [EQUIVALENT, 0 ms] (52) QDP (53) TransformationProof [EQUIVALENT, 0 ms] (54) QDP (55) TransformationProof [EQUIVALENT, 0 ms] (56) QDP (57) TransformationProof [EQUIVALENT, 0 ms] (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) TransformationProof [EQUIVALENT, 0 ms] (62) QDP (63) TransformationProof [EQUIVALENT, 0 ms] (64) QDP (65) TransformationProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 1 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) TransformationProof [EQUIVALENT, 0 ms] (76) QDP (77) TransformationProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 0 ms] (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) TransformationProof [EQUIVALENT, 0 ms] (84) QDP (85) TransformationProof [EQUIVALENT, 0 ms] (86) QDP (87) TransformationProof [EQUIVALENT, 0 ms] (88) QDP (89) UsableRulesProof [EQUIVALENT, 0 ms] (90) QDP (91) QReductionProof [EQUIVALENT, 0 ms] (92) QDP (93) TransformationProof [EQUIVALENT, 0 ms] (94) QDP (95) TransformationProof [EQUIVALENT, 0 ms] (96) QDP (97) TransformationProof [EQUIVALENT, 0 ms] (98) QDP (99) TransformationProof [EQUIVALENT, 0 ms] (100) QDP (101) TransformationProof [EQUIVALENT, 0 ms] (102) QDP (103) TransformationProof [EQUIVALENT, 0 ms] (104) QDP (105) TransformationProof [EQUIVALENT, 0 ms] (106) QDP (107) TransformationProof [EQUIVALENT, 0 ms] (108) QDP (109) TransformationProof [EQUIVALENT, 0 ms] (110) QDP (111) TransformationProof [EQUIVALENT, 0 ms] (112) QDP (113) TransformationProof [EQUIVALENT, 0 ms] (114) QDP (115) TransformationProof [EQUIVALENT, 0 ms] (116) QDP (117) DependencyGraphProof [EQUIVALENT, 0 ms] (118) AND (119) QDP (120) UsableRulesProof [EQUIVALENT, 0 ms] (121) QDP (122) TransformationProof [EQUIVALENT, 0 ms] (123) QDP (124) UsableRulesProof [EQUIVALENT, 0 ms] (125) QDP (126) TransformationProof [EQUIVALENT, 0 ms] (127) QDP (128) TransformationProof [EQUIVALENT, 0 ms] (129) QDP (130) UsableRulesProof [EQUIVALENT, 0 ms] (131) QDP (132) QReductionProof [EQUIVALENT, 0 ms] (133) QDP (134) TransformationProof [EQUIVALENT, 0 ms] (135) QDP (136) TransformationProof [EQUIVALENT, 0 ms] (137) QDP (138) DependencyGraphProof [EQUIVALENT, 0 ms] (139) TRUE (140) QDP (141) TransformationProof [EQUIVALENT, 0 ms] (142) QDP (143) TransformationProof [EQUIVALENT, 0 ms] (144) QDP (145) TransformationProof [EQUIVALENT, 0 ms] (146) QDP (147) TransformationProof [EQUIVALENT, 0 ms] (148) QDP (149) TransformationProof [EQUIVALENT, 0 ms] (150) QDP (151) TransformationProof [EQUIVALENT, 0 ms] (152) QDP (153) TransformationProof [EQUIVALENT, 0 ms] (154) QDP (155) DependencyGraphProof [EQUIVALENT, 0 ms] (156) AND (157) QDP (158) UsableRulesProof [EQUIVALENT, 0 ms] (159) QDP (160) TransformationProof [EQUIVALENT, 0 ms] (161) QDP (162) UsableRulesProof [EQUIVALENT, 0 ms] (163) QDP (164) TransformationProof [EQUIVALENT, 0 ms] (165) QDP (166) TransformationProof [EQUIVALENT, 0 ms] (167) QDP (168) UsableRulesProof [EQUIVALENT, 0 ms] (169) QDP (170) QReductionProof [EQUIVALENT, 0 ms] (171) QDP (172) TransformationProof [EQUIVALENT, 0 ms] (173) QDP (174) TransformationProof [EQUIVALENT, 0 ms] (175) QDP (176) DependencyGraphProof [EQUIVALENT, 0 ms] (177) TRUE (178) QDP (179) TransformationProof [EQUIVALENT, 0 ms] (180) QDP (181) TransformationProof [EQUIVALENT, 0 ms] (182) QDP (183) TransformationProof [EQUIVALENT, 0 ms] (184) QDP (185) TransformationProof [EQUIVALENT, 0 ms] (186) QDP (187) TransformationProof [EQUIVALENT, 0 ms] (188) QDP (189) TransformationProof [EQUIVALENT, 0 ms] (190) QDP (191) TransformationProof [EQUIVALENT, 0 ms] (192) QDP (193) DependencyGraphProof [EQUIVALENT, 0 ms] (194) QDP (195) TransformationProof [EQUIVALENT, 0 ms] (196) QDP (197) TransformationProof [EQUIVALENT, 0 ms] (198) QDP (199) TransformationProof [EQUIVALENT, 0 ms] (200) QDP (201) TransformationProof [EQUIVALENT, 0 ms] (202) QDP (203) TransformationProof [EQUIVALENT, 0 ms] (204) QDP (205) TransformationProof [EQUIVALENT, 0 ms] (206) QDP (207) TransformationProof [EQUIVALENT, 0 ms] (208) QDP (209) TransformationProof [EQUIVALENT, 0 ms] (210) QDP (211) QDPOrderProof [EQUIVALENT, 40 ms] (212) QDP (213) QDPOrderProof [EQUIVALENT, 45 ms] (214) QDP (215) DependencyGraphProof [EQUIVALENT, 0 ms] (216) AND (217) QDP (218) QDPOrderProof [EQUIVALENT, 0 ms] (219) QDP (220) QDPPairToRuleProof [EQUIVALENT, 0 ms] (221) AND (222) QDP (223) TransformationProof [EQUIVALENT, 0 ms] (224) QDP (225) TransformationProof [EQUIVALENT, 0 ms] (226) QDP (227) TransformationProof [EQUIVALENT, 0 ms] (228) QDP (229) TransformationProof [EQUIVALENT, 0 ms] (230) QDP (231) TransformationProof [EQUIVALENT, 0 ms] (232) QDP (233) TransformationProof [EQUIVALENT, 0 ms] (234) QDP (235) TransformationProof [EQUIVALENT, 0 ms] (236) QDP (237) TransformationProof [EQUIVALENT, 0 ms] (238) QDP (239) TransformationProof [EQUIVALENT, 0 ms] (240) QDP (241) TransformationProof [EQUIVALENT, 0 ms] (242) QDP (243) TransformationProof [EQUIVALENT, 0 ms] (244) QDP (245) TransformationProof [EQUIVALENT, 0 ms] (246) QDP (247) TransformationProof [EQUIVALENT, 0 ms] (248) QDP (249) TransformationProof [EQUIVALENT, 0 ms] (250) QDP (251) TransformationProof [EQUIVALENT, 0 ms] (252) QDP (253) TransformationProof [EQUIVALENT, 0 ms] (254) QDP (255) UsableRulesProof [EQUIVALENT, 0 ms] (256) QDP (257) TransformationProof [EQUIVALENT, 0 ms] (258) QDP (259) InductionCalculusProof [EQUIVALENT, 0 ms] (260) QDP (261) NonInfProof [EQUIVALENT, 3381 ms] (262) AND (263) QDP (264) DependencyGraphProof [EQUIVALENT, 0 ms] (265) TRUE (266) QDP (267) InductionCalculusProof [EQUIVALENT, 0 ms] (268) QDP (269) NonInfProof [EQUIVALENT, 16.6 s] (270) AND (271) QDP (272) DependencyGraphProof [EQUIVALENT, 0 ms] (273) TRUE (274) QDP (275) DependencyGraphProof [EQUIVALENT, 0 ms] (276) TRUE (277) QDP (278) QDPSizeChangeProof [EQUIVALENT, 0 ms] (279) YES (280) QDP (281) TransformationProof [EQUIVALENT, 0 ms] (282) QDP (283) UsableRulesProof [EQUIVALENT, 0 ms] (284) QDP (285) QReductionProof [EQUIVALENT, 0 ms] (286) QDP (287) TransformationProof [EQUIVALENT, 0 ms] (288) QDP (289) TransformationProof [EQUIVALENT, 0 ms] (290) QDP (291) TransformationProof [EQUIVALENT, 0 ms] (292) QDP (293) QDPOrderProof [EQUIVALENT, 11 ms] (294) QDP (295) QDPSizeChangeProof [EQUIVALENT, 0 ms] (296) YES (297) QDP (298) QDPSizeChangeProof [EQUIVALENT, 0 ms] (299) YES (300) QDP (301) QDPSizeChangeProof [EQUIVALENT, 0 ms] (302) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\(q,_)->q" is transformed to "q1 (q,_) = q; " The following Lambda expression "\(_,r)->r" is transformed to "r0 (_,r) = r; " The following Lambda expression "\(m,_)->m" is transformed to "m0 (m,_) = m; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) IFR (EQUIVALENT) If Reductions: The following If expression "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" is transformed to "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); primDivNatS0 x y False = Zero; " The following If expression "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" is transformed to "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); primModNatS0 x y False = Succ x; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. Binding Reductions: The bind variable of the following binding Pattern "frac@(Float vv vw)" is replaced by the following term "Float vv vw" The bind variable of the following binding Pattern "frac@(Double ww wx)" is replaced by the following term "Double ww wx" ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "takeWhile p [] = []; takeWhile p (x : xs)|p xx : takeWhile p xs|otherwise[]; " is transformed to "takeWhile p [] = takeWhile3 p []; takeWhile p (x : xs) = takeWhile2 p (x : xs); " "takeWhile1 p x xs True = x : takeWhile p xs; takeWhile1 p x xs False = takeWhile0 p x xs otherwise; " "takeWhile0 p x xs True = []; " "takeWhile2 p (x : xs) = takeWhile1 p x xs (p x); " "takeWhile3 p [] = []; takeWhile3 xy xz = takeWhile2 xy xz; " The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (8) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (9) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "(fromIntegral q,r :% y) where { q = q1 vu30; ; q1 (q,wy) = q; ; r = r0 vu30; ; r0 (wz,r) = r; ; vu30 = quotRem x y; } " are unpacked to the following functions on top level "properFractionQ1 yu yv (q,wy) = q; " "properFractionVu30 yu yv = quotRem yu yv; " "properFractionQ yu yv = properFractionQ1 yu yv (properFractionVu30 yu yv); " "properFractionR yu yv = properFractionR0 yu yv (properFractionVu30 yu yv); " "properFractionR0 yu yv (wz,r) = r; " The bindings of the following Let/Where expression "m where { m = m0 vu6; ; m0 (m,xv) = m; ; vu6 = properFraction x; } " are unpacked to the following functions on top level "truncateVu6 yw = properFraction yw; " "truncateM0 yw (m,xv) = m; " "truncateM yw = truncateM0 yw (truncateVu6 yw); " ---------------------------------------- (10) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (11) NumRed (SOUND) Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. ---------------------------------------- (12) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (13) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="enumFromTo",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="enumFromTo yx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="enumFromTo yx3 yx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="map toEnum (enumFromTo (fromEnum yx3) (fromEnum yx4))",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="map toEnum (numericEnumFromTo (fromEnum yx3) (fromEnum yx4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="map toEnum (takeWhile (flip (<=) (fromEnum yx4)) (numericEnumFrom (fromEnum yx3)))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 8[label="map toEnum (takeWhile (flip (<=) (fromEnum yx4)) (fromEnum yx3 : (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9[label="map toEnum (takeWhile2 (flip (<=) (fromEnum yx4)) (fromEnum yx3 : (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 10[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (flip (<=) (fromEnum yx4) (fromEnum yx3)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 11[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) ((<=) fromEnum yx3 fromEnum yx4))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 12[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (compare (fromEnum yx3) (fromEnum yx4) /= GT))",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 13[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (not (compare (fromEnum yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3]; 14[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromEnum yx3) (numericEnumFrom $! fromEnum yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromEnum yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 15[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncate yx3) (numericEnumFrom $! truncate yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncate yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 16[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM yx3) (numericEnumFrom $! truncateM yx3 + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM yx3) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];16 -> 17[label="",style="solid", color="black", weight=3]; 17[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 yx3 (truncateVu6 yx3)) (numericEnumFrom $! truncateM0 yx3 (truncateVu6 yx3) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 yx3 (truncateVu6 yx3)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];17 -> 18[label="",style="solid", color="black", weight=3]; 18[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 yx3 (properFraction yx3)) (numericEnumFrom $! truncateM0 yx3 (properFraction yx3) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 yx3 (properFraction yx3)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4564[label="yx3/yx30 :% yx31",fontsize=10,color="white",style="solid",shape="box"];18 -> 4564[label="",style="solid", color="burlywood", weight=9]; 4564 -> 19[label="",style="solid", color="burlywood", weight=3]; 19[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31))) (numericEnumFrom $! truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 (yx30 :% yx31) (properFraction (yx30 :% yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 20[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31)) (numericEnumFrom $! truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (truncateM0 (yx30 :% yx31) (fromIntegral (properFractionQ yx30 yx31),properFractionR yx30 yx31 :% yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3]; 21[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromIntegral (properFractionQ yx30 yx31)) (numericEnumFrom $! fromIntegral (properFractionQ yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromIntegral (properFractionQ yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3]; 22[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger . toInteger) (numericEnumFrom $! fromInteger . toInteger + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger . toInteger) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3]; 23[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger (toInteger (properFractionQ yx30 yx31))) (numericEnumFrom $! fromInteger (toInteger (properFractionQ yx30 yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger (toInteger (properFractionQ yx30 yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];23 -> 24[label="",style="solid", color="black", weight=3]; 24[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (fromInteger (Integer (properFractionQ yx30 yx31))) (numericEnumFrom $! fromInteger (Integer (properFractionQ yx30 yx31)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (fromInteger (Integer (properFractionQ yx30 yx31))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];24 -> 25[label="",style="solid", color="black", weight=3]; 25[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ yx30 yx31) (numericEnumFrom $! properFractionQ yx30 yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ yx30 yx31) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3]; 26[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (properFractionVu30 yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];26 -> 27[label="",style="solid", color="black", weight=3]; 27[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (quotRem yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (quotRem yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (quotRem yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];27 -> 28[label="",style="solid", color="black", weight=3]; 28[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (primQrmInt yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3]; 29[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31)) (numericEnumFrom $! properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31) + fromInt (Pos (Succ Zero))) (not (primCmpInt (properFractionQ1 yx30 yx31 (primQuotInt yx30 yx31,primRemInt yx30 yx31)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3]; 30[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt yx30 yx31) (numericEnumFrom $! primQuotInt yx30 yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt yx30 yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4565[label="yx30/Pos yx300",fontsize=10,color="white",style="solid",shape="box"];30 -> 4565[label="",style="solid", color="burlywood", weight=9]; 4565 -> 31[label="",style="solid", color="burlywood", weight=3]; 4566[label="yx30/Neg yx300",fontsize=10,color="white",style="solid",shape="box"];30 -> 4566[label="",style="solid", color="burlywood", weight=9]; 4566 -> 32[label="",style="solid", color="burlywood", weight=3]; 31[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) yx31) (numericEnumFrom $! primQuotInt (Pos yx300) yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4567[label="yx31/Pos yx310",fontsize=10,color="white",style="solid",shape="box"];31 -> 4567[label="",style="solid", color="burlywood", weight=9]; 4567 -> 33[label="",style="solid", color="burlywood", weight=3]; 4568[label="yx31/Neg yx310",fontsize=10,color="white",style="solid",shape="box"];31 -> 4568[label="",style="solid", color="burlywood", weight=9]; 4568 -> 34[label="",style="solid", color="burlywood", weight=3]; 32[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) yx31) (numericEnumFrom $! primQuotInt (Neg yx300) yx31 + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) yx31) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4569[label="yx31/Pos yx310",fontsize=10,color="white",style="solid",shape="box"];32 -> 4569[label="",style="solid", color="burlywood", weight=9]; 4569 -> 35[label="",style="solid", color="burlywood", weight=3]; 4570[label="yx31/Neg yx310",fontsize=10,color="white",style="solid",shape="box"];32 -> 4570[label="",style="solid", color="burlywood", weight=9]; 4570 -> 36[label="",style="solid", color="burlywood", weight=3]; 33[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos yx310)) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4571[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];33 -> 4571[label="",style="solid", color="burlywood", weight=9]; 4571 -> 37[label="",style="solid", color="burlywood", weight=3]; 4572[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 4572[label="",style="solid", color="burlywood", weight=9]; 4572 -> 38[label="",style="solid", color="burlywood", weight=3]; 34[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg yx310)) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4573[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];34 -> 4573[label="",style="solid", color="burlywood", weight=9]; 4573 -> 39[label="",style="solid", color="burlywood", weight=3]; 4574[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 4574[label="",style="solid", color="burlywood", weight=9]; 4574 -> 40[label="",style="solid", color="burlywood", weight=3]; 35[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos yx310)) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4575[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];35 -> 4575[label="",style="solid", color="burlywood", weight=9]; 4575 -> 41[label="",style="solid", color="burlywood", weight=3]; 4576[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 4576[label="",style="solid", color="burlywood", weight=9]; 4576 -> 42[label="",style="solid", color="burlywood", weight=3]; 36[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg yx310)) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg yx310) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg yx310)) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4577[label="yx310/Succ yx3100",fontsize=10,color="white",style="solid",shape="box"];36 -> 4577[label="",style="solid", color="burlywood", weight=9]; 4577 -> 43[label="",style="solid", color="burlywood", weight=3]; 4578[label="yx310/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 4578[label="",style="solid", color="burlywood", weight=9]; 4578 -> 44[label="",style="solid", color="burlywood", weight=3]; 37[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos (Succ yx3100))) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];37 -> 45[label="",style="solid", color="black", weight=3]; 38[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Pos Zero)) (numericEnumFrom $! primQuotInt (Pos yx300) (Pos Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Pos Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3]; 39[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg (Succ yx3100))) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3]; 40[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Pos yx300) (Neg Zero)) (numericEnumFrom $! primQuotInt (Pos yx300) (Neg Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Pos yx300) (Neg Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3]; 41[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos (Succ yx3100))) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3]; 42[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Pos Zero)) (numericEnumFrom $! primQuotInt (Neg yx300) (Pos Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Pos Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3]; 43[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg (Succ yx3100))) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3]; 44[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (primQuotInt (Neg yx300) (Neg Zero)) (numericEnumFrom $! primQuotInt (Neg yx300) (Neg Zero) + fromInt (Pos (Succ Zero))) (not (primCmpInt (primQuotInt (Neg yx300) (Neg Zero)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3]; 45 -> 3761[label="",style="dashed", color="red", weight=0]; 45[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Pos (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Pos (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];45 -> 3762[label="",style="dashed", color="magenta", weight=3]; 45 -> 3763[label="",style="dashed", color="magenta", weight=3]; 45 -> 3764[label="",style="dashed", color="magenta", weight=3]; 45 -> 3765[label="",style="dashed", color="magenta", weight=3]; 45 -> 3766[label="",style="dashed", color="magenta", weight=3]; 46[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="triangle"];46 -> 55[label="",style="solid", color="black", weight=3]; 47[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4579[label="yx300/Succ yx3000",fontsize=10,color="white",style="solid",shape="box"];47 -> 4579[label="",style="solid", color="burlywood", weight=9]; 4579 -> 56[label="",style="solid", color="burlywood", weight=3]; 4580[label="yx300/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 4580[label="",style="solid", color="burlywood", weight=9]; 4580 -> 57[label="",style="solid", color="burlywood", weight=3]; 48 -> 46[label="",style="dashed", color="red", weight=0]; 48[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];49 -> 47[label="",style="dashed", color="red", weight=0]; 49[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];49 -> 58[label="",style="dashed", color="magenta", weight=3]; 49 -> 59[label="",style="dashed", color="magenta", weight=3]; 50 -> 46[label="",style="dashed", color="red", weight=0]; 50[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];51 -> 3761[label="",style="dashed", color="red", weight=0]; 51[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Pos (primDivNatS yx300 (Succ yx3100))) (numericEnumFrom $! Pos (primDivNatS yx300 (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Pos (primDivNatS yx300 (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];51 -> 3767[label="",style="dashed", color="magenta", weight=3]; 51 -> 3768[label="",style="dashed", color="magenta", weight=3]; 51 -> 3769[label="",style="dashed", color="magenta", weight=3]; 51 -> 3770[label="",style="dashed", color="magenta", weight=3]; 51 -> 3771[label="",style="dashed", color="magenta", weight=3]; 52 -> 46[label="",style="dashed", color="red", weight=0]; 52[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (error []) (numericEnumFrom $! error [] + fromInt (Pos (Succ Zero))) (not (primCmpInt (error []) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];3762 -> 576[label="",style="dashed", color="red", weight=0]; 3762[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3762 -> 3997[label="",style="dashed", color="magenta", weight=3]; 3763 -> 1689[label="",style="dashed", color="red", weight=0]; 3763[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3763 -> 3998[label="",style="dashed", color="magenta", weight=3]; 3763 -> 3999[label="",style="dashed", color="magenta", weight=3]; 3764 -> 1689[label="",style="dashed", color="red", weight=0]; 3764[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3764 -> 4000[label="",style="dashed", color="magenta", weight=3]; 3764 -> 4001[label="",style="dashed", color="magenta", weight=3]; 3765 -> 460[label="",style="dashed", color="red", weight=0]; 3765[label="fromEnum yx4",fontsize=16,color="magenta"];3765 -> 4002[label="",style="dashed", color="magenta", weight=3]; 3766 -> 1689[label="",style="dashed", color="red", weight=0]; 3766[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3766 -> 4003[label="",style="dashed", color="magenta", weight=3]; 3766 -> 4004[label="",style="dashed", color="magenta", weight=3]; 3761[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos yx251) yx199 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4581[label="yx251/Succ yx2510",fontsize=10,color="white",style="solid",shape="box"];3761 -> 4581[label="",style="solid", color="burlywood", weight=9]; 4581 -> 4005[label="",style="solid", color="burlywood", weight=3]; 4582[label="yx251/Zero",fontsize=10,color="white",style="solid",shape="box"];3761 -> 4582[label="",style="solid", color="burlywood", weight=9]; 4582 -> 4006[label="",style="solid", color="burlywood", weight=3]; 55[label="error []",fontsize=16,color="red",shape="box"];56[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS (Succ yx3000) (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS (Succ yx3000) (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS (Succ yx3000) (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3]; 57[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS Zero (Succ yx3100))) (numericEnumFrom $! Neg (primDivNatS Zero (Succ yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS Zero (Succ yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];57 -> 65[label="",style="solid", color="black", weight=3]; 58[label="yx3100",fontsize=16,color="green",shape="box"];59[label="yx300",fontsize=16,color="green",shape="box"];3767 -> 576[label="",style="dashed", color="red", weight=0]; 3767[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3767 -> 4007[label="",style="dashed", color="magenta", weight=3]; 3768 -> 1689[label="",style="dashed", color="red", weight=0]; 3768[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3768 -> 4008[label="",style="dashed", color="magenta", weight=3]; 3768 -> 4009[label="",style="dashed", color="magenta", weight=3]; 3769 -> 1689[label="",style="dashed", color="red", weight=0]; 3769[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3769 -> 4010[label="",style="dashed", color="magenta", weight=3]; 3769 -> 4011[label="",style="dashed", color="magenta", weight=3]; 3770 -> 460[label="",style="dashed", color="red", weight=0]; 3770[label="fromEnum yx4",fontsize=16,color="magenta"];3770 -> 4012[label="",style="dashed", color="magenta", weight=3]; 3771 -> 1689[label="",style="dashed", color="red", weight=0]; 3771[label="primDivNatS yx300 (Succ yx3100)",fontsize=16,color="magenta"];3771 -> 4013[label="",style="dashed", color="magenta", weight=3]; 3771 -> 4014[label="",style="dashed", color="magenta", weight=3]; 3997[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];576[label="fromInt yx8",fontsize=16,color="black",shape="triangle"];576 -> 650[label="",style="solid", color="black", weight=3]; 3998[label="yx3100",fontsize=16,color="green",shape="box"];3999[label="yx300",fontsize=16,color="green",shape="box"];1689[label="primDivNatS yx400 (Succ yx4100)",fontsize=16,color="burlywood",shape="triangle"];4583[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];1689 -> 4583[label="",style="solid", color="burlywood", weight=9]; 4583 -> 1725[label="",style="solid", color="burlywood", weight=3]; 4584[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];1689 -> 4584[label="",style="solid", color="burlywood", weight=9]; 4584 -> 1726[label="",style="solid", color="burlywood", weight=3]; 4000[label="yx3100",fontsize=16,color="green",shape="box"];4001[label="yx300",fontsize=16,color="green",shape="box"];4002[label="yx4",fontsize=16,color="green",shape="box"];460[label="fromEnum yx7",fontsize=16,color="black",shape="triangle"];460 -> 519[label="",style="solid", color="black", weight=3]; 4003[label="yx3100",fontsize=16,color="green",shape="box"];4004[label="yx300",fontsize=16,color="green",shape="box"];4005[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos (Succ yx2510)) yx199 == GT)))",fontsize=16,color="burlywood",shape="box"];4585[label="yx199/Pos yx1990",fontsize=10,color="white",style="solid",shape="box"];4005 -> 4585[label="",style="solid", color="burlywood", weight=9]; 4585 -> 4025[label="",style="solid", color="burlywood", weight=3]; 4586[label="yx199/Neg yx1990",fontsize=10,color="white",style="solid",shape="box"];4005 -> 4586[label="",style="solid", color="burlywood", weight=9]; 4586 -> 4026[label="",style="solid", color="burlywood", weight=3]; 4006[label="map toEnum (takeWhile1 (flip (<=) yx199) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) yx199 == GT)))",fontsize=16,color="burlywood",shape="box"];4587[label="yx199/Pos yx1990",fontsize=10,color="white",style="solid",shape="box"];4006 -> 4587[label="",style="solid", color="burlywood", weight=9]; 4587 -> 4027[label="",style="solid", color="burlywood", weight=3]; 4588[label="yx199/Neg yx1990",fontsize=10,color="white",style="solid",shape="box"];4006 -> 4588[label="",style="solid", color="burlywood", weight=9]; 4588 -> 4028[label="",style="solid", color="burlywood", weight=3]; 64[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100))) (numericEnumFrom $! Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 yx3000 yx3100 (primGEqNatS yx3000 yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4589[label="yx3000/Succ yx30000",fontsize=10,color="white",style="solid",shape="box"];64 -> 4589[label="",style="solid", color="burlywood", weight=9]; 4589 -> 69[label="",style="solid", color="burlywood", weight=3]; 4590[label="yx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 4590[label="",style="solid", color="burlywood", weight=9]; 4590 -> 70[label="",style="solid", color="burlywood", weight=3]; 65[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="triangle"];65 -> 71[label="",style="solid", color="black", weight=3]; 4007[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4008[label="yx3100",fontsize=16,color="green",shape="box"];4009[label="yx300",fontsize=16,color="green",shape="box"];4010[label="yx3100",fontsize=16,color="green",shape="box"];4011[label="yx300",fontsize=16,color="green",shape="box"];4012[label="yx4",fontsize=16,color="green",shape="box"];4013[label="yx3100",fontsize=16,color="green",shape="box"];4014[label="yx300",fontsize=16,color="green",shape="box"];650[label="yx8",fontsize=16,color="green",shape="box"];1725[label="primDivNatS (Succ yx4000) (Succ yx4100)",fontsize=16,color="black",shape="box"];1725 -> 1745[label="",style="solid", color="black", weight=3]; 1726[label="primDivNatS Zero (Succ yx4100)",fontsize=16,color="black",shape="box"];1726 -> 1746[label="",style="solid", color="black", weight=3]; 519[label="truncate yx7",fontsize=16,color="black",shape="box"];519 -> 578[label="",style="solid", color="black", weight=3]; 4025[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos (Succ yx2510)) (Pos yx1990) == GT)))",fontsize=16,color="black",shape="box"];4025 -> 4044[label="",style="solid", color="black", weight=3]; 4026[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos (Succ yx2510)) (Neg yx1990) == GT)))",fontsize=16,color="black",shape="box"];4026 -> 4045[label="",style="solid", color="black", weight=3]; 4027[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Pos yx1990) == GT)))",fontsize=16,color="burlywood",shape="box"];4591[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];4027 -> 4591[label="",style="solid", color="burlywood", weight=9]; 4591 -> 4046[label="",style="solid", color="burlywood", weight=3]; 4592[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];4027 -> 4592[label="",style="solid", color="burlywood", weight=9]; 4592 -> 4047[label="",style="solid", color="burlywood", weight=3]; 4028[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Neg yx1990) == GT)))",fontsize=16,color="burlywood",shape="box"];4593[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];4028 -> 4593[label="",style="solid", color="burlywood", weight=9]; 4593 -> 4048[label="",style="solid", color="burlywood", weight=3]; 4594[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];4028 -> 4594[label="",style="solid", color="burlywood", weight=9]; 4594 -> 4049[label="",style="solid", color="burlywood", weight=3]; 69[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) yx3100 (primGEqNatS (Succ yx30000) yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4595[label="yx3100/Succ yx31000",fontsize=10,color="white",style="solid",shape="box"];69 -> 4595[label="",style="solid", color="burlywood", weight=9]; 4595 -> 77[label="",style="solid", color="burlywood", weight=3]; 4596[label="yx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 4596[label="",style="solid", color="burlywood", weight=9]; 4596 -> 78[label="",style="solid", color="burlywood", weight=3]; 70[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100))) (numericEnumFrom $! Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero yx3100 (primGEqNatS Zero yx3100))) (fromEnum yx4) == GT)))",fontsize=16,color="burlywood",shape="box"];4597[label="yx3100/Succ yx31000",fontsize=10,color="white",style="solid",shape="box"];70 -> 4597[label="",style="solid", color="burlywood", weight=9]; 4597 -> 79[label="",style="solid", color="burlywood", weight=3]; 4598[label="yx3100/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 4598[label="",style="solid", color="burlywood", weight=9]; 4598 -> 80[label="",style="solid", color="burlywood", weight=3]; 71[label="map toEnum (takeWhile1 (flip (<=) (truncate yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncate yx4) == GT)))",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3]; 1745 -> 1151[label="",style="dashed", color="red", weight=0]; 1745[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1746[label="Zero",fontsize=16,color="green",shape="box"];578[label="truncateM yx7",fontsize=16,color="black",shape="box"];578 -> 652[label="",style="solid", color="black", weight=3]; 4044[label="map toEnum (takeWhile1 (flip (<=) (Pos yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpNat (Succ yx2510) yx1990 == GT)))",fontsize=16,color="burlywood",shape="box"];4599[label="yx1990/Succ yx19900",fontsize=10,color="white",style="solid",shape="box"];4044 -> 4599[label="",style="solid", color="burlywood", weight=9]; 4599 -> 4052[label="",style="solid", color="burlywood", weight=3]; 4600[label="yx1990/Zero",fontsize=10,color="white",style="solid",shape="box"];4044 -> 4600[label="",style="solid", color="burlywood", weight=9]; 4600 -> 4053[label="",style="solid", color="burlywood", weight=3]; 4045[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (GT == GT)))",fontsize=16,color="black",shape="triangle"];4045 -> 4054[label="",style="solid", color="black", weight=3]; 4046[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Pos (Succ yx19900)) == GT)))",fontsize=16,color="black",shape="box"];4046 -> 4055[label="",style="solid", color="black", weight=3]; 4047[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];4047 -> 4056[label="",style="solid", color="black", weight=3]; 4048[label="map toEnum (takeWhile1 (flip (<=) (Neg (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Neg (Succ yx19900)) == GT)))",fontsize=16,color="black",shape="box"];4048 -> 4057[label="",style="solid", color="black", weight=3]; 4049[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpInt (Pos Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];4049 -> 4058[label="",style="solid", color="black", weight=3]; 77[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS (Succ yx30000) (Succ yx31000)))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];77 -> 87[label="",style="solid", color="black", weight=3]; 78[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) Zero (primGEqNatS (Succ yx30000) Zero))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];78 -> 88[label="",style="solid", color="black", weight=3]; 79[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000)))) (numericEnumFrom $! Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero (Succ yx31000) (primGEqNatS Zero (Succ yx31000)))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];79 -> 89[label="",style="solid", color="black", weight=3]; 80[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero))) (numericEnumFrom $! Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero Zero (primGEqNatS Zero Zero))) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];80 -> 90[label="",style="solid", color="black", weight=3]; 81[label="map toEnum (takeWhile1 (flip (<=) (truncateM yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM yx4) == GT)))",fontsize=16,color="black",shape="box"];81 -> 91[label="",style="solid", color="black", weight=3]; 1151[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="burlywood",shape="triangle"];4601[label="yx4000/Succ yx40000",fontsize=10,color="white",style="solid",shape="box"];1151 -> 4601[label="",style="solid", color="burlywood", weight=9]; 4601 -> 1276[label="",style="solid", color="burlywood", weight=3]; 4602[label="yx4000/Zero",fontsize=10,color="white",style="solid",shape="box"];1151 -> 4602[label="",style="solid", color="burlywood", weight=9]; 4602 -> 1277[label="",style="solid", color="burlywood", weight=3]; 652[label="truncateM0 yx7 (truncateVu6 yx7)",fontsize=16,color="black",shape="box"];652 -> 821[label="",style="solid", color="black", weight=3]; 4052[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpNat (Succ yx2510) (Succ yx19900) == GT)))",fontsize=16,color="black",shape="box"];4052 -> 4063[label="",style="solid", color="black", weight=3]; 4053[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpNat (Succ yx2510) Zero == GT)))",fontsize=16,color="black",shape="box"];4053 -> 4064[label="",style="solid", color="black", weight=3]; 4054[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not True))",fontsize=16,color="black",shape="box"];4054 -> 4065[label="",style="solid", color="black", weight=3]; 4055 -> 4428[label="",style="dashed", color="red", weight=0]; 4055[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpNat Zero (Succ yx19900) == GT)))",fontsize=16,color="magenta"];4055 -> 4429[label="",style="dashed", color="magenta", weight=3]; 4055 -> 4430[label="",style="dashed", color="magenta", weight=3]; 4055 -> 4431[label="",style="dashed", color="magenta", weight=3]; 4055 -> 4432[label="",style="dashed", color="magenta", weight=3]; 4055 -> 4433[label="",style="dashed", color="magenta", weight=3]; 4055 -> 4434[label="",style="dashed", color="magenta", weight=3]; 4056[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];4056 -> 4067[label="",style="solid", color="black", weight=3]; 4057 -> 4045[label="",style="dashed", color="red", weight=0]; 4057[label="map toEnum (takeWhile1 (flip (<=) (Neg (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (GT == GT)))",fontsize=16,color="magenta"];4057 -> 4068[label="",style="dashed", color="magenta", weight=3]; 4058[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];4058 -> 4069[label="",style="solid", color="black", weight=3]; 87 -> 2234[label="",style="dashed", color="red", weight=0]; 87[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000)) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) (Succ yx31000) (primGEqNatS yx30000 yx31000))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];87 -> 2235[label="",style="dashed", color="magenta", weight=3]; 87 -> 2236[label="",style="dashed", color="magenta", weight=3]; 87 -> 2237[label="",style="dashed", color="magenta", weight=3]; 87 -> 2238[label="",style="dashed", color="magenta", weight=3]; 87 -> 2239[label="",style="dashed", color="magenta", weight=3]; 87 -> 2240[label="",style="dashed", color="magenta", weight=3]; 87 -> 2241[label="",style="dashed", color="magenta", weight=3]; 88[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 (Succ yx30000) Zero True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx30000) Zero True) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 (Succ yx30000) Zero True)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];88 -> 100[label="",style="solid", color="black", weight=3]; 89[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero (Succ yx31000) False)) (numericEnumFrom $! Neg (primDivNatS0 Zero (Succ yx31000) False) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero (Succ yx31000) False)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];89 -> 101[label="",style="solid", color="black", weight=3]; 90[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (primDivNatS0 Zero Zero True)) (numericEnumFrom $! Neg (primDivNatS0 Zero Zero True) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (primDivNatS0 Zero Zero True)) (fromEnum yx4) == GT)))",fontsize=16,color="black",shape="box"];90 -> 102[label="",style="solid", color="black", weight=3]; 91[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 yx4 (truncateVu6 yx4))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 yx4 (truncateVu6 yx4)) == GT)))",fontsize=16,color="black",shape="box"];91 -> 103[label="",style="solid", color="black", weight=3]; 1276[label="primDivNatS0 (Succ yx40000) yx4100 (primGEqNatS (Succ yx40000) yx4100)",fontsize=16,color="burlywood",shape="box"];4603[label="yx4100/Succ yx41000",fontsize=10,color="white",style="solid",shape="box"];1276 -> 4603[label="",style="solid", color="burlywood", weight=9]; 4603 -> 1405[label="",style="solid", color="burlywood", weight=3]; 4604[label="yx4100/Zero",fontsize=10,color="white",style="solid",shape="box"];1276 -> 4604[label="",style="solid", color="burlywood", weight=9]; 4604 -> 1406[label="",style="solid", color="burlywood", weight=3]; 1277[label="primDivNatS0 Zero yx4100 (primGEqNatS Zero yx4100)",fontsize=16,color="burlywood",shape="box"];4605[label="yx4100/Succ yx41000",fontsize=10,color="white",style="solid",shape="box"];1277 -> 4605[label="",style="solid", color="burlywood", weight=9]; 4605 -> 1407[label="",style="solid", color="burlywood", weight=3]; 4606[label="yx4100/Zero",fontsize=10,color="white",style="solid",shape="box"];1277 -> 4606[label="",style="solid", color="burlywood", weight=9]; 4606 -> 1408[label="",style="solid", color="burlywood", weight=3]; 821[label="truncateM0 yx7 (properFraction yx7)",fontsize=16,color="burlywood",shape="box"];4607[label="yx7/yx70 :% yx71",fontsize=10,color="white",style="solid",shape="box"];821 -> 4607[label="",style="solid", color="burlywood", weight=9]; 4607 -> 1033[label="",style="solid", color="burlywood", weight=3]; 4063 -> 4428[label="",style="dashed", color="red", weight=0]; 4063[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx19900))) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (primCmpNat yx2510 yx19900 == GT)))",fontsize=16,color="magenta"];4063 -> 4435[label="",style="dashed", color="magenta", weight=3]; 4063 -> 4436[label="",style="dashed", color="magenta", weight=3]; 4063 -> 4437[label="",style="dashed", color="magenta", weight=3]; 4063 -> 4438[label="",style="dashed", color="magenta", weight=3]; 4063 -> 4439[label="",style="dashed", color="magenta", weight=3]; 4063 -> 4440[label="",style="dashed", color="magenta", weight=3]; 4064[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not (GT == GT)))",fontsize=16,color="black",shape="box"];4064 -> 4074[label="",style="solid", color="black", weight=3]; 4065[label="map toEnum (takeWhile1 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) False)",fontsize=16,color="black",shape="box"];4065 -> 4075[label="",style="solid", color="black", weight=3]; 4429[label="yx249",fontsize=16,color="green",shape="box"];4430[label="yx237",fontsize=16,color="green",shape="box"];4431[label="Zero",fontsize=16,color="green",shape="box"];4432[label="Succ yx19900",fontsize=16,color="green",shape="box"];4433[label="yx250",fontsize=16,color="green",shape="box"];4434[label="yx19900",fontsize=16,color="green",shape="box"];4428[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat yx276 yx277 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4608[label="yx276/Succ yx2760",fontsize=10,color="white",style="solid",shape="box"];4428 -> 4608[label="",style="solid", color="burlywood", weight=9]; 4608 -> 4489[label="",style="solid", color="burlywood", weight=3]; 4609[label="yx276/Zero",fontsize=10,color="white",style="solid",shape="box"];4428 -> 4609[label="",style="solid", color="burlywood", weight=9]; 4609 -> 4490[label="",style="solid", color="burlywood", weight=3]; 4067[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not False))",fontsize=16,color="black",shape="box"];4067 -> 4077[label="",style="solid", color="black", weight=3]; 4068[label="Succ yx19900",fontsize=16,color="green",shape="box"];4069[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not False))",fontsize=16,color="black",shape="box"];4069 -> 4078[label="",style="solid", color="black", weight=3]; 2235[label="yx31000",fontsize=16,color="green",shape="box"];2236[label="yx31000",fontsize=16,color="green",shape="box"];2237 -> 460[label="",style="dashed", color="red", weight=0]; 2237[label="fromEnum yx4",fontsize=16,color="magenta"];2237 -> 2312[label="",style="dashed", color="magenta", weight=3]; 2238[label="yx30000",fontsize=16,color="green",shape="box"];2239 -> 460[label="",style="dashed", color="red", weight=0]; 2239[label="fromEnum yx4",fontsize=16,color="magenta"];2239 -> 2313[label="",style="dashed", color="magenta", weight=3]; 2240 -> 576[label="",style="dashed", color="red", weight=0]; 2240[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2240 -> 2314[label="",style="dashed", color="magenta", weight=3]; 2241[label="yx30000",fontsize=16,color="green",shape="box"];2234[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx210 yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4610[label="yx210/Succ yx2100",fontsize=10,color="white",style="solid",shape="box"];2234 -> 4610[label="",style="solid", color="burlywood", weight=9]; 4610 -> 2315[label="",style="solid", color="burlywood", weight=3]; 4611[label="yx210/Zero",fontsize=10,color="white",style="solid",shape="box"];2234 -> 4611[label="",style="solid", color="burlywood", weight=9]; 4611 -> 2316[label="",style="solid", color="burlywood", weight=3]; 100 -> 2772[label="",style="dashed", color="red", weight=0]; 100[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];100 -> 2773[label="",style="dashed", color="magenta", weight=3]; 100 -> 2774[label="",style="dashed", color="magenta", weight=3]; 100 -> 2775[label="",style="dashed", color="magenta", weight=3]; 100 -> 2776[label="",style="dashed", color="magenta", weight=3]; 100 -> 2777[label="",style="dashed", color="magenta", weight=3]; 100 -> 2778[label="",style="dashed", color="magenta", weight=3]; 101 -> 65[label="",style="dashed", color="red", weight=0]; 101[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];102 -> 2772[label="",style="dashed", color="red", weight=0]; 102[label="map toEnum (takeWhile1 (flip (<=) (fromEnum yx4)) (Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))) + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (fromEnum yx4) == GT)))",fontsize=16,color="magenta"];102 -> 2779[label="",style="dashed", color="magenta", weight=3]; 102 -> 2780[label="",style="dashed", color="magenta", weight=3]; 102 -> 2781[label="",style="dashed", color="magenta", weight=3]; 102 -> 2782[label="",style="dashed", color="magenta", weight=3]; 102 -> 2783[label="",style="dashed", color="magenta", weight=3]; 102 -> 2784[label="",style="dashed", color="magenta", weight=3]; 103[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 yx4 (properFraction yx4))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 yx4 (properFraction yx4)) == GT)))",fontsize=16,color="burlywood",shape="box"];4612[label="yx4/yx40 :% yx41",fontsize=10,color="white",style="solid",shape="box"];103 -> 4612[label="",style="solid", color="burlywood", weight=9]; 4612 -> 117[label="",style="solid", color="burlywood", weight=3]; 1405[label="primDivNatS0 (Succ yx40000) (Succ yx41000) (primGEqNatS (Succ yx40000) (Succ yx41000))",fontsize=16,color="black",shape="box"];1405 -> 1541[label="",style="solid", color="black", weight=3]; 1406[label="primDivNatS0 (Succ yx40000) Zero (primGEqNatS (Succ yx40000) Zero)",fontsize=16,color="black",shape="box"];1406 -> 1542[label="",style="solid", color="black", weight=3]; 1407[label="primDivNatS0 Zero (Succ yx41000) (primGEqNatS Zero (Succ yx41000))",fontsize=16,color="black",shape="box"];1407 -> 1543[label="",style="solid", color="black", weight=3]; 1408[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1408 -> 1544[label="",style="solid", color="black", weight=3]; 1033[label="truncateM0 (yx70 :% yx71) (properFraction (yx70 :% yx71))",fontsize=16,color="black",shape="box"];1033 -> 1174[label="",style="solid", color="black", weight=3]; 4435[label="yx249",fontsize=16,color="green",shape="box"];4436[label="yx237",fontsize=16,color="green",shape="box"];4437[label="yx2510",fontsize=16,color="green",shape="box"];4438[label="yx19900",fontsize=16,color="green",shape="box"];4439[label="yx250",fontsize=16,color="green",shape="box"];4440[label="yx19900",fontsize=16,color="green",shape="box"];4074[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) (not True))",fontsize=16,color="black",shape="box"];4074 -> 4083[label="",style="solid", color="black", weight=3]; 4075[label="map toEnum (takeWhile0 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) otherwise)",fontsize=16,color="black",shape="box"];4075 -> 4084[label="",style="solid", color="black", weight=3]; 4489[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat (Succ yx2760) yx277 == GT)))",fontsize=16,color="burlywood",shape="box"];4613[label="yx277/Succ yx2770",fontsize=10,color="white",style="solid",shape="box"];4489 -> 4613[label="",style="solid", color="burlywood", weight=9]; 4613 -> 4491[label="",style="solid", color="burlywood", weight=3]; 4614[label="yx277/Zero",fontsize=10,color="white",style="solid",shape="box"];4489 -> 4614[label="",style="solid", color="burlywood", weight=9]; 4614 -> 4492[label="",style="solid", color="burlywood", weight=3]; 4490[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat Zero yx277 == GT)))",fontsize=16,color="burlywood",shape="box"];4615[label="yx277/Succ yx2770",fontsize=10,color="white",style="solid",shape="box"];4490 -> 4615[label="",style="solid", color="burlywood", weight=9]; 4615 -> 4493[label="",style="solid", color="burlywood", weight=3]; 4616[label="yx277/Zero",fontsize=10,color="white",style="solid",shape="box"];4490 -> 4616[label="",style="solid", color="burlywood", weight=9]; 4616 -> 4494[label="",style="solid", color="burlywood", weight=3]; 4077[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) True)",fontsize=16,color="black",shape="box"];4077 -> 4086[label="",style="solid", color="black", weight=3]; 4078[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) True)",fontsize=16,color="black",shape="box"];4078 -> 4087[label="",style="solid", color="black", weight=3]; 2312[label="yx4",fontsize=16,color="green",shape="box"];2313[label="yx4",fontsize=16,color="green",shape="box"];2314[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];2315[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="box"];4617[label="yx211/Succ yx2110",fontsize=10,color="white",style="solid",shape="box"];2315 -> 4617[label="",style="solid", color="burlywood", weight=9]; 4617 -> 2345[label="",style="solid", color="burlywood", weight=3]; 4618[label="yx211/Zero",fontsize=10,color="white",style="solid",shape="box"];2315 -> 4618[label="",style="solid", color="burlywood", weight=9]; 4618 -> 2346[label="",style="solid", color="burlywood", weight=3]; 2316[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero yx211))) yx213 == GT)))",fontsize=16,color="burlywood",shape="box"];4619[label="yx211/Succ yx2110",fontsize=10,color="white",style="solid",shape="box"];2316 -> 4619[label="",style="solid", color="burlywood", weight=9]; 4619 -> 2347[label="",style="solid", color="burlywood", weight=3]; 4620[label="yx211/Zero",fontsize=10,color="white",style="solid",shape="box"];2316 -> 4620[label="",style="solid", color="burlywood", weight=9]; 4620 -> 2348[label="",style="solid", color="burlywood", weight=3]; 2773 -> 1689[label="",style="dashed", color="red", weight=0]; 2773[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2773 -> 2992[label="",style="dashed", color="magenta", weight=3]; 2773 -> 2993[label="",style="dashed", color="magenta", weight=3]; 2774 -> 460[label="",style="dashed", color="red", weight=0]; 2774[label="fromEnum yx4",fontsize=16,color="magenta"];2774 -> 2994[label="",style="dashed", color="magenta", weight=3]; 2775 -> 1689[label="",style="dashed", color="red", weight=0]; 2775[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2775 -> 2995[label="",style="dashed", color="magenta", weight=3]; 2775 -> 2996[label="",style="dashed", color="magenta", weight=3]; 2776 -> 1689[label="",style="dashed", color="red", weight=0]; 2776[label="primDivNatS (primMinusNatS (Succ yx30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2776 -> 2997[label="",style="dashed", color="magenta", weight=3]; 2776 -> 2998[label="",style="dashed", color="magenta", weight=3]; 2777 -> 460[label="",style="dashed", color="red", weight=0]; 2777[label="fromEnum yx4",fontsize=16,color="magenta"];2777 -> 2999[label="",style="dashed", color="magenta", weight=3]; 2778 -> 576[label="",style="dashed", color="red", weight=0]; 2778[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2778 -> 3000[label="",style="dashed", color="magenta", weight=3]; 2772[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4621[label="yx213/Pos yx2130",fontsize=10,color="white",style="solid",shape="box"];2772 -> 4621[label="",style="solid", color="burlywood", weight=9]; 4621 -> 3001[label="",style="solid", color="burlywood", weight=3]; 4622[label="yx213/Neg yx2130",fontsize=10,color="white",style="solid",shape="box"];2772 -> 4622[label="",style="solid", color="burlywood", weight=9]; 4622 -> 3002[label="",style="solid", color="burlywood", weight=3]; 2779 -> 1689[label="",style="dashed", color="red", weight=0]; 2779[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2779 -> 3003[label="",style="dashed", color="magenta", weight=3]; 2779 -> 3004[label="",style="dashed", color="magenta", weight=3]; 2780 -> 460[label="",style="dashed", color="red", weight=0]; 2780[label="fromEnum yx4",fontsize=16,color="magenta"];2780 -> 3005[label="",style="dashed", color="magenta", weight=3]; 2781 -> 1689[label="",style="dashed", color="red", weight=0]; 2781[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2781 -> 3006[label="",style="dashed", color="magenta", weight=3]; 2781 -> 3007[label="",style="dashed", color="magenta", weight=3]; 2782 -> 1689[label="",style="dashed", color="red", weight=0]; 2782[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2782 -> 3008[label="",style="dashed", color="magenta", weight=3]; 2782 -> 3009[label="",style="dashed", color="magenta", weight=3]; 2783 -> 460[label="",style="dashed", color="red", weight=0]; 2783[label="fromEnum yx4",fontsize=16,color="magenta"];2783 -> 3010[label="",style="dashed", color="magenta", weight=3]; 2784 -> 576[label="",style="dashed", color="red", weight=0]; 2784[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];2784 -> 3011[label="",style="dashed", color="magenta", weight=3]; 117[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 (yx40 :% yx41) (properFraction (yx40 :% yx41)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 (yx40 :% yx41) (properFraction (yx40 :% yx41))) == GT)))",fontsize=16,color="black",shape="box"];117 -> 131[label="",style="solid", color="black", weight=3]; 1541 -> 3700[label="",style="dashed", color="red", weight=0]; 1541[label="primDivNatS0 (Succ yx40000) (Succ yx41000) (primGEqNatS yx40000 yx41000)",fontsize=16,color="magenta"];1541 -> 3701[label="",style="dashed", color="magenta", weight=3]; 1541 -> 3702[label="",style="dashed", color="magenta", weight=3]; 1541 -> 3703[label="",style="dashed", color="magenta", weight=3]; 1541 -> 3704[label="",style="dashed", color="magenta", weight=3]; 1542[label="primDivNatS0 (Succ yx40000) Zero True",fontsize=16,color="black",shape="box"];1542 -> 1752[label="",style="solid", color="black", weight=3]; 1543[label="primDivNatS0 Zero (Succ yx41000) False",fontsize=16,color="black",shape="box"];1543 -> 1753[label="",style="solid", color="black", weight=3]; 1544[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];1544 -> 1754[label="",style="solid", color="black", weight=3]; 1174[label="truncateM0 (yx70 :% yx71) (fromIntegral (properFractionQ yx70 yx71),properFractionR yx70 yx71 :% yx71)",fontsize=16,color="black",shape="box"];1174 -> 1302[label="",style="solid", color="black", weight=3]; 4083[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) False)",fontsize=16,color="black",shape="box"];4083 -> 4092[label="",style="solid", color="black", weight=3]; 4084[label="map toEnum (takeWhile0 (flip (<=) (Neg yx1990)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) True)",fontsize=16,color="black",shape="box"];4084 -> 4093[label="",style="solid", color="black", weight=3]; 4491[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat (Succ yx2760) (Succ yx2770) == GT)))",fontsize=16,color="black",shape="box"];4491 -> 4495[label="",style="solid", color="black", weight=3]; 4492[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat (Succ yx2760) Zero == GT)))",fontsize=16,color="black",shape="box"];4492 -> 4496[label="",style="solid", color="black", weight=3]; 4493[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat Zero (Succ yx2770) == GT)))",fontsize=16,color="black",shape="box"];4493 -> 4497[label="",style="solid", color="black", weight=3]; 4494[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat Zero Zero == GT)))",fontsize=16,color="black",shape="box"];4494 -> 4498[label="",style="solid", color="black", weight=3]; 4086[label="map toEnum (Pos yx249 : takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos yx250 + yx237))",fontsize=16,color="black",shape="box"];4086 -> 4095[label="",style="solid", color="black", weight=3]; 4087[label="map toEnum (Pos yx249 : takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom $! Pos yx250 + yx237))",fontsize=16,color="black",shape="box"];4087 -> 4096[label="",style="solid", color="black", weight=3]; 2345[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110))) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) (Succ yx2110)))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2345 -> 2382[label="",style="solid", color="black", weight=3]; 2346[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS (Succ yx2100) Zero))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2346 -> 2383[label="",style="solid", color="black", weight=3]; 2347[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110)))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110))) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero (Succ yx2110)))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2347 -> 2384[label="",style="solid", color="black", weight=3]; 2348[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS Zero Zero))) yx213 == GT)))",fontsize=16,color="black",shape="box"];2348 -> 2385[label="",style="solid", color="black", weight=3]; 2992[label="Zero",fontsize=16,color="green",shape="box"];2993 -> 1978[label="",style="dashed", color="red", weight=0]; 2993[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2994[label="yx4",fontsize=16,color="green",shape="box"];2995[label="Zero",fontsize=16,color="green",shape="box"];2996 -> 1978[label="",style="dashed", color="red", weight=0]; 2996[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2997[label="Zero",fontsize=16,color="green",shape="box"];2998 -> 1978[label="",style="dashed", color="red", weight=0]; 2998[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="magenta"];2999[label="yx4",fontsize=16,color="green",shape="box"];3000[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3001[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) (Pos yx2130) == GT)))",fontsize=16,color="black",shape="box"];3001 -> 3046[label="",style="solid", color="black", weight=3]; 3002[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpInt (Neg (Succ yx219)) (Neg yx2130) == GT)))",fontsize=16,color="black",shape="box"];3002 -> 3047[label="",style="solid", color="black", weight=3]; 3003[label="Zero",fontsize=16,color="green",shape="box"];3004 -> 1730[label="",style="dashed", color="red", weight=0]; 3004[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3005[label="yx4",fontsize=16,color="green",shape="box"];3006[label="Zero",fontsize=16,color="green",shape="box"];3007 -> 1730[label="",style="dashed", color="red", weight=0]; 3007[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3008[label="Zero",fontsize=16,color="green",shape="box"];3009 -> 1730[label="",style="dashed", color="red", weight=0]; 3009[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];3010[label="yx4",fontsize=16,color="green",shape="box"];3011[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];131[label="map toEnum (takeWhile1 (flip (<=) (truncateM0 (yx40 :% yx41) (fromIntegral (properFractionQ yx40 yx41),properFractionR yx40 yx41 :% yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (truncateM0 (yx40 :% yx41) (fromIntegral (properFractionQ yx40 yx41),properFractionR yx40 yx41 :% yx41)) == GT)))",fontsize=16,color="black",shape="box"];131 -> 147[label="",style="solid", color="black", weight=3]; 3701[label="yx40000",fontsize=16,color="green",shape="box"];3702[label="yx41000",fontsize=16,color="green",shape="box"];3703[label="yx40000",fontsize=16,color="green",shape="box"];3704[label="yx41000",fontsize=16,color="green",shape="box"];3700[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS yx247 yx248)",fontsize=16,color="burlywood",shape="triangle"];4623[label="yx247/Succ yx2470",fontsize=10,color="white",style="solid",shape="box"];3700 -> 4623[label="",style="solid", color="burlywood", weight=9]; 4623 -> 3741[label="",style="solid", color="burlywood", weight=3]; 4624[label="yx247/Zero",fontsize=10,color="white",style="solid",shape="box"];3700 -> 4624[label="",style="solid", color="burlywood", weight=9]; 4624 -> 3742[label="",style="solid", color="burlywood", weight=3]; 1752[label="Succ (primDivNatS (primMinusNatS (Succ yx40000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];1752 -> 2069[label="",style="dashed", color="green", weight=3]; 1753[label="Zero",fontsize=16,color="green",shape="box"];1754[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];1754 -> 2070[label="",style="dashed", color="green", weight=3]; 1302[label="fromIntegral (properFractionQ yx70 yx71)",fontsize=16,color="black",shape="box"];1302 -> 1447[label="",style="solid", color="black", weight=3]; 4092[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) otherwise)",fontsize=16,color="black",shape="box"];4092 -> 4102[label="",style="solid", color="black", weight=3]; 4093 -> 3176[label="",style="dashed", color="red", weight=0]; 4093[label="map toEnum []",fontsize=16,color="magenta"];4495 -> 4428[label="",style="dashed", color="red", weight=0]; 4495[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (primCmpNat yx2760 yx2770 == GT)))",fontsize=16,color="magenta"];4495 -> 4499[label="",style="dashed", color="magenta", weight=3]; 4495 -> 4500[label="",style="dashed", color="magenta", weight=3]; 4496[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not (GT == GT)))",fontsize=16,color="black",shape="box"];4496 -> 4501[label="",style="solid", color="black", weight=3]; 4497[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! 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Pos yx250 + yx237))",fontsize=16,color="green",shape="box"];4096 -> 4106[label="",style="dashed", color="green", weight=3]; 4096 -> 4107[label="",style="dashed", color="green", weight=3]; 2382 -> 2234[label="",style="dashed", color="red", weight=0]; 2382[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110))) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110)) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) (primGEqNatS yx2100 yx2110))) yx213 == GT)))",fontsize=16,color="magenta"];2382 -> 2418[label="",style="dashed", color="magenta", weight=3]; 2382 -> 2419[label="",style="dashed", color="magenta", weight=3]; 2383[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) True) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) yx213 == GT)))",fontsize=16,color="black",shape="triangle"];2383 -> 2420[label="",style="solid", color="black", weight=3]; 2384[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) False)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) False) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) False)) yx213 == GT)))",fontsize=16,color="black",shape="box"];2384 -> 2421[label="",style="solid", color="black", weight=3]; 2385 -> 2383[label="",style="dashed", color="red", weight=0]; 2385[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) (numericEnumFrom $! Neg (primDivNatS0 (Succ yx208) (Succ yx209) True) + yx212) (not (primCmpInt (Neg (primDivNatS0 (Succ yx208) (Succ yx209) True)) yx213 == GT)))",fontsize=16,color="magenta"];1978[label="primMinusNatS (Succ yx30000) Zero",fontsize=16,color="black",shape="triangle"];1978 -> 2041[label="",style="solid", color="black", weight=3]; 3046[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (LT == GT)))",fontsize=16,color="black",shape="triangle"];3046 -> 3081[label="",style="solid", color="black", weight=3]; 3047[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat yx2130 (Succ yx219) == GT)))",fontsize=16,color="burlywood",shape="box"];4625[label="yx2130/Succ yx21300",fontsize=10,color="white",style="solid",shape="box"];3047 -> 4625[label="",style="solid", color="burlywood", weight=9]; 4625 -> 3082[label="",style="solid", color="burlywood", weight=3]; 4626[label="yx2130/Zero",fontsize=10,color="white",style="solid",shape="box"];3047 -> 4626[label="",style="solid", color="burlywood", weight=9]; 4626 -> 3083[label="",style="solid", color="burlywood", weight=3]; 1730[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];1730 -> 1749[label="",style="solid", color="black", weight=3]; 147[label="map toEnum (takeWhile1 (flip (<=) (fromIntegral (properFractionQ yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromIntegral (properFractionQ yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];147 -> 165[label="",style="solid", color="black", weight=3]; 3741[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS (Succ yx2470) yx248)",fontsize=16,color="burlywood",shape="box"];4627[label="yx248/Succ yx2480",fontsize=10,color="white",style="solid",shape="box"];3741 -> 4627[label="",style="solid", color="burlywood", weight=9]; 4627 -> 3753[label="",style="solid", color="burlywood", weight=3]; 4628[label="yx248/Zero",fontsize=10,color="white",style="solid",shape="box"];3741 -> 4628[label="",style="solid", color="burlywood", weight=9]; 4628 -> 3754[label="",style="solid", color="burlywood", weight=3]; 3742[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS Zero yx248)",fontsize=16,color="burlywood",shape="box"];4629[label="yx248/Succ yx2480",fontsize=10,color="white",style="solid",shape="box"];3742 -> 4629[label="",style="solid", color="burlywood", weight=9]; 4629 -> 3755[label="",style="solid", color="burlywood", weight=3]; 4630[label="yx248/Zero",fontsize=10,color="white",style="solid",shape="box"];3742 -> 4630[label="",style="solid", color="burlywood", weight=9]; 4630 -> 3756[label="",style="solid", color="burlywood", weight=3]; 2069 -> 1689[label="",style="dashed", color="red", weight=0]; 2069[label="primDivNatS (primMinusNatS (Succ yx40000) Zero) (Succ Zero)",fontsize=16,color="magenta"];2069 -> 2097[label="",style="dashed", color="magenta", weight=3]; 2069 -> 2098[label="",style="dashed", color="magenta", weight=3]; 2070 -> 1689[label="",style="dashed", color="red", weight=0]; 2070[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];2070 -> 2099[label="",style="dashed", color="magenta", weight=3]; 2070 -> 2100[label="",style="dashed", color="magenta", weight=3]; 1447[label="fromInteger . toInteger",fontsize=16,color="black",shape="box"];1447 -> 1584[label="",style="solid", color="black", weight=3]; 4102[label="map toEnum (takeWhile0 (flip (<=) (Pos Zero)) (Pos yx249) (numericEnumFrom $! Pos yx250 + yx237) True)",fontsize=16,color="black",shape="box"];4102 -> 4114[label="",style="solid", color="black", weight=3]; 3176[label="map toEnum []",fontsize=16,color="black",shape="triangle"];3176 -> 3212[label="",style="solid", color="black", weight=3]; 4499[label="yx2760",fontsize=16,color="green",shape="box"];4500[label="yx2770",fontsize=16,color="green",shape="box"];4501[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not True))",fontsize=16,color="black",shape="box"];4501 -> 4504[label="",style="solid", color="black", weight=3]; 4502[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not False))",fontsize=16,color="black",shape="triangle"];4502 -> 4505[label="",style="solid", color="black", weight=3]; 4503 -> 4502[label="",style="dashed", color="red", weight=0]; 4503[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) (not False))",fontsize=16,color="magenta"];4104[label="toEnum (Pos yx249)",fontsize=16,color="blue",shape="box"];4631[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4631[label="",style="solid", color="blue", weight=9]; 4631 -> 4117[label="",style="solid", color="blue", weight=3]; 4632[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4632[label="",style="solid", color="blue", weight=9]; 4632 -> 4118[label="",style="solid", color="blue", weight=3]; 4633[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4633[label="",style="solid", color="blue", weight=9]; 4633 -> 4119[label="",style="solid", color="blue", weight=3]; 4634[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4634[label="",style="solid", color="blue", weight=9]; 4634 -> 4120[label="",style="solid", color="blue", weight=3]; 4635[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4635[label="",style="solid", color="blue", weight=9]; 4635 -> 4121[label="",style="solid", color="blue", weight=3]; 4636[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4636[label="",style="solid", color="blue", weight=9]; 4636 -> 4122[label="",style="solid", color="blue", weight=3]; 4637[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4637[label="",style="solid", color="blue", weight=9]; 4637 -> 4123[label="",style="solid", color="blue", weight=3]; 4638[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4638[label="",style="solid", color="blue", weight=9]; 4638 -> 4124[label="",style="solid", color="blue", weight=3]; 4639[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];4104 -> 4639[label="",style="solid", color="blue", weight=9]; 4639 -> 4125[label="",style="solid", color="blue", weight=3]; 4105[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom $! Pos yx250 + yx237))",fontsize=16,color="black",shape="box"];4105 -> 4126[label="",style="solid", color="black", weight=3]; 4106[label="toEnum (Pos yx249)",fontsize=16,color="blue",shape="box"];4640[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4640[label="",style="solid", color="blue", weight=9]; 4640 -> 4127[label="",style="solid", color="blue", weight=3]; 4641[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4641[label="",style="solid", color="blue", weight=9]; 4641 -> 4128[label="",style="solid", color="blue", weight=3]; 4642[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4642[label="",style="solid", color="blue", weight=9]; 4642 -> 4129[label="",style="solid", color="blue", weight=3]; 4643[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4643[label="",style="solid", color="blue", weight=9]; 4643 -> 4130[label="",style="solid", color="blue", weight=3]; 4644[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4644[label="",style="solid", color="blue", weight=9]; 4644 -> 4131[label="",style="solid", color="blue", weight=3]; 4645[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4645[label="",style="solid", color="blue", weight=9]; 4645 -> 4132[label="",style="solid", color="blue", weight=3]; 4646[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4646[label="",style="solid", color="blue", weight=9]; 4646 -> 4133[label="",style="solid", color="blue", weight=3]; 4647[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4647[label="",style="solid", color="blue", weight=9]; 4647 -> 4134[label="",style="solid", color="blue", weight=3]; 4648[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];4106 -> 4648[label="",style="solid", color="blue", weight=9]; 4648 -> 4135[label="",style="solid", color="blue", weight=3]; 4107[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom $! Pos yx250 + yx237))",fontsize=16,color="black",shape="box"];4107 -> 4136[label="",style="solid", color="black", weight=3]; 2418[label="yx2110",fontsize=16,color="green",shape="box"];2419[label="yx2100",fontsize=16,color="green",shape="box"];2420 -> 2772[label="",style="dashed", color="red", weight=0]; 2420[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))))) (numericEnumFrom $! Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209)))) + yx212) (not (primCmpInt (Neg (Succ (primDivNatS (primMinusNatS (Succ yx208) (Succ yx209)) (Succ (Succ yx209))))) yx213 == GT)))",fontsize=16,color="magenta"];2420 -> 2821[label="",style="dashed", color="magenta", weight=3]; 2420 -> 2822[label="",style="dashed", color="magenta", weight=3]; 2420 -> 2823[label="",style="dashed", color="magenta", weight=3]; 2421[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) yx213 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4649[label="yx213/Pos yx2130",fontsize=10,color="white",style="solid",shape="box"];2421 -> 4649[label="",style="solid", color="burlywood", weight=9]; 4649 -> 3014[label="",style="solid", color="burlywood", weight=3]; 4650[label="yx213/Neg yx2130",fontsize=10,color="white",style="solid",shape="box"];2421 -> 4650[label="",style="solid", color="burlywood", weight=9]; 4650 -> 3015[label="",style="solid", color="burlywood", weight=3]; 2041[label="Succ yx30000",fontsize=16,color="green",shape="box"];3081[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not False))",fontsize=16,color="black",shape="triangle"];3081 -> 3112[label="",style="solid", color="black", weight=3]; 3082[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx21300) (Succ yx219) == GT)))",fontsize=16,color="black",shape="box"];3082 -> 3113[label="",style="solid", color="black", weight=3]; 3083[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat Zero (Succ yx219) == GT)))",fontsize=16,color="black",shape="box"];3083 -> 3114[label="",style="solid", color="black", weight=3]; 1749[label="Zero",fontsize=16,color="green",shape="box"];165[label="map toEnum (takeWhile1 (flip (<=) (fromInteger . toInteger)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromInteger . toInteger) == GT)))",fontsize=16,color="black",shape="box"];165 -> 183[label="",style="solid", color="black", weight=3]; 3753[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS (Succ yx2470) (Succ yx2480))",fontsize=16,color="black",shape="box"];3753 -> 4015[label="",style="solid", color="black", weight=3]; 3754[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS (Succ yx2470) Zero)",fontsize=16,color="black",shape="box"];3754 -> 4016[label="",style="solid", color="black", weight=3]; 3755[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS Zero (Succ yx2480))",fontsize=16,color="black",shape="box"];3755 -> 4017[label="",style="solid", color="black", weight=3]; 3756[label="primDivNatS0 (Succ yx245) (Succ yx246) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];3756 -> 4018[label="",style="solid", color="black", weight=3]; 2097[label="Zero",fontsize=16,color="green",shape="box"];2098 -> 1978[label="",style="dashed", color="red", weight=0]; 2098[label="primMinusNatS (Succ yx40000) Zero",fontsize=16,color="magenta"];2098 -> 2120[label="",style="dashed", color="magenta", weight=3]; 2099[label="Zero",fontsize=16,color="green",shape="box"];2100 -> 1730[label="",style="dashed", color="red", weight=0]; 2100[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];1584[label="fromInteger (toInteger (properFractionQ yx70 yx71))",fontsize=16,color="blue",shape="box"];4651[label="toInteger :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];1584 -> 4651[label="",style="solid", color="blue", weight=9]; 4651 -> 2034[label="",style="solid", color="blue", weight=3]; 4652[label="toInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1584 -> 4652[label="",style="solid", color="blue", weight=9]; 4652 -> 2035[label="",style="solid", color="blue", weight=3]; 4114 -> 3176[label="",style="dashed", color="red", weight=0]; 4114[label="map toEnum []",fontsize=16,color="magenta"];3212[label="[]",fontsize=16,color="green",shape="box"];4504[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) False)",fontsize=16,color="black",shape="box"];4504 -> 4506[label="",style="solid", color="black", weight=3]; 4505[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! 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Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos yx2130) == GT)))",fontsize=16,color="burlywood",shape="box"];4653[label="yx2130/Succ yx21300",fontsize=10,color="white",style="solid",shape="box"];3014 -> 4653[label="",style="solid", color="burlywood", weight=9]; 4653 -> 3051[label="",style="solid", color="burlywood", weight=3]; 4654[label="yx2130/Zero",fontsize=10,color="white",style="solid",shape="box"];3014 -> 4654[label="",style="solid", color="burlywood", weight=9]; 4654 -> 3052[label="",style="solid", color="burlywood", weight=3]; 3015[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! 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Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos (Succ yx21300)) == GT)))",fontsize=16,color="black",shape="box"];3051 -> 3087[label="",style="solid", color="black", weight=3]; 3052[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];3052 -> 3088[label="",style="solid", color="black", weight=3]; 3053[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Neg (Succ yx21300)) == GT)))",fontsize=16,color="black",shape="box"];3053 -> 3089[label="",style="solid", color="black", weight=3]; 3054[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];3054 -> 3090[label="",style="solid", color="black", weight=3]; 3146[label="map toEnum (Neg (Succ yx217) : takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg (Succ yx218) + yx212))",fontsize=16,color="black",shape="box"];3146 -> 3167[label="",style="solid", color="black", weight=3]; 3147[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! 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Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (fromInteger (Integer (properFractionQ yx40 yx41))) == GT)))",fontsize=16,color="black",shape="box"];203 -> 225[label="",style="solid", color="black", weight=3]; 4029[label="yx2480",fontsize=16,color="green",shape="box"];4030[label="yx2470",fontsize=16,color="green",shape="box"];4031[label="Succ (primDivNatS (primMinusNatS (Succ yx245) (Succ yx246)) (Succ (Succ yx246)))",fontsize=16,color="green",shape="box"];4031 -> 4050[label="",style="dashed", color="green", weight=3]; 4032[label="Zero",fontsize=16,color="green",shape="box"];2321 -> 3164[label="",style="dashed", color="red", weight=0]; 2321[label="fromInteger (Integer (properFractionQ yx70 yx71))",fontsize=16,color="magenta"];2321 -> 3165[label="",style="dashed", color="magenta", weight=3]; 2322[label="fromInteger (properFractionQ yx70 yx71)",fontsize=16,color="black",shape="box"];2322 -> 2354[label="",style="solid", color="black", weight=3]; 4508[label="map toEnum (takeWhile0 (flip (<=) (Pos (Succ yx272))) (Pos yx273) (numericEnumFrom $! Pos yx274 + yx275) True)",fontsize=16,color="black",shape="box"];4508 -> 4510[label="",style="solid", color="black", weight=3]; 4509[label="toEnum (Pos yx273) : map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (numericEnumFrom $! Pos yx274 + yx275))",fontsize=16,color="green",shape="box"];4509 -> 4511[label="",style="dashed", color="green", weight=3]; 4509 -> 4512[label="",style="dashed", color="green", weight=3]; 4170[label="intToRatio (Pos yx249)",fontsize=16,color="black",shape="box"];4170 -> 4181[label="",style="solid", color="black", weight=3]; 4171[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) yx237)) (numericEnumFrom (primPlusInt (Pos yx250) yx237))))",fontsize=16,color="burlywood",shape="box"];4663[label="yx237/Pos yx2370",fontsize=10,color="white",style="solid",shape="box"];4171 -> 4663[label="",style="solid", color="burlywood", weight=9]; 4663 -> 4182[label="",style="solid", color="burlywood", weight=3]; 4664[label="yx237/Neg yx2370",fontsize=10,color="white",style="solid",shape="box"];4171 -> 4664[label="",style="solid", color="burlywood", weight=9]; 4664 -> 4183[label="",style="solid", color="burlywood", weight=3]; 4172[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) yx237)) (numericEnumFrom (primPlusInt (Pos yx250) yx237))))",fontsize=16,color="burlywood",shape="box"];4665[label="yx237/Pos yx2370",fontsize=10,color="white",style="solid",shape="box"];4172 -> 4665[label="",style="solid", color="burlywood", weight=9]; 4665 -> 4184[label="",style="solid", color="burlywood", weight=3]; 4666[label="yx237/Neg yx2370",fontsize=10,color="white",style="solid",shape="box"];4172 -> 4666[label="",style="solid", color="burlywood", weight=9]; 4666 -> 4185[label="",style="solid", color="burlywood", weight=3]; 3060[label="yx209",fontsize=16,color="green",shape="box"];3061[label="yx208",fontsize=16,color="green",shape="box"];3017[label="primMinusNatS (Succ yx200) (Succ yx201)",fontsize=16,color="black",shape="triangle"];3017 -> 3055[label="",style="solid", color="black", weight=3]; 3062[label="yx209",fontsize=16,color="green",shape="box"];3063[label="yx208",fontsize=16,color="green",shape="box"];3064[label="yx209",fontsize=16,color="green",shape="box"];3065[label="yx208",fontsize=16,color="green",shape="box"];3087[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! 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Neg (Succ yx218) + yx212))",fontsize=16,color="green",shape="box"];3167 -> 3202[label="",style="dashed", color="green", weight=3]; 3167 -> 3203[label="",style="dashed", color="green", weight=3]; 3168[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx213000) (Succ yx2190) == GT)))",fontsize=16,color="black",shape="box"];3168 -> 3204[label="",style="solid", color="black", weight=3]; 3169[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat (Succ yx213000) Zero == GT)))",fontsize=16,color="black",shape="box"];3169 -> 3205[label="",style="solid", color="black", weight=3]; 3170[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! 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Pos yx274 + yx275))",fontsize=16,color="black",shape="box"];4512 -> 4522[label="",style="solid", color="black", weight=3]; 4181[label="fromInt (Pos yx249) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];4181 -> 4193[label="",style="dashed", color="green", weight=3]; 4181 -> 4194[label="",style="dashed", color="green", weight=3]; 4182[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Pos yx2370))) (numericEnumFrom (primPlusInt (Pos yx250) (Pos yx2370)))))",fontsize=16,color="black",shape="box"];4182 -> 4195[label="",style="solid", color="black", weight=3]; 4183[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Neg yx2370))) (numericEnumFrom (primPlusInt (Pos yx250) (Neg yx2370)))))",fontsize=16,color="black",shape="box"];4183 -> 4196[label="",style="solid", color="black", weight=3]; 4184[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Pos yx2370))) (numericEnumFrom (primPlusInt (Pos yx250) (Pos yx2370)))))",fontsize=16,color="black",shape="box"];4184 -> 4197[label="",style="solid", color="black", weight=3]; 4185[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primPlusInt (Pos yx250) (Neg yx2370))) (numericEnumFrom (primPlusInt (Pos yx250) (Neg yx2370)))))",fontsize=16,color="black",shape="box"];4185 -> 4198[label="",style="solid", color="black", weight=3]; 3055[label="primMinusNatS yx200 yx201",fontsize=16,color="burlywood",shape="triangle"];4676[label="yx200/Succ yx2000",fontsize=10,color="white",style="solid",shape="box"];3055 -> 4676[label="",style="solid", color="burlywood", weight=9]; 4676 -> 3091[label="",style="solid", color="burlywood", weight=3]; 4677[label="yx200/Zero",fontsize=10,color="white",style="solid",shape="box"];3055 -> 4677[label="",style="solid", color="burlywood", weight=9]; 4677 -> 3092[label="",style="solid", color="burlywood", weight=3]; 3118[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not False))",fontsize=16,color="black",shape="triangle"];3118 -> 3154[label="",style="solid", color="black", weight=3]; 3119 -> 3118[label="",style="dashed", color="red", weight=0]; 3119[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not False))",fontsize=16,color="magenta"];3120[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not (GT == GT)))",fontsize=16,color="black",shape="box"];3120 -> 3155[label="",style="solid", color="black", weight=3]; 3202[label="toEnum (Neg (Succ yx217))",fontsize=16,color="blue",shape="box"];4678[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4678[label="",style="solid", color="blue", weight=9]; 4678 -> 3235[label="",style="solid", color="blue", weight=3]; 4679[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4679[label="",style="solid", color="blue", weight=9]; 4679 -> 3236[label="",style="solid", color="blue", weight=3]; 4680[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4680[label="",style="solid", color="blue", weight=9]; 4680 -> 3237[label="",style="solid", color="blue", weight=3]; 4681[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4681[label="",style="solid", color="blue", weight=9]; 4681 -> 3238[label="",style="solid", color="blue", weight=3]; 4682[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4682[label="",style="solid", color="blue", weight=9]; 4682 -> 3239[label="",style="solid", color="blue", weight=3]; 4683[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4683[label="",style="solid", color="blue", weight=9]; 4683 -> 3240[label="",style="solid", color="blue", weight=3]; 4684[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4684[label="",style="solid", color="blue", weight=9]; 4684 -> 3241[label="",style="solid", color="blue", weight=3]; 4685[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4685[label="",style="solid", color="blue", weight=9]; 4685 -> 3242[label="",style="solid", color="blue", weight=3]; 4686[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];3202 -> 4686[label="",style="solid", color="blue", weight=9]; 4686 -> 3243[label="",style="solid", color="blue", weight=3]; 3203[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg (Succ yx218) + yx212))",fontsize=16,color="black",shape="box"];3203 -> 3244[label="",style="solid", color="black", weight=3]; 3204 -> 3113[label="",style="dashed", color="red", weight=0]; 3204[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (primCmpNat yx213000 yx2190 == GT)))",fontsize=16,color="magenta"];3204 -> 3245[label="",style="dashed", color="magenta", weight=3]; 3204 -> 3246[label="",style="dashed", color="magenta", weight=3]; 3205[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not (GT == GT)))",fontsize=16,color="black",shape="box"];3205 -> 3247[label="",style="solid", color="black", weight=3]; 3206 -> 3046[label="",style="dashed", color="red", weight=0]; 3206[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! 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Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (properFractionVu30 yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];247 -> 271[label="",style="solid", color="black", weight=3]; 4059[label="Succ yx246",fontsize=16,color="green",shape="box"];4060 -> 3055[label="",style="dashed", color="red", weight=0]; 4060[label="primMinusNatS (Succ yx245) (Succ yx246)",fontsize=16,color="magenta"];4060 -> 4070[label="",style="dashed", color="magenta", weight=3]; 4060 -> 4071[label="",style="dashed", color="magenta", weight=3]; 3183[label="properFractionQ1 yx70 yx71 (properFractionVu30 yx70 yx71)",fontsize=16,color="black",shape="box"];3183 -> 3218[label="",style="solid", color="black", weight=3]; 3184[label="yx220",fontsize=16,color="green",shape="box"];2391[label="fromInteger (properFractionQ1 yx70 yx71 (quotRem yx70 yx71))",fontsize=16,color="burlywood",shape="box"];4687[label="yx70/Integer yx700",fontsize=10,color="white",style="solid",shape="box"];2391 -> 4687[label="",style="solid", color="burlywood", weight=9]; 4687 -> 3033[label="",style="solid", color="burlywood", weight=3]; 4513 -> 4117[label="",style="dashed", color="red", weight=0]; 4513[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4513 -> 4523[label="",style="dashed", color="magenta", weight=3]; 4514 -> 4118[label="",style="dashed", color="red", weight=0]; 4514[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4514 -> 4524[label="",style="dashed", color="magenta", weight=3]; 4515 -> 4119[label="",style="dashed", color="red", weight=0]; 4515[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4515 -> 4525[label="",style="dashed", color="magenta", weight=3]; 4516 -> 4120[label="",style="dashed", color="red", weight=0]; 4516[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4516 -> 4526[label="",style="dashed", color="magenta", weight=3]; 4517 -> 4121[label="",style="dashed", color="red", weight=0]; 4517[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4517 -> 4527[label="",style="dashed", color="magenta", weight=3]; 4518 -> 4122[label="",style="dashed", color="red", weight=0]; 4518[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4518 -> 4528[label="",style="dashed", color="magenta", weight=3]; 4519 -> 4123[label="",style="dashed", color="red", weight=0]; 4519[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4519 -> 4529[label="",style="dashed", color="magenta", weight=3]; 4520 -> 4124[label="",style="dashed", color="red", weight=0]; 4520[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4520 -> 4530[label="",style="dashed", color="magenta", weight=3]; 4521 -> 4125[label="",style="dashed", color="red", weight=0]; 4521[label="toEnum (Pos yx273)",fontsize=16,color="magenta"];4521 -> 4531[label="",style="dashed", color="magenta", weight=3]; 4522[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (Pos yx274 + yx275 `seq` numericEnumFrom (Pos yx274 + yx275)))",fontsize=16,color="black",shape="box"];4522 -> 4532[label="",style="solid", color="black", weight=3]; 4193[label="fromInt (Pos yx249)",fontsize=16,color="blue",shape="box"];4688[label="fromInt :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4193 -> 4688[label="",style="solid", color="blue", weight=9]; 4688 -> 4207[label="",style="solid", color="blue", weight=3]; 4689[label="fromInt :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];4193 -> 4689[label="",style="solid", color="blue", weight=9]; 4689 -> 4208[label="",style="solid", color="blue", weight=3]; 4194[label="fromInt (Pos (Succ Zero))",fontsize=16,color="blue",shape="box"];4690[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];4194 -> 4690[label="",style="solid", color="blue", weight=9]; 4690 -> 4209[label="",style="solid", color="blue", weight=3]; 4691[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];4194 -> 4691[label="",style="solid", color="blue", weight=9]; 4691 -> 4210[label="",style="solid", color="blue", weight=3]; 4195 -> 4211[label="",style="dashed", color="red", weight=0]; 4195[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos (primPlusNat yx250 yx2370))) (numericEnumFrom (Pos (primPlusNat yx250 yx2370)))))",fontsize=16,color="magenta"];4195 -> 4212[label="",style="dashed", color="magenta", weight=3]; 4195 -> 4213[label="",style="dashed", color="magenta", weight=3]; 4196 -> 3481[label="",style="dashed", color="red", weight=0]; 4196[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (primMinusNat yx250 yx2370)) (numericEnumFrom (primMinusNat yx250 yx2370))))",fontsize=16,color="magenta"];4196 -> 4214[label="",style="dashed", color="magenta", weight=3]; 4196 -> 4215[label="",style="dashed", color="magenta", weight=3]; 4196 -> 4216[label="",style="dashed", color="magenta", weight=3]; 4197 -> 4217[label="",style="dashed", color="red", weight=0]; 4197[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (Pos (primPlusNat yx250 yx2370))) (numericEnumFrom (Pos (primPlusNat yx250 yx2370)))))",fontsize=16,color="magenta"];4197 -> 4218[label="",style="dashed", color="magenta", weight=3]; 4197 -> 4219[label="",style="dashed", color="magenta", weight=3]; 4198 -> 3481[label="",style="dashed", color="red", weight=0]; 4198[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (primMinusNat yx250 yx2370)) (numericEnumFrom (primMinusNat yx250 yx2370))))",fontsize=16,color="magenta"];4198 -> 4220[label="",style="dashed", color="magenta", weight=3]; 4198 -> 4221[label="",style="dashed", color="magenta", weight=3]; 4198 -> 4222[label="",style="dashed", color="magenta", weight=3]; 3091[label="primMinusNatS (Succ yx2000) yx201",fontsize=16,color="burlywood",shape="box"];4692[label="yx201/Succ yx2010",fontsize=10,color="white",style="solid",shape="box"];3091 -> 4692[label="",style="solid", color="burlywood", weight=9]; 4692 -> 3121[label="",style="solid", color="burlywood", weight=3]; 4693[label="yx201/Zero",fontsize=10,color="white",style="solid",shape="box"];3091 -> 4693[label="",style="solid", color="burlywood", weight=9]; 4693 -> 3122[label="",style="solid", color="burlywood", weight=3]; 3092[label="primMinusNatS Zero yx201",fontsize=16,color="burlywood",shape="box"];4694[label="yx201/Succ yx2010",fontsize=10,color="white",style="solid",shape="box"];3092 -> 4694[label="",style="solid", color="burlywood", weight=9]; 4694 -> 3123[label="",style="solid", color="burlywood", weight=3]; 4695[label="yx201/Zero",fontsize=10,color="white",style="solid",shape="box"];3092 -> 4695[label="",style="solid", color="burlywood", weight=9]; 4695 -> 3124[label="",style="solid", color="burlywood", weight=3]; 3154[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) True)",fontsize=16,color="black",shape="box"];3154 -> 3177[label="",style="solid", color="black", weight=3]; 3155[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) (not True))",fontsize=16,color="black",shape="box"];3155 -> 3178[label="",style="solid", color="black", weight=3]; 3235[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3235 -> 3282[label="",style="solid", color="black", weight=3]; 3236[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3236 -> 3283[label="",style="solid", color="black", weight=3]; 3237[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3237 -> 3284[label="",style="solid", color="black", weight=3]; 3238[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3238 -> 3285[label="",style="solid", color="black", weight=3]; 3239[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3239 -> 3286[label="",style="solid", color="black", weight=3]; 3240[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3240 -> 3287[label="",style="solid", color="black", weight=3]; 3241[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3241 -> 3288[label="",style="solid", color="black", weight=3]; 3242[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3242 -> 3289[label="",style="solid", color="black", weight=3]; 3243[label="toEnum (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3243 -> 3290[label="",style="solid", color="black", weight=3]; 3244[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) + yx212 `seq` numericEnumFrom (Neg (Succ yx218) + yx212)))",fontsize=16,color="black",shape="box"];3244 -> 3291[label="",style="solid", color="black", weight=3]; 3245[label="yx2190",fontsize=16,color="green",shape="box"];3246[label="yx213000",fontsize=16,color="green",shape="box"];3247[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not True))",fontsize=16,color="black",shape="box"];3247 -> 3292[label="",style="solid", color="black", weight=3]; 3248 -> 3081[label="",style="dashed", color="red", weight=0]; 3248[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) (not False))",fontsize=16,color="magenta"];271[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (quotRem yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (quotRem yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];271 -> 297[label="",style="solid", color="black", weight=3]; 4070[label="Succ yx246",fontsize=16,color="green",shape="box"];4071[label="Succ yx245",fontsize=16,color="green",shape="box"];3218[label="properFractionQ1 yx70 yx71 (quotRem yx70 yx71)",fontsize=16,color="black",shape="box"];3218 -> 3263[label="",style="solid", color="black", weight=3]; 3033[label="fromInteger (properFractionQ1 (Integer yx700) yx71 (quotRem (Integer yx700) yx71))",fontsize=16,color="burlywood",shape="box"];4696[label="yx71/Integer yx710",fontsize=10,color="white",style="solid",shape="box"];3033 -> 4696[label="",style="solid", color="burlywood", weight=9]; 4696 -> 3067[label="",style="solid", color="burlywood", weight=3]; 4523[label="yx273",fontsize=16,color="green",shape="box"];4524[label="yx273",fontsize=16,color="green",shape="box"];4525[label="yx273",fontsize=16,color="green",shape="box"];4526[label="yx273",fontsize=16,color="green",shape="box"];4527[label="yx273",fontsize=16,color="green",shape="box"];4528[label="yx273",fontsize=16,color="green",shape="box"];4529[label="yx273",fontsize=16,color="green",shape="box"];4530[label="yx273",fontsize=16,color="green",shape="box"];4531[label="yx273",fontsize=16,color="green",shape="box"];4532[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (enforceWHNF (WHNF (Pos yx274 + yx275)) (numericEnumFrom (Pos yx274 + yx275))))",fontsize=16,color="black",shape="box"];4532 -> 4533[label="",style="solid", color="black", weight=3]; 4207 -> 576[label="",style="dashed", color="red", weight=0]; 4207[label="fromInt (Pos yx249)",fontsize=16,color="magenta"];4207 -> 4236[label="",style="dashed", color="magenta", weight=3]; 4208[label="fromInt (Pos yx249)",fontsize=16,color="black",shape="triangle"];4208 -> 4237[label="",style="solid", color="black", weight=3]; 4209 -> 576[label="",style="dashed", color="red", weight=0]; 4209[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4209 -> 4238[label="",style="dashed", color="magenta", weight=3]; 4210 -> 4208[label="",style="dashed", color="red", weight=0]; 4210[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4210 -> 4239[label="",style="dashed", color="magenta", weight=3]; 4212 -> 3743[label="",style="dashed", color="red", weight=0]; 4212[label="primPlusNat yx250 yx2370",fontsize=16,color="magenta"];4212 -> 4240[label="",style="dashed", color="magenta", weight=3]; 4212 -> 4241[label="",style="dashed", color="magenta", weight=3]; 4213 -> 3743[label="",style="dashed", color="red", weight=0]; 4213[label="primPlusNat yx250 yx2370",fontsize=16,color="magenta"];4213 -> 4242[label="",style="dashed", color="magenta", weight=3]; 4213 -> 4243[label="",style="dashed", color="magenta", weight=3]; 4211[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (enforceWHNF (WHNF (Pos yx253)) (numericEnumFrom (Pos yx252))))",fontsize=16,color="black",shape="triangle"];4211 -> 4244[label="",style="solid", color="black", weight=3]; 4214[label="Pos Zero",fontsize=16,color="green",shape="box"];4215[label="yx250",fontsize=16,color="green",shape="box"];4216[label="yx2370",fontsize=16,color="green",shape="box"];3481[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx21200 yx218)) (numericEnumFrom (primMinusNat yx21200 yx218))))",fontsize=16,color="burlywood",shape="triangle"];4697[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3481 -> 4697[label="",style="solid", color="burlywood", weight=9]; 4697 -> 3515[label="",style="solid", color="burlywood", weight=3]; 4698[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3481 -> 4698[label="",style="solid", color="burlywood", weight=9]; 4698 -> 3516[label="",style="solid", color="burlywood", weight=3]; 4218 -> 3743[label="",style="dashed", color="red", weight=0]; 4218[label="primPlusNat yx250 yx2370",fontsize=16,color="magenta"];4218 -> 4245[label="",style="dashed", color="magenta", weight=3]; 4218 -> 4246[label="",style="dashed", color="magenta", weight=3]; 4219 -> 3743[label="",style="dashed", color="red", weight=0]; 4219[label="primPlusNat yx250 yx2370",fontsize=16,color="magenta"];4219 -> 4247[label="",style="dashed", color="magenta", weight=3]; 4219 -> 4248[label="",style="dashed", color="magenta", weight=3]; 4217[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (enforceWHNF (WHNF (Pos yx255)) (numericEnumFrom (Pos yx254))))",fontsize=16,color="black",shape="triangle"];4217 -> 4249[label="",style="solid", color="black", weight=3]; 4220[label="Neg Zero",fontsize=16,color="green",shape="box"];4221[label="yx250",fontsize=16,color="green",shape="box"];4222[label="yx2370",fontsize=16,color="green",shape="box"];3121[label="primMinusNatS (Succ yx2000) (Succ yx2010)",fontsize=16,color="black",shape="box"];3121 -> 3156[label="",style="solid", color="black", weight=3]; 3122[label="primMinusNatS (Succ yx2000) Zero",fontsize=16,color="black",shape="box"];3122 -> 3157[label="",style="solid", color="black", weight=3]; 3123[label="primMinusNatS Zero (Succ yx2010)",fontsize=16,color="black",shape="box"];3123 -> 3158[label="",style="solid", color="black", weight=3]; 3124[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];3124 -> 3159[label="",style="solid", color="black", weight=3]; 3177[label="map toEnum (Neg Zero : takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="black",shape="box"];3177 -> 3213[label="",style="solid", color="black", weight=3]; 3178[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) False)",fontsize=16,color="black",shape="box"];3178 -> 3214[label="",style="solid", color="black", weight=3]; 3282[label="error []",fontsize=16,color="red",shape="box"];3283[label="error []",fontsize=16,color="red",shape="box"];3284[label="error []",fontsize=16,color="red",shape="box"];3285[label="error []",fontsize=16,color="red",shape="box"];3286[label="fromInt (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3286 -> 3329[label="",style="solid", color="black", weight=3]; 3287[label="error []",fontsize=16,color="red",shape="box"];3288[label="error []",fontsize=16,color="red",shape="box"];3289[label="error []",fontsize=16,color="red",shape="box"];3290[label="error []",fontsize=16,color="red",shape="box"];3291[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (Succ yx218) + yx212)) (numericEnumFrom (Neg (Succ yx218) + yx212))))",fontsize=16,color="black",shape="box"];3291 -> 3330[label="",style="solid", color="black", weight=3]; 3292[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) False)",fontsize=16,color="black",shape="box"];3292 -> 3331[label="",style="solid", color="black", weight=3]; 297[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (primQrmInt yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (primQrmInt yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];297 -> 323[label="",style="solid", color="black", weight=3]; 3263[label="properFractionQ1 yx70 yx71 (primQrmInt yx70 yx71)",fontsize=16,color="black",shape="box"];3263 -> 3315[label="",style="solid", color="black", weight=3]; 3067[label="fromInteger (properFractionQ1 (Integer yx700) (Integer yx710) (quotRem (Integer yx700) (Integer yx710)))",fontsize=16,color="black",shape="box"];3067 -> 3097[label="",style="solid", color="black", weight=3]; 4533[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (enforceWHNF (WHNF (primPlusInt (Pos yx274) yx275)) (numericEnumFrom (primPlusInt (Pos yx274) yx275))))",fontsize=16,color="burlywood",shape="box"];4699[label="yx275/Pos yx2750",fontsize=10,color="white",style="solid",shape="box"];4533 -> 4699[label="",style="solid", color="burlywood", weight=9]; 4699 -> 4534[label="",style="solid", color="burlywood", weight=3]; 4700[label="yx275/Neg yx2750",fontsize=10,color="white",style="solid",shape="box"];4533 -> 4700[label="",style="solid", color="burlywood", weight=9]; 4700 -> 4535[label="",style="solid", color="burlywood", weight=3]; 4236[label="Pos yx249",fontsize=16,color="green",shape="box"];4237[label="Integer (Pos yx249)",fontsize=16,color="green",shape="box"];4238[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4239[label="Succ Zero",fontsize=16,color="green",shape="box"];4240[label="yx2370",fontsize=16,color="green",shape="box"];4241[label="yx250",fontsize=16,color="green",shape="box"];3743[label="primPlusNat yx218 yx21200",fontsize=16,color="burlywood",shape="triangle"];4701[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3743 -> 4701[label="",style="solid", color="burlywood", weight=9]; 4701 -> 3757[label="",style="solid", color="burlywood", weight=3]; 4702[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3743 -> 4702[label="",style="solid", color="burlywood", weight=9]; 4702 -> 3758[label="",style="solid", color="burlywood", weight=3]; 4242[label="yx2370",fontsize=16,color="green",shape="box"];4243[label="yx250",fontsize=16,color="green",shape="box"];4244[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (numericEnumFrom (Pos yx252)))",fontsize=16,color="black",shape="box"];4244 -> 4260[label="",style="solid", color="black", weight=3]; 3515[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) yx218)) (numericEnumFrom (primMinusNat (Succ yx212000) yx218))))",fontsize=16,color="burlywood",shape="box"];4703[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3515 -> 4703[label="",style="solid", color="burlywood", weight=9]; 4703 -> 3523[label="",style="solid", color="burlywood", weight=3]; 4704[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3515 -> 4704[label="",style="solid", color="burlywood", weight=9]; 4704 -> 3524[label="",style="solid", color="burlywood", weight=3]; 3516[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero yx218)) (numericEnumFrom (primMinusNat Zero yx218))))",fontsize=16,color="burlywood",shape="box"];4705[label="yx218/Succ yx2180",fontsize=10,color="white",style="solid",shape="box"];3516 -> 4705[label="",style="solid", color="burlywood", weight=9]; 4705 -> 3525[label="",style="solid", color="burlywood", weight=3]; 4706[label="yx218/Zero",fontsize=10,color="white",style="solid",shape="box"];3516 -> 4706[label="",style="solid", color="burlywood", weight=9]; 4706 -> 3526[label="",style="solid", color="burlywood", weight=3]; 4245[label="yx2370",fontsize=16,color="green",shape="box"];4246[label="yx250",fontsize=16,color="green",shape="box"];4247[label="yx2370",fontsize=16,color="green",shape="box"];4248[label="yx250",fontsize=16,color="green",shape="box"];4249[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (numericEnumFrom (Pos yx254)))",fontsize=16,color="black",shape="box"];4249 -> 4261[label="",style="solid", color="black", weight=3]; 3156 -> 3055[label="",style="dashed", color="red", weight=0]; 3156[label="primMinusNatS yx2000 yx2010",fontsize=16,color="magenta"];3156 -> 3179[label="",style="dashed", color="magenta", weight=3]; 3156 -> 3180[label="",style="dashed", color="magenta", weight=3]; 3157[label="Succ yx2000",fontsize=16,color="green",shape="box"];3158[label="Zero",fontsize=16,color="green",shape="box"];3159[label="Zero",fontsize=16,color="green",shape="box"];3213[label="toEnum (Neg Zero) : map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="green",shape="box"];3213 -> 3250[label="",style="dashed", color="green", weight=3]; 3213 -> 3251[label="",style="dashed", color="green", weight=3]; 3214[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) otherwise)",fontsize=16,color="black",shape="box"];3214 -> 3252[label="",style="solid", color="black", weight=3]; 3329[label="intToRatio (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3329 -> 3347[label="",style="solid", color="black", weight=3]; 3330[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) yx212)) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) yx212))))",fontsize=16,color="burlywood",shape="box"];4707[label="yx212/Pos yx2120",fontsize=10,color="white",style="solid",shape="box"];3330 -> 4707[label="",style="solid", color="burlywood", weight=9]; 4707 -> 3348[label="",style="solid", color="burlywood", weight=3]; 4708[label="yx212/Neg yx2120",fontsize=10,color="white",style="solid",shape="box"];3330 -> 4708[label="",style="solid", color="burlywood", weight=9]; 4708 -> 3349[label="",style="solid", color="burlywood", weight=3]; 3331[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) otherwise)",fontsize=16,color="black",shape="box"];3331 -> 3350[label="",style="solid", color="black", weight=3]; 323[label="map toEnum (takeWhile1 (flip (<=) (properFractionQ1 yx40 yx41 (primQuotInt yx40 yx41,primRemInt yx40 yx41))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (properFractionQ1 yx40 yx41 (primQuotInt yx40 yx41,primRemInt yx40 yx41)) == GT)))",fontsize=16,color="black",shape="box"];323 -> 353[label="",style="solid", color="black", weight=3]; 3315 -> 3345[label="",style="dashed", color="red", weight=0]; 3315[label="properFractionQ1 yx70 yx71 (primQuotInt yx70 yx71,primRemInt yx70 yx71)",fontsize=16,color="magenta"];3315 -> 3346[label="",style="dashed", color="magenta", weight=3]; 3097[label="fromInteger (properFractionQ1 (Integer yx700) (Integer yx710) (Integer (primQuotInt yx700 yx710),Integer (primRemInt yx700 yx710)))",fontsize=16,color="black",shape="box"];3097 -> 3129[label="",style="solid", color="black", weight=3]; 4534[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (enforceWHNF (WHNF (primPlusInt (Pos yx274) (Pos yx2750))) (numericEnumFrom (primPlusInt (Pos yx274) (Pos yx2750)))))",fontsize=16,color="black",shape="box"];4534 -> 4536[label="",style="solid", color="black", weight=3]; 4535[label="map toEnum (takeWhile (flip (<=) (Pos (Succ yx272))) (enforceWHNF (WHNF (primPlusInt (Pos yx274) (Neg yx2750))) (numericEnumFrom (primPlusInt (Pos yx274) (Neg yx2750)))))",fontsize=16,color="black",shape="box"];4535 -> 4537[label="",style="solid", color="black", weight=3]; 3757[label="primPlusNat (Succ yx2180) yx21200",fontsize=16,color="burlywood",shape="box"];4709[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3757 -> 4709[label="",style="solid", color="burlywood", weight=9]; 4709 -> 4021[label="",style="solid", color="burlywood", weight=3]; 4710[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3757 -> 4710[label="",style="solid", color="burlywood", weight=9]; 4710 -> 4022[label="",style="solid", color="burlywood", weight=3]; 3758[label="primPlusNat Zero yx21200",fontsize=16,color="burlywood",shape="box"];4711[label="yx21200/Succ yx212000",fontsize=10,color="white",style="solid",shape="box"];3758 -> 4711[label="",style="solid", color="burlywood", weight=9]; 4711 -> 4023[label="",style="solid", color="burlywood", weight=3]; 4712[label="yx21200/Zero",fontsize=10,color="white",style="solid",shape="box"];3758 -> 4712[label="",style="solid", color="burlywood", weight=9]; 4712 -> 4024[label="",style="solid", color="burlywood", weight=3]; 4260 -> 4269[label="",style="dashed", color="red", weight=0]; 4260[label="map toEnum (takeWhile (flip (<=) (Pos Zero)) (Pos yx252 : (numericEnumFrom $! Pos yx252 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4260 -> 4270[label="",style="dashed", color="magenta", weight=3]; 3523[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) (Succ yx2180))) (numericEnumFrom (primMinusNat (Succ yx212000) (Succ yx2180)))))",fontsize=16,color="black",shape="box"];3523 -> 3548[label="",style="solid", color="black", weight=3]; 3524[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx212000) Zero)) (numericEnumFrom (primMinusNat (Succ yx212000) Zero))))",fontsize=16,color="black",shape="box"];3524 -> 3549[label="",style="solid", color="black", weight=3]; 3525[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero (Succ yx2180))) (numericEnumFrom (primMinusNat Zero (Succ yx2180)))))",fontsize=16,color="black",shape="box"];3525 -> 3550[label="",style="solid", color="black", weight=3]; 3526[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3526 -> 3551[label="",style="solid", color="black", weight=3]; 4261 -> 4271[label="",style="dashed", color="red", weight=0]; 4261[label="map toEnum (takeWhile (flip (<=) (Neg Zero)) (Pos yx254 : (numericEnumFrom $! Pos yx254 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4261 -> 4272[label="",style="dashed", color="magenta", weight=3]; 3179[label="yx2010",fontsize=16,color="green",shape="box"];3180[label="yx2000",fontsize=16,color="green",shape="box"];3250[label="toEnum (Neg Zero)",fontsize=16,color="blue",shape="box"];4713[label="toEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4713[label="",style="solid", color="blue", weight=9]; 4713 -> 3294[label="",style="solid", color="blue", weight=3]; 4714[label="toEnum :: Int -> ()",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4714[label="",style="solid", color="blue", weight=9]; 4714 -> 3295[label="",style="solid", color="blue", weight=3]; 4715[label="toEnum :: Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4715[label="",style="solid", color="blue", weight=9]; 4715 -> 3296[label="",style="solid", color="blue", weight=3]; 4716[label="toEnum :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4716[label="",style="solid", color="blue", weight=9]; 4716 -> 3297[label="",style="solid", color="blue", weight=3]; 4717[label="toEnum :: Int -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4717[label="",style="solid", color="blue", weight=9]; 4717 -> 3298[label="",style="solid", color="blue", weight=3]; 4718[label="toEnum :: Int -> Char",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4718[label="",style="solid", color="blue", weight=9]; 4718 -> 3299[label="",style="solid", color="blue", weight=3]; 4719[label="toEnum :: Int -> Double",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4719[label="",style="solid", color="blue", weight=9]; 4719 -> 3300[label="",style="solid", color="blue", weight=3]; 4720[label="toEnum :: Int -> Bool",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4720[label="",style="solid", color="blue", weight=9]; 4720 -> 3301[label="",style="solid", color="blue", weight=3]; 4721[label="toEnum :: Int -> Float",fontsize=10,color="white",style="solid",shape="box"];3250 -> 4721[label="",style="solid", color="blue", weight=9]; 4721 -> 3302[label="",style="solid", color="blue", weight=3]; 3251[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom $! Neg Zero + yx212))",fontsize=16,color="black",shape="box"];3251 -> 3303[label="",style="solid", color="black", weight=3]; 3252[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx212) True)",fontsize=16,color="black",shape="box"];3252 -> 3304[label="",style="solid", color="black", weight=3]; 3347[label="fromInt (Neg (Succ yx217)) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3347 -> 3399[label="",style="dashed", color="green", weight=3]; 3347 -> 3400[label="",style="dashed", color="green", weight=3]; 3348[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) (Pos yx2120))) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) (Pos yx2120)))))",fontsize=16,color="black",shape="box"];3348 -> 3401[label="",style="solid", color="black", weight=3]; 3349[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg (Succ yx218)) (Neg yx2120))) (numericEnumFrom (primPlusInt (Neg (Succ yx218)) (Neg yx2120)))))",fontsize=16,color="black",shape="box"];3349 -> 3402[label="",style="solid", color="black", weight=3]; 3350[label="map toEnum (takeWhile0 (flip (<=) yx207) (Neg (Succ yx217)) (numericEnumFrom $! Neg (Succ yx218) + yx212) True)",fontsize=16,color="black",shape="box"];3350 -> 3403[label="",style="solid", color="black", weight=3]; 353[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt yx40 yx41)) (Neg Zero) (numericEnumFrom $! 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Pos yx252 + yx258)))",fontsize=16,color="black",shape="box"];4283 -> 4293[label="",style="solid", color="black", weight=3]; 3566[label="yx212000",fontsize=16,color="green",shape="box"];3567[label="yx2180",fontsize=16,color="green",shape="box"];3568[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Pos (Succ yx212000))))",fontsize=16,color="black",shape="box"];3568 -> 3591[label="",style="solid", color="black", weight=3]; 3569[label="yx2180",fontsize=16,color="green",shape="box"];3482[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (Succ yx218))) (numericEnumFrom (Neg (Succ yx218)))))",fontsize=16,color="black",shape="triangle"];3482 -> 3517[label="",style="solid", color="black", weight=3]; 3570[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];3570 -> 3592[label="",style="solid", color="black", weight=3]; 4284[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4285[label="map toEnum (takeWhile2 (flip (<=) (Neg Zero)) (Pos yx254 : (numericEnumFrom $! Pos yx254 + yx259)))",fontsize=16,color="black",shape="box"];4285 -> 4294[label="",style="solid", color="black", weight=3]; 3333[label="error []",fontsize=16,color="red",shape="box"];3334[label="error []",fontsize=16,color="red",shape="box"];3335[label="error []",fontsize=16,color="red",shape="box"];3336[label="error []",fontsize=16,color="red",shape="box"];3337[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];3337 -> 3353[label="",style="solid", color="black", weight=3]; 3338[label="error []",fontsize=16,color="red",shape="box"];3339[label="error []",fontsize=16,color="red",shape="box"];3340[label="error []",fontsize=16,color="red",shape="box"];3341[label="error []",fontsize=16,color="red",shape="box"];3342[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg Zero + yx212)) (numericEnumFrom (Neg Zero + yx212))))",fontsize=16,color="black",shape="box"];3342 -> 3354[label="",style="solid", color="black", weight=3]; 3440 -> 576[label="",style="dashed", color="red", weight=0]; 3440[label="fromInt (Neg (Succ yx217))",fontsize=16,color="magenta"];3440 -> 3477[label="",style="dashed", color="magenta", weight=3]; 3441[label="fromInt (Neg (Succ yx217))",fontsize=16,color="black",shape="box"];3441 -> 3478[label="",style="solid", color="black", weight=3]; 3442 -> 576[label="",style="dashed", color="red", weight=0]; 3442[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3442 -> 3479[label="",style="dashed", color="magenta", weight=3]; 3443[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];3443 -> 3480[label="",style="solid", color="black", weight=3]; 3444[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx21200) (Succ yx218))) (numericEnumFrom (primMinusNat (Succ yx21200) (Succ yx218)))))",fontsize=16,color="black",shape="box"];3444 -> 3481[label="",style="solid", color="black", weight=3]; 3445[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero (Succ yx218))) (numericEnumFrom (primMinusNat Zero (Succ yx218)))))",fontsize=16,color="black",shape="box"];3445 -> 3482[label="",style="solid", color="black", weight=3]; 3446[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (primPlusNat (Succ yx218) yx2120))))",fontsize=16,color="black",shape="box"];3446 -> 3483[label="",style="solid", color="black", weight=3]; 422[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Pos yx410))) (Neg Zero) (numericEnumFrom $! 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Pos yx252 + yx258) (flip (<=) (Pos Zero) (Pos yx252)))",fontsize=16,color="black",shape="box"];4293 -> 4302[label="",style="solid", color="black", weight=3]; 3591 -> 3609[label="",style="dashed", color="red", weight=0]; 3591[label="map toEnum (takeWhile (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3591 -> 3610[label="",style="dashed", color="magenta", weight=3]; 3517[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (Succ yx218))))",fontsize=16,color="black",shape="box"];3517 -> 3527[label="",style="solid", color="black", weight=3]; 3592 -> 3614[label="",style="dashed", color="red", weight=0]; 3592[label="map toEnum (takeWhile (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3592 -> 3615[label="",style="dashed", color="magenta", weight=3]; 4294[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx254) (numericEnumFrom $! Pos yx254 + yx259) (flip (<=) (Neg Zero) (Pos yx254)))",fontsize=16,color="black",shape="box"];4294 -> 4303[label="",style="solid", color="black", weight=3]; 3353[label="intToRatio (Neg Zero)",fontsize=16,color="black",shape="box"];3353 -> 3414[label="",style="solid", color="black", weight=3]; 3354[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) yx212)) (numericEnumFrom (primPlusInt (Neg Zero) yx212))))",fontsize=16,color="burlywood",shape="box"];4748[label="yx212/Pos yx2120",fontsize=10,color="white",style="solid",shape="box"];3354 -> 4748[label="",style="solid", color="burlywood", weight=9]; 4748 -> 3415[label="",style="solid", color="burlywood", weight=3]; 4749[label="yx212/Neg yx2120",fontsize=10,color="white",style="solid",shape="box"];3354 -> 4749[label="",style="solid", color="burlywood", weight=9]; 4749 -> 3416[label="",style="solid", color="burlywood", weight=3]; 3477[label="Neg (Succ yx217)",fontsize=16,color="green",shape="box"];3478[label="Integer (Neg (Succ yx217))",fontsize=16,color="green",shape="box"];3479[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3480[label="Integer (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3483 -> 3518[label="",style="dashed", color="red", weight=0]; 3483[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! 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Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Neg (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];477 -> 536[label="",style="solid", color="black", weight=3]; 478[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Pos yx400) (Neg Zero))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (primQuotInt (Pos yx400) (Neg Zero)) == GT)))",fontsize=16,color="black",shape="box"];478 -> 537[label="",style="solid", color="black", weight=3]; 479[label="map toEnum (takeWhile1 (flip (<=) (primQuotInt (Neg yx400) (Pos (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! 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Pos yx278 + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];4549 -> 4551[label="",style="dashed", color="magenta", weight=3]; 4061[label="yx212000",fontsize=16,color="green",shape="box"];4062[label="yx2180",fontsize=16,color="green",shape="box"];4302[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx252) (numericEnumFrom $! Pos yx252 + yx258) ((<=) Pos yx252 Pos Zero))",fontsize=16,color="black",shape="box"];4302 -> 4312[label="",style="solid", color="black", weight=3]; 3610 -> 576[label="",style="dashed", color="red", weight=0]; 3610[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3610 -> 3626[label="",style="dashed", color="magenta", weight=3]; 3609[label="map toEnum (takeWhile (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + yx238)))",fontsize=16,color="black",shape="triangle"];3609 -> 3627[label="",style="solid", color="black", weight=3]; 3527 -> 3552[label="",style="dashed", color="red", weight=0]; 3527[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3527 -> 3553[label="",style="dashed", color="magenta", weight=3]; 3615 -> 576[label="",style="dashed", color="red", weight=0]; 3615[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3615 -> 3628[label="",style="dashed", color="magenta", weight=3]; 3614[label="map toEnum (takeWhile (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + yx239)))",fontsize=16,color="black",shape="triangle"];3614 -> 3629[label="",style="solid", color="black", weight=3]; 4303[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx254) (numericEnumFrom $! Pos yx254 + yx259) ((<=) Pos yx254 Neg Zero))",fontsize=16,color="black",shape="box"];4303 -> 4313[label="",style="solid", color="black", weight=3]; 3414[label="fromInt (Neg Zero) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3414 -> 3457[label="",style="dashed", color="green", weight=3]; 3414 -> 3458[label="",style="dashed", color="green", weight=3]; 3415[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Pos yx2120))) (numericEnumFrom (primPlusInt (Neg Zero) (Pos yx2120)))))",fontsize=16,color="black",shape="box"];3415 -> 3459[label="",style="solid", color="black", weight=3]; 3416[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Neg yx2120))) (numericEnumFrom (primPlusInt (Neg Zero) (Neg yx2120)))))",fontsize=16,color="black",shape="box"];3416 -> 3460[label="",style="solid", color="black", weight=3]; 3519 -> 576[label="",style="dashed", color="red", weight=0]; 3519[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3519 -> 3528[label="",style="dashed", color="magenta", weight=3]; 3518[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232)))",fontsize=16,color="black",shape="triangle"];3518 -> 3529[label="",style="solid", color="black", weight=3]; 534[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4758[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];534 -> 4758[label="",style="solid", color="burlywood", weight=9]; 4758 -> 592[label="",style="solid", color="burlywood", weight=3]; 4759[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];534 -> 4759[label="",style="solid", color="burlywood", weight=9]; 4759 -> 593[label="",style="solid", color="burlywood", weight=3]; 535[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="black",shape="triangle"];535 -> 594[label="",style="solid", color="black", weight=3]; 536[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="burlywood",shape="triangle"];4760[label="yx400/Succ yx4000",fontsize=10,color="white",style="solid",shape="box"];536 -> 4760[label="",style="solid", color="burlywood", weight=9]; 4760 -> 595[label="",style="solid", color="burlywood", weight=3]; 4761[label="yx400/Zero",fontsize=10,color="white",style="solid",shape="box"];536 -> 4761[label="",style="solid", color="burlywood", weight=9]; 4761 -> 596[label="",style="solid", color="burlywood", weight=3]; 537 -> 535[label="",style="dashed", color="red", weight=0]; 537[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (error []) == GT)))",fontsize=16,color="magenta"];538 -> 536[label="",style="dashed", color="red", weight=0]; 538[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS yx400 (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS yx400 (Succ yx4100))) == GT)))",fontsize=16,color="magenta"];538 -> 597[label="",style="dashed", color="magenta", weight=3]; 538 -> 598[label="",style="dashed", color="magenta", weight=3]; 539 -> 535[label="",style="dashed", color="red", weight=0]; 539[label="map toEnum (takeWhile1 (flip (<=) (error [])) (Neg Zero) (numericEnumFrom $! 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Pos yx278 + yx280)))",fontsize=16,color="black",shape="triangle"];4550 -> 4553[label="",style="solid", color="black", weight=3]; 4312[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx252) (numericEnumFrom $! Pos yx252 + yx258) (compare (Pos yx252) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];4312 -> 4341[label="",style="solid", color="black", weight=3]; 3626[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3627[label="map toEnum (takeWhile2 (flip (<=) yx207) (Pos (Succ yx212000) : (numericEnumFrom $! Pos (Succ yx212000) + yx238)))",fontsize=16,color="black",shape="box"];3627 -> 3646[label="",style="solid", color="black", weight=3]; 3553 -> 576[label="",style="dashed", color="red", weight=0]; 3553[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3553 -> 3571[label="",style="dashed", color="magenta", weight=3]; 3552[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + yx234)))",fontsize=16,color="black",shape="triangle"];3552 -> 3572[label="",style="solid", color="black", weight=3]; 3628[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3629[label="map toEnum (takeWhile2 (flip (<=) yx207) (Pos Zero : (numericEnumFrom $! Pos Zero + yx239)))",fontsize=16,color="black",shape="box"];3629 -> 3647[label="",style="solid", color="black", weight=3]; 4313[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx254) (numericEnumFrom $! Pos yx254 + yx259) (compare (Pos yx254) (Neg Zero) /= GT))",fontsize=16,color="black",shape="box"];4313 -> 4342[label="",style="solid", color="black", weight=3]; 3457[label="fromInt (Neg Zero)",fontsize=16,color="blue",shape="box"];4762[label="fromInt :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3457 -> 4762[label="",style="solid", color="blue", weight=9]; 4762 -> 3486[label="",style="solid", color="blue", weight=3]; 4763[label="fromInt :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];3457 -> 4763[label="",style="solid", color="blue", weight=9]; 4763 -> 3487[label="",style="solid", color="blue", weight=3]; 3458[label="fromInt (Pos (Succ Zero))",fontsize=16,color="blue",shape="box"];4764[label="fromInt :: -> Int Int",fontsize=10,color="white",style="solid",shape="box"];3458 -> 4764[label="",style="solid", color="blue", weight=9]; 4764 -> 3488[label="",style="solid", color="blue", weight=3]; 4765[label="fromInt :: -> Int Integer",fontsize=10,color="white",style="solid",shape="box"];3458 -> 4765[label="",style="solid", color="blue", weight=9]; 4765 -> 3489[label="",style="solid", color="blue", weight=3]; 3459[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat yx2120 Zero)) (numericEnumFrom (primMinusNat yx2120 Zero))))",fontsize=16,color="burlywood",shape="box"];4766[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3459 -> 4766[label="",style="solid", color="burlywood", weight=9]; 4766 -> 3490[label="",style="solid", color="burlywood", weight=3]; 4767[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3459 -> 4767[label="",style="solid", color="burlywood", weight=9]; 4767 -> 3491[label="",style="solid", color="burlywood", weight=3]; 3460[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Neg (primPlusNat Zero yx2120))) (numericEnumFrom (Neg (primPlusNat Zero yx2120)))))",fontsize=16,color="black",shape="box"];3460 -> 3492[label="",style="solid", color="black", weight=3]; 3528[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3529[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120) : (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232)))",fontsize=16,color="black",shape="box"];3529 -> 3557[label="",style="solid", color="black", weight=3]; 592[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS (Succ yx4000) (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS (Succ yx4000) (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];592 -> 677[label="",style="solid", color="black", weight=3]; 593[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS Zero (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS Zero (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];593 -> 678[label="",style="solid", color="black", weight=3]; 594[label="error []",fontsize=16,color="red",shape="box"];595[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS (Succ yx4000) (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS (Succ yx4000) (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];595 -> 679[label="",style="solid", color="black", weight=3]; 596[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS Zero (Succ yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS Zero (Succ yx4100))) == GT)))",fontsize=16,color="black",shape="box"];596 -> 680[label="",style="solid", color="black", weight=3]; 597[label="yx4100",fontsize=16,color="green",shape="box"];598[label="yx400",fontsize=16,color="green",shape="box"];599[label="yx4100",fontsize=16,color="green",shape="box"];600[label="yx400",fontsize=16,color="green",shape="box"];3316[label="Pos (primDivNatS yx7000 (Succ yx71000))",fontsize=16,color="green",shape="box"];3316 -> 3361[label="",style="dashed", color="green", weight=3]; 3317[label="error []",fontsize=16,color="black",shape="triangle"];3317 -> 3362[label="",style="solid", color="black", weight=3]; 3318[label="Neg (primDivNatS yx7000 (Succ yx71000))",fontsize=16,color="green",shape="box"];3318 -> 3363[label="",style="dashed", color="green", weight=3]; 3319 -> 3317[label="",style="dashed", color="red", weight=0]; 3319[label="error []",fontsize=16,color="magenta"];3320[label="Neg (primDivNatS yx7000 (Succ yx71000))",fontsize=16,color="green",shape="box"];3320 -> 3364[label="",style="dashed", color="green", weight=3]; 3321 -> 3317[label="",style="dashed", color="red", weight=0]; 3321[label="error []",fontsize=16,color="magenta"];3322[label="Pos (primDivNatS yx7000 (Succ yx71000))",fontsize=16,color="green",shape="box"];3322 -> 3365[label="",style="dashed", color="green", weight=3]; 3323 -> 3317[label="",style="dashed", color="red", weight=0]; 3323[label="error []",fontsize=16,color="magenta"];4552[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];4553[label="map toEnum (takeWhile2 (flip (<=) (Pos (Succ yx272))) (Pos yx278 : (numericEnumFrom $! Pos yx278 + yx280)))",fontsize=16,color="black",shape="box"];4553 -> 4554[label="",style="solid", color="black", weight=3]; 4341[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx252) (numericEnumFrom $! Pos yx252 + yx258) (not (compare (Pos yx252) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];4341 -> 4350[label="",style="solid", color="black", weight=3]; 3646[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx238) (flip (<=) yx207 (Pos (Succ yx212000))))",fontsize=16,color="black",shape="box"];3646 -> 3658[label="",style="solid", color="black", weight=3]; 3571[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3572[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (Succ yx218) : (numericEnumFrom $! Neg (Succ yx218) + yx234)))",fontsize=16,color="black",shape="box"];3572 -> 3593[label="",style="solid", color="black", weight=3]; 3647[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx239) (flip (<=) yx207 (Pos Zero)))",fontsize=16,color="black",shape="box"];3647 -> 3659[label="",style="solid", color="black", weight=3]; 4342[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx254) (numericEnumFrom $! Pos yx254 + yx259) (not (compare (Pos yx254) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];4342 -> 4351[label="",style="solid", color="black", weight=3]; 3486 -> 576[label="",style="dashed", color="red", weight=0]; 3486[label="fromInt (Neg Zero)",fontsize=16,color="magenta"];3486 -> 3530[label="",style="dashed", color="magenta", weight=3]; 3487[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];3487 -> 3531[label="",style="solid", color="black", weight=3]; 3488 -> 576[label="",style="dashed", color="red", weight=0]; 3488[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3488 -> 3532[label="",style="dashed", color="magenta", weight=3]; 3489 -> 3443[label="",style="dashed", color="red", weight=0]; 3489[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3490[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat (Succ yx21200) Zero)) (numericEnumFrom (primMinusNat (Succ yx21200) Zero))))",fontsize=16,color="black",shape="box"];3490 -> 3533[label="",style="solid", color="black", weight=3]; 3491[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3491 -> 3534[label="",style="solid", color="black", weight=3]; 3492[label="map toEnum (takeWhile (flip (<=) yx207) (numericEnumFrom (Neg (primPlusNat Zero yx2120))))",fontsize=16,color="black",shape="box"];3492 -> 3535[label="",style="solid", color="black", weight=3]; 3557[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232) (flip (<=) yx207 (Neg (primPlusNat (Succ yx218) yx2120))))",fontsize=16,color="black",shape="box"];3557 -> 3585[label="",style="solid", color="black", weight=3]; 677 -> 856[label="",style="dashed", color="red", weight=0]; 677[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];677 -> 857[label="",style="dashed", color="magenta", weight=3]; 678 -> 858[label="",style="dashed", color="red", weight=0]; 678[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="magenta"];678 -> 859[label="",style="dashed", color="magenta", weight=3]; 679 -> 860[label="",style="dashed", color="red", weight=0]; 679[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];679 -> 861[label="",style="dashed", color="magenta", weight=3]; 680 -> 862[label="",style="dashed", color="red", weight=0]; 680[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + fromInt (Pos (Succ Zero))) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="magenta"];680 -> 863[label="",style="dashed", color="magenta", weight=3]; 3361 -> 1689[label="",style="dashed", color="red", weight=0]; 3361[label="primDivNatS yx7000 (Succ yx71000)",fontsize=16,color="magenta"];3361 -> 3421[label="",style="dashed", color="magenta", weight=3]; 3361 -> 3422[label="",style="dashed", color="magenta", weight=3]; 3362[label="error []",fontsize=16,color="red",shape="box"];3363 -> 1689[label="",style="dashed", color="red", weight=0]; 3363[label="primDivNatS yx7000 (Succ yx71000)",fontsize=16,color="magenta"];3363 -> 3423[label="",style="dashed", color="magenta", weight=3]; 3363 -> 3424[label="",style="dashed", color="magenta", weight=3]; 3364 -> 1689[label="",style="dashed", color="red", weight=0]; 3364[label="primDivNatS yx7000 (Succ yx71000)",fontsize=16,color="magenta"];3364 -> 3425[label="",style="dashed", color="magenta", weight=3]; 3364 -> 3426[label="",style="dashed", color="magenta", weight=3]; 3365 -> 1689[label="",style="dashed", color="red", weight=0]; 3365[label="primDivNatS yx7000 (Succ yx71000)",fontsize=16,color="magenta"];3365 -> 3427[label="",style="dashed", color="magenta", weight=3]; 3365 -> 3428[label="",style="dashed", color="magenta", weight=3]; 4554[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx278) (numericEnumFrom $! Pos yx278 + yx280) (flip (<=) (Pos (Succ yx272)) (Pos yx278)))",fontsize=16,color="black",shape="box"];4554 -> 4555[label="",style="solid", color="black", weight=3]; 4350 -> 3761[label="",style="dashed", color="red", weight=0]; 4350[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Pos yx252) (numericEnumFrom $! Pos yx252 + yx258) (not (primCmpInt (Pos yx252) (Pos Zero) == GT)))",fontsize=16,color="magenta"];4350 -> 4360[label="",style="dashed", color="magenta", weight=3]; 4350 -> 4361[label="",style="dashed", color="magenta", weight=3]; 4350 -> 4362[label="",style="dashed", color="magenta", weight=3]; 4350 -> 4363[label="",style="dashed", color="magenta", weight=3]; 4350 -> 4364[label="",style="dashed", color="magenta", weight=3]; 3658[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx238) ((<=) Pos (Succ yx212000) yx207))",fontsize=16,color="black",shape="box"];3658 -> 3670[label="",style="solid", color="black", weight=3]; 3593[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx234) (flip (<=) yx207 (Neg (Succ yx218))))",fontsize=16,color="black",shape="box"];3593 -> 3620[label="",style="solid", color="black", weight=3]; 3659[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx239) ((<=) Pos Zero yx207))",fontsize=16,color="black",shape="box"];3659 -> 3671[label="",style="solid", color="black", weight=3]; 4351 -> 3761[label="",style="dashed", color="red", weight=0]; 4351[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Pos yx254) (numericEnumFrom $! Pos yx254 + yx259) (not (primCmpInt (Pos yx254) (Neg Zero) == GT)))",fontsize=16,color="magenta"];4351 -> 4365[label="",style="dashed", color="magenta", weight=3]; 4351 -> 4366[label="",style="dashed", color="magenta", weight=3]; 4351 -> 4367[label="",style="dashed", color="magenta", weight=3]; 4351 -> 4368[label="",style="dashed", color="magenta", weight=3]; 4351 -> 4369[label="",style="dashed", color="magenta", weight=3]; 3530[label="Neg Zero",fontsize=16,color="green",shape="box"];3531[label="Integer (Neg Zero)",fontsize=16,color="green",shape="box"];3532[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3533 -> 3549[label="",style="dashed", color="red", weight=0]; 3533[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos (Succ yx21200))) (numericEnumFrom (Pos (Succ yx21200)))))",fontsize=16,color="magenta"];3533 -> 3576[label="",style="dashed", color="magenta", weight=3]; 3534 -> 3551[label="",style="dashed", color="red", weight=0]; 3534[label="map toEnum (takeWhile (flip (<=) yx207) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="magenta"];3535 -> 3577[label="",style="dashed", color="red", weight=0]; 3535[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3535 -> 3578[label="",style="dashed", color="magenta", weight=3]; 3585[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232) ((<=) Neg (primPlusNat (Succ yx218) yx2120) yx207))",fontsize=16,color="black",shape="box"];3585 -> 3606[label="",style="solid", color="black", weight=3]; 857 -> 576[label="",style="dashed", color="red", weight=0]; 857[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];857 -> 1062[label="",style="dashed", color="magenta", weight=3]; 856 -> 803[label="",style="dashed", color="red", weight=0]; 856[label="map toEnum (takeWhile1 (flip (<=) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + yx64) (not (primCmpInt (Neg Zero) (Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];856 -> 1063[label="",style="dashed", color="magenta", weight=3]; 856 -> 1064[label="",style="dashed", color="magenta", weight=3]; 856 -> 1065[label="",style="dashed", color="magenta", weight=3]; 859 -> 576[label="",style="dashed", color="red", weight=0]; 859[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];859 -> 1066[label="",style="dashed", color="magenta", weight=3]; 858 -> 803[label="",style="dashed", color="red", weight=0]; 858[label="map toEnum (takeWhile1 (flip (<=) (Pos Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + yx65) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="magenta"];858 -> 1067[label="",style="dashed", color="magenta", weight=3]; 858 -> 1068[label="",style="dashed", color="magenta", weight=3]; 858 -> 1069[label="",style="dashed", color="magenta", weight=3]; 861 -> 576[label="",style="dashed", color="red", weight=0]; 861[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];861 -> 1070[label="",style="dashed", color="magenta", weight=3]; 860 -> 803[label="",style="dashed", color="red", weight=0]; 860[label="map toEnum (takeWhile1 (flip (<=) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)))) (Neg Zero) (numericEnumFrom $! Neg Zero + yx66) (not (primCmpInt (Neg Zero) (Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))) == GT)))",fontsize=16,color="magenta"];860 -> 1071[label="",style="dashed", color="magenta", weight=3]; 860 -> 1072[label="",style="dashed", color="magenta", weight=3]; 860 -> 1073[label="",style="dashed", color="magenta", weight=3]; 863 -> 576[label="",style="dashed", color="red", weight=0]; 863[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];863 -> 1074[label="",style="dashed", color="magenta", weight=3]; 862 -> 803[label="",style="dashed", color="red", weight=0]; 862[label="map toEnum (takeWhile1 (flip (<=) (Neg Zero)) (Neg Zero) (numericEnumFrom $! Neg Zero + yx67) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="magenta"];862 -> 1075[label="",style="dashed", color="magenta", weight=3]; 862 -> 1076[label="",style="dashed", color="magenta", weight=3]; 862 -> 1077[label="",style="dashed", color="magenta", weight=3]; 3421[label="yx71000",fontsize=16,color="green",shape="box"];3422[label="yx7000",fontsize=16,color="green",shape="box"];3423[label="yx71000",fontsize=16,color="green",shape="box"];3424[label="yx7000",fontsize=16,color="green",shape="box"];3425[label="yx71000",fontsize=16,color="green",shape="box"];3426[label="yx7000",fontsize=16,color="green",shape="box"];3427[label="yx71000",fontsize=16,color="green",shape="box"];3428[label="yx7000",fontsize=16,color="green",shape="box"];4555[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx278) (numericEnumFrom $! Pos yx278 + yx280) ((<=) Pos yx278 Pos (Succ yx272)))",fontsize=16,color="black",shape="box"];4555 -> 4556[label="",style="solid", color="black", weight=3]; 4360[label="yx258",fontsize=16,color="green",shape="box"];4361[label="yx252",fontsize=16,color="green",shape="box"];4362[label="yx252",fontsize=16,color="green",shape="box"];4363[label="Pos Zero",fontsize=16,color="green",shape="box"];4364[label="yx252",fontsize=16,color="green",shape="box"];3670[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx238) (compare (Pos (Succ yx212000)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3670 -> 3692[label="",style="solid", color="black", weight=3]; 3620[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx234) ((<=) Neg (Succ yx218) yx207))",fontsize=16,color="black",shape="box"];3620 -> 3641[label="",style="solid", color="black", weight=3]; 3671[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx239) (compare (Pos Zero) yx207 /= GT))",fontsize=16,color="black",shape="box"];3671 -> 3693[label="",style="solid", color="black", weight=3]; 4365[label="yx259",fontsize=16,color="green",shape="box"];4366[label="yx254",fontsize=16,color="green",shape="box"];4367[label="yx254",fontsize=16,color="green",shape="box"];4368[label="Neg Zero",fontsize=16,color="green",shape="box"];4369[label="yx254",fontsize=16,color="green",shape="box"];3576[label="yx21200",fontsize=16,color="green",shape="box"];3578 -> 576[label="",style="dashed", color="red", weight=0]; 3578[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3578 -> 3596[label="",style="dashed", color="magenta", weight=3]; 3577[label="map toEnum (takeWhile (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235)))",fontsize=16,color="black",shape="triangle"];3577 -> 3597[label="",style="solid", color="black", weight=3]; 3606[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232) (compare (Neg (primPlusNat (Succ yx218) yx2120)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3606 -> 3621[label="",style="solid", color="black", weight=3]; 1062[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1063[label="Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1063 -> 1217[label="",style="dashed", color="green", weight=3]; 1064[label="Pos (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1064 -> 1218[label="",style="dashed", color="green", weight=3]; 1065[label="yx64",fontsize=16,color="green",shape="box"];803[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) yx27 == GT)))",fontsize=16,color="burlywood",shape="triangle"];4768[label="yx27/Pos yx270",fontsize=10,color="white",style="solid",shape="box"];803 -> 4768[label="",style="solid", color="burlywood", weight=9]; 4768 -> 972[label="",style="solid", color="burlywood", weight=3]; 4769[label="yx27/Neg yx270",fontsize=10,color="white",style="solid",shape="box"];803 -> 4769[label="",style="solid", color="burlywood", weight=9]; 4769 -> 973[label="",style="solid", color="burlywood", weight=3]; 1066[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1067[label="Pos Zero",fontsize=16,color="green",shape="box"];1068[label="Pos Zero",fontsize=16,color="green",shape="box"];1069[label="yx65",fontsize=16,color="green",shape="box"];1070[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1071[label="Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1071 -> 1219[label="",style="dashed", color="green", weight=3]; 1072[label="Neg (primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100))",fontsize=16,color="green",shape="box"];1072 -> 1220[label="",style="dashed", color="green", weight=3]; 1073[label="yx66",fontsize=16,color="green",shape="box"];1074[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];1075[label="Neg Zero",fontsize=16,color="green",shape="box"];1076[label="Neg Zero",fontsize=16,color="green",shape="box"];1077[label="yx67",fontsize=16,color="green",shape="box"];4556[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx278) (numericEnumFrom $! Pos yx278 + yx280) (compare (Pos yx278) (Pos (Succ yx272)) /= GT))",fontsize=16,color="black",shape="box"];4556 -> 4557[label="",style="solid", color="black", weight=3]; 3692[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx238) (not (compare (Pos (Succ yx212000)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3692 -> 3751[label="",style="solid", color="black", weight=3]; 3641[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx234) (compare (Neg (Succ yx218)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3641 -> 3652[label="",style="solid", color="black", weight=3]; 3693[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx239) (not (compare (Pos Zero) yx207 == GT)))",fontsize=16,color="black",shape="box"];3693 -> 3752[label="",style="solid", color="black", weight=3]; 3596[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3597[label="map toEnum (takeWhile2 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120) : (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235)))",fontsize=16,color="black",shape="box"];3597 -> 3625[label="",style="solid", color="black", weight=3]; 3621[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232) (not (compare (Neg (primPlusNat (Succ yx218) yx2120)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3621 -> 3642[label="",style="solid", color="black", weight=3]; 1217 -> 1151[label="",style="dashed", color="red", weight=0]; 1217[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1218 -> 1151[label="",style="dashed", color="red", weight=0]; 1218[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];972[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos yx270) == GT)))",fontsize=16,color="burlywood",shape="box"];4770[label="yx270/Succ yx2700",fontsize=10,color="white",style="solid",shape="box"];972 -> 4770[label="",style="solid", color="burlywood", weight=9]; 4770 -> 1161[label="",style="solid", color="burlywood", weight=3]; 4771[label="yx270/Zero",fontsize=10,color="white",style="solid",shape="box"];972 -> 4771[label="",style="solid", color="burlywood", weight=9]; 4771 -> 1162[label="",style="solid", color="burlywood", weight=3]; 973[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg yx270) == GT)))",fontsize=16,color="burlywood",shape="box"];4772[label="yx270/Succ yx2700",fontsize=10,color="white",style="solid",shape="box"];973 -> 4772[label="",style="solid", color="burlywood", weight=9]; 4772 -> 1163[label="",style="solid", color="burlywood", weight=3]; 4773[label="yx270/Zero",fontsize=10,color="white",style="solid",shape="box"];973 -> 4773[label="",style="solid", color="burlywood", weight=9]; 4773 -> 1164[label="",style="solid", color="burlywood", weight=3]; 1219 -> 1151[label="",style="dashed", color="red", weight=0]; 1219[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1219 -> 1342[label="",style="dashed", color="magenta", weight=3]; 1220 -> 1151[label="",style="dashed", color="red", weight=0]; 1220[label="primDivNatS0 yx4000 yx4100 (primGEqNatS yx4000 yx4100)",fontsize=16,color="magenta"];1220 -> 1343[label="",style="dashed", color="magenta", weight=3]; 4557[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx278) (numericEnumFrom $! Pos yx278 + yx280) (not (compare (Pos yx278) (Pos (Succ yx272)) == GT)))",fontsize=16,color="black",shape="box"];4557 -> 4558[label="",style="solid", color="black", weight=3]; 3751 -> 3761[label="",style="dashed", color="red", weight=0]; 3751[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos (Succ yx212000)) (numericEnumFrom $! Pos (Succ yx212000) + yx238) (not (primCmpInt (Pos (Succ yx212000)) yx207 == GT)))",fontsize=16,color="magenta"];3751 -> 3979[label="",style="dashed", color="magenta", weight=3]; 3751 -> 3980[label="",style="dashed", color="magenta", weight=3]; 3751 -> 3981[label="",style="dashed", color="magenta", weight=3]; 3751 -> 3982[label="",style="dashed", color="magenta", weight=3]; 3751 -> 3983[label="",style="dashed", color="magenta", weight=3]; 3652[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx234) (not (compare (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3652 -> 3664[label="",style="solid", color="black", weight=3]; 3752 -> 3761[label="",style="dashed", color="red", weight=0]; 3752[label="map toEnum (takeWhile1 (flip (<=) yx207) (Pos Zero) (numericEnumFrom $! Pos Zero + yx239) (not (primCmpInt (Pos Zero) yx207 == GT)))",fontsize=16,color="magenta"];3752 -> 3984[label="",style="dashed", color="magenta", weight=3]; 3752 -> 3985[label="",style="dashed", color="magenta", weight=3]; 3752 -> 3986[label="",style="dashed", color="magenta", weight=3]; 3752 -> 3987[label="",style="dashed", color="magenta", weight=3]; 3752 -> 3988[label="",style="dashed", color="magenta", weight=3]; 3625[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235) (flip (<=) yx207 (Neg (primPlusNat Zero yx2120))))",fontsize=16,color="black",shape="box"];3625 -> 3645[label="",style="solid", color="black", weight=3]; 3642[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) yx2120)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) yx2120) + yx232) (not (primCmpInt (Neg (primPlusNat (Succ yx218) yx2120)) yx207 == GT)))",fontsize=16,color="burlywood",shape="box"];4774[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3642 -> 4774[label="",style="solid", color="burlywood", weight=9]; 4774 -> 3653[label="",style="solid", color="burlywood", weight=3]; 4775[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3642 -> 4775[label="",style="solid", color="burlywood", weight=9]; 4775 -> 3654[label="",style="solid", color="burlywood", weight=3]; 1161[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos (Succ yx2700)) == GT)))",fontsize=16,color="black",shape="box"];1161 -> 1288[label="",style="solid", color="black", weight=3]; 1162[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];1162 -> 1289[label="",style="solid", color="black", weight=3]; 1163[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg (Succ yx2700)) == GT)))",fontsize=16,color="black",shape="box"];1163 -> 1290[label="",style="solid", color="black", weight=3]; 1164[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpInt (Neg Zero) (Neg Zero) == GT)))",fontsize=16,color="black",shape="box"];1164 -> 1291[label="",style="solid", color="black", weight=3]; 1342[label="yx4100",fontsize=16,color="green",shape="box"];1343[label="yx4100",fontsize=16,color="green",shape="box"];4558 -> 3761[label="",style="dashed", color="red", weight=0]; 4558[label="map toEnum (takeWhile1 (flip (<=) (Pos (Succ yx272))) (Pos yx278) (numericEnumFrom $! Pos yx278 + yx280) (not (primCmpInt (Pos yx278) (Pos (Succ yx272)) == GT)))",fontsize=16,color="magenta"];4558 -> 4559[label="",style="dashed", color="magenta", weight=3]; 4558 -> 4560[label="",style="dashed", color="magenta", weight=3]; 4558 -> 4561[label="",style="dashed", color="magenta", weight=3]; 4558 -> 4562[label="",style="dashed", color="magenta", weight=3]; 4558 -> 4563[label="",style="dashed", color="magenta", weight=3]; 3979[label="yx238",fontsize=16,color="green",shape="box"];3980[label="Succ yx212000",fontsize=16,color="green",shape="box"];3981[label="Succ yx212000",fontsize=16,color="green",shape="box"];3982[label="yx207",fontsize=16,color="green",shape="box"];3983[label="Succ yx212000",fontsize=16,color="green",shape="box"];3664 -> 2772[label="",style="dashed", color="red", weight=0]; 3664[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx234) (not (primCmpInt (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="magenta"];3664 -> 3676[label="",style="dashed", color="magenta", weight=3]; 3664 -> 3677[label="",style="dashed", color="magenta", weight=3]; 3664 -> 3678[label="",style="dashed", color="magenta", weight=3]; 3664 -> 3679[label="",style="dashed", color="magenta", weight=3]; 3984[label="yx239",fontsize=16,color="green",shape="box"];3985[label="Zero",fontsize=16,color="green",shape="box"];3986[label="Zero",fontsize=16,color="green",shape="box"];3987[label="yx207",fontsize=16,color="green",shape="box"];3988[label="Zero",fontsize=16,color="green",shape="box"];3645[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235) ((<=) Neg (primPlusNat Zero yx2120) yx207))",fontsize=16,color="black",shape="box"];3645 -> 3657[label="",style="solid", color="black", weight=3]; 3653[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) (Succ yx21200))) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) (Succ yx21200)) + yx232) (not (primCmpInt (Neg (primPlusNat (Succ yx218) (Succ yx21200))) yx207 == GT)))",fontsize=16,color="black",shape="box"];3653 -> 3665[label="",style="solid", color="black", weight=3]; 3654[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat (Succ yx218) Zero)) (numericEnumFrom $! Neg (primPlusNat (Succ yx218) Zero) + yx232) (not (primCmpInt (Neg (primPlusNat (Succ yx218) Zero)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3654 -> 3666[label="",style="solid", color="black", weight=3]; 1288[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (LT == GT)))",fontsize=16,color="black",shape="box"];1288 -> 1432[label="",style="solid", color="black", weight=3]; 1289[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (EQ == GT)))",fontsize=16,color="black",shape="triangle"];1289 -> 1433[label="",style="solid", color="black", weight=3]; 1290[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (primCmpNat (Succ yx2700) Zero == GT)))",fontsize=16,color="black",shape="box"];1290 -> 1434[label="",style="solid", color="black", weight=3]; 1291 -> 1289[label="",style="dashed", color="red", weight=0]; 1291[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (EQ == GT)))",fontsize=16,color="magenta"];4559[label="yx280",fontsize=16,color="green",shape="box"];4560[label="yx278",fontsize=16,color="green",shape="box"];4561[label="yx278",fontsize=16,color="green",shape="box"];4562[label="Pos (Succ yx272)",fontsize=16,color="green",shape="box"];4563[label="yx278",fontsize=16,color="green",shape="box"];3676[label="yx218",fontsize=16,color="green",shape="box"];3677[label="yx218",fontsize=16,color="green",shape="box"];3678[label="yx207",fontsize=16,color="green",shape="box"];3679[label="yx234",fontsize=16,color="green",shape="box"];3657[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235) (compare (Neg (primPlusNat Zero yx2120)) yx207 /= GT))",fontsize=16,color="black",shape="box"];3657 -> 3669[label="",style="solid", color="black", weight=3]; 3665 -> 2772[label="",style="dashed", color="red", weight=0]; 3665[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ (Succ (primPlusNat yx218 yx21200)))) (numericEnumFrom $! Neg (Succ (Succ (primPlusNat yx218 yx21200))) + yx232) (not (primCmpInt (Neg (Succ (Succ (primPlusNat yx218 yx21200)))) yx207 == GT)))",fontsize=16,color="magenta"];3665 -> 3680[label="",style="dashed", color="magenta", weight=3]; 3665 -> 3681[label="",style="dashed", color="magenta", weight=3]; 3665 -> 3682[label="",style="dashed", color="magenta", weight=3]; 3665 -> 3683[label="",style="dashed", color="magenta", weight=3]; 3665 -> 3684[label="",style="dashed", color="magenta", weight=3]; 3666 -> 2772[label="",style="dashed", color="red", weight=0]; 3666[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx218)) (numericEnumFrom $! Neg (Succ yx218) + yx232) (not (primCmpInt (Neg (Succ yx218)) yx207 == GT)))",fontsize=16,color="magenta"];3666 -> 3685[label="",style="dashed", color="magenta", weight=3]; 3666 -> 3686[label="",style="dashed", color="magenta", weight=3]; 3666 -> 3687[label="",style="dashed", color="magenta", weight=3]; 3666 -> 3688[label="",style="dashed", color="magenta", weight=3]; 1432[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not False))",fontsize=16,color="black",shape="triangle"];1432 -> 1570[label="",style="solid", color="black", weight=3]; 1433 -> 1432[label="",style="dashed", color="red", weight=0]; 1433[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not False))",fontsize=16,color="magenta"];1434[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not (GT == GT)))",fontsize=16,color="black",shape="box"];1434 -> 1571[label="",style="solid", color="black", weight=3]; 3669[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235) (not (compare (Neg (primPlusNat Zero yx2120)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3669 -> 3691[label="",style="solid", color="black", weight=3]; 3680[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3680 -> 3743[label="",style="dashed", color="green", weight=3]; 3681[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3681 -> 3744[label="",style="dashed", color="green", weight=3]; 3682[label="Succ (primPlusNat yx218 yx21200)",fontsize=16,color="green",shape="box"];3682 -> 3745[label="",style="dashed", color="green", weight=3]; 3683[label="yx207",fontsize=16,color="green",shape="box"];3684[label="yx232",fontsize=16,color="green",shape="box"];3685[label="yx218",fontsize=16,color="green",shape="box"];3686[label="yx218",fontsize=16,color="green",shape="box"];3687[label="yx207",fontsize=16,color="green",shape="box"];3688[label="yx232",fontsize=16,color="green",shape="box"];1570[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) True)",fontsize=16,color="black",shape="box"];1570 -> 1781[label="",style="solid", color="black", weight=3]; 1571[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) (not True))",fontsize=16,color="black",shape="box"];1571 -> 1782[label="",style="solid", color="black", weight=3]; 3691[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero yx2120)) (numericEnumFrom $! Neg (primPlusNat Zero yx2120) + yx235) (not (primCmpInt (Neg (primPlusNat Zero yx2120)) yx207 == GT)))",fontsize=16,color="burlywood",shape="box"];4776[label="yx2120/Succ yx21200",fontsize=10,color="white",style="solid",shape="box"];3691 -> 4776[label="",style="solid", color="burlywood", weight=9]; 4776 -> 3749[label="",style="solid", color="burlywood", weight=3]; 4777[label="yx2120/Zero",fontsize=10,color="white",style="solid",shape="box"];3691 -> 4777[label="",style="solid", color="burlywood", weight=9]; 4777 -> 3750[label="",style="solid", color="burlywood", weight=3]; 3744 -> 3743[label="",style="dashed", color="red", weight=0]; 3744[label="primPlusNat yx218 yx21200",fontsize=16,color="magenta"];3745 -> 3743[label="",style="dashed", color="red", weight=0]; 3745[label="primPlusNat yx218 yx21200",fontsize=16,color="magenta"];1781[label="map toEnum (Neg Zero : takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="black",shape="box"];1781 -> 2343[label="",style="solid", color="black", weight=3]; 1782[label="map toEnum (takeWhile1 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) False)",fontsize=16,color="black",shape="box"];1782 -> 2344[label="",style="solid", color="black", weight=3]; 3749[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero (Succ yx21200))) (numericEnumFrom $! Neg (primPlusNat Zero (Succ yx21200)) + yx235) (not (primCmpInt (Neg (primPlusNat Zero (Succ yx21200))) yx207 == GT)))",fontsize=16,color="black",shape="box"];3749 -> 4019[label="",style="solid", color="black", weight=3]; 3750[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (primPlusNat Zero Zero)) (numericEnumFrom $! Neg (primPlusNat Zero Zero) + yx235) (not (primCmpInt (Neg (primPlusNat Zero Zero)) yx207 == GT)))",fontsize=16,color="black",shape="box"];3750 -> 4020[label="",style="solid", color="black", weight=3]; 2343[label="toEnum (Neg Zero) : map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="green",shape="box"];2343 -> 2379[label="",style="dashed", color="green", weight=3]; 2343 -> 2380[label="",style="dashed", color="green", weight=3]; 2344[label="map toEnum (takeWhile0 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) otherwise)",fontsize=16,color="black",shape="box"];2344 -> 2381[label="",style="solid", color="black", weight=3]; 4019 -> 2772[label="",style="dashed", color="red", weight=0]; 4019[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg (Succ yx21200)) (numericEnumFrom $! Neg (Succ yx21200) + yx235) (not (primCmpInt (Neg (Succ yx21200)) yx207 == GT)))",fontsize=16,color="magenta"];4019 -> 4033[label="",style="dashed", color="magenta", weight=3]; 4019 -> 4034[label="",style="dashed", color="magenta", weight=3]; 4019 -> 4035[label="",style="dashed", color="magenta", weight=3]; 4019 -> 4036[label="",style="dashed", color="magenta", weight=3]; 4019 -> 4037[label="",style="dashed", color="magenta", weight=3]; 4020 -> 2421[label="",style="dashed", color="red", weight=0]; 4020[label="map toEnum (takeWhile1 (flip (<=) yx207) (Neg Zero) (numericEnumFrom $! Neg Zero + yx235) (not (primCmpInt (Neg Zero) yx207 == GT)))",fontsize=16,color="magenta"];4020 -> 4038[label="",style="dashed", color="magenta", weight=3]; 4020 -> 4039[label="",style="dashed", color="magenta", weight=3]; 2379[label="toEnum (Neg Zero)",fontsize=16,color="black",shape="box"];2379 -> 2415[label="",style="solid", color="black", weight=3]; 2380[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom $! Neg Zero + yx57))",fontsize=16,color="black",shape="box"];2380 -> 2416[label="",style="solid", color="black", weight=3]; 2381[label="map toEnum (takeWhile0 (flip (<=) yx26) (Neg Zero) (numericEnumFrom $! Neg Zero + yx57) True)",fontsize=16,color="black",shape="box"];2381 -> 2417[label="",style="solid", color="black", weight=3]; 4033[label="yx21200",fontsize=16,color="green",shape="box"];4034[label="yx21200",fontsize=16,color="green",shape="box"];4035[label="yx21200",fontsize=16,color="green",shape="box"];4036[label="yx207",fontsize=16,color="green",shape="box"];4037[label="yx235",fontsize=16,color="green",shape="box"];4038[label="yx207",fontsize=16,color="green",shape="box"];4039[label="yx235",fontsize=16,color="green",shape="box"];2415[label="fromInt (Neg Zero)",fontsize=16,color="black",shape="box"];2415 -> 3044[label="",style="solid", color="black", weight=3]; 2416[label="map toEnum (takeWhile (flip (<=) yx26) (Neg Zero + yx57 `seq` numericEnumFrom (Neg Zero + yx57)))",fontsize=16,color="black",shape="box"];2416 -> 3045[label="",style="solid", color="black", weight=3]; 2417 -> 2375[label="",style="dashed", color="red", weight=0]; 2417[label="map toEnum []",fontsize=16,color="magenta"];3044[label="intToRatio (Neg Zero)",fontsize=16,color="black",shape="box"];3044 -> 3079[label="",style="solid", color="black", weight=3]; 3045[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Neg Zero + yx57)) (numericEnumFrom (Neg Zero + yx57))))",fontsize=16,color="black",shape="box"];3045 -> 3080[label="",style="solid", color="black", weight=3]; 2375[label="map toEnum []",fontsize=16,color="black",shape="triangle"];2375 -> 2412[label="",style="solid", color="black", weight=3]; 3079[label="fromInt (Neg Zero) :% fromInt (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];3079 -> 3108[label="",style="dashed", color="green", weight=3]; 3079 -> 3109[label="",style="dashed", color="green", weight=3]; 3080[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) yx57)) (numericEnumFrom (primPlusInt (Neg Zero) yx57))))",fontsize=16,color="burlywood",shape="box"];4778[label="yx57/Pos yx570",fontsize=10,color="white",style="solid",shape="box"];3080 -> 4778[label="",style="solid", color="burlywood", weight=9]; 4778 -> 3110[label="",style="solid", color="burlywood", weight=3]; 4779[label="yx57/Neg yx570",fontsize=10,color="white",style="solid",shape="box"];3080 -> 4779[label="",style="solid", color="burlywood", weight=9]; 4779 -> 3111[label="",style="solid", color="burlywood", weight=3]; 2412[label="[]",fontsize=16,color="green",shape="box"];3108 -> 576[label="",style="dashed", color="red", weight=0]; 3108[label="fromInt (Neg Zero)",fontsize=16,color="magenta"];3108 -> 3142[label="",style="dashed", color="magenta", weight=3]; 3109 -> 576[label="",style="dashed", color="red", weight=0]; 3109[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3109 -> 3143[label="",style="dashed", color="magenta", weight=3]; 3110[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Pos yx570))) (numericEnumFrom (primPlusInt (Neg Zero) (Pos yx570)))))",fontsize=16,color="black",shape="box"];3110 -> 3144[label="",style="solid", color="black", weight=3]; 3111[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primPlusInt (Neg Zero) (Neg yx570))) (numericEnumFrom (primPlusInt (Neg Zero) (Neg yx570)))))",fontsize=16,color="black",shape="box"];3111 -> 3145[label="",style="solid", color="black", weight=3]; 3142[label="Neg Zero",fontsize=16,color="green",shape="box"];3143[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3144[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat yx570 Zero)) (numericEnumFrom (primMinusNat yx570 Zero))))",fontsize=16,color="burlywood",shape="box"];4780[label="yx570/Succ yx5700",fontsize=10,color="white",style="solid",shape="box"];3144 -> 4780[label="",style="solid", color="burlywood", weight=9]; 4780 -> 3199[label="",style="solid", color="burlywood", weight=3]; 4781[label="yx570/Zero",fontsize=10,color="white",style="solid",shape="box"];3144 -> 4781[label="",style="solid", color="burlywood", weight=9]; 4781 -> 3200[label="",style="solid", color="burlywood", weight=3]; 3145[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Neg (primPlusNat Zero yx570))) (numericEnumFrom (Neg (primPlusNat Zero yx570)))))",fontsize=16,color="black",shape="box"];3145 -> 3201[label="",style="solid", color="black", weight=3]; 3199[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat (Succ yx5700) Zero)) (numericEnumFrom (primMinusNat (Succ yx5700) Zero))))",fontsize=16,color="black",shape="box"];3199 -> 3232[label="",style="solid", color="black", weight=3]; 3200[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (primMinusNat Zero Zero)) (numericEnumFrom (primMinusNat Zero Zero))))",fontsize=16,color="black",shape="box"];3200 -> 3233[label="",style="solid", color="black", weight=3]; 3201[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom (Neg (primPlusNat Zero yx570))))",fontsize=16,color="black",shape="box"];3201 -> 3234[label="",style="solid", color="black", weight=3]; 3232[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Pos (Succ yx5700))) (numericEnumFrom (Pos (Succ yx5700)))))",fontsize=16,color="black",shape="box"];3232 -> 3435[label="",style="solid", color="black", weight=3]; 3233 -> 3231[label="",style="dashed", color="red", weight=0]; 3233[label="map toEnum (takeWhile (flip (<=) yx26) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="magenta"];3233 -> 3436[label="",style="dashed", color="magenta", weight=3]; 3234 -> 3577[label="",style="dashed", color="red", weight=0]; 3234[label="map toEnum (takeWhile (flip (<=) yx26) (Neg (primPlusNat Zero yx570) : (numericEnumFrom $! Neg (primPlusNat Zero yx570) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3234 -> 3579[label="",style="dashed", color="magenta", weight=3]; 3234 -> 3580[label="",style="dashed", color="magenta", weight=3]; 3234 -> 3581[label="",style="dashed", color="magenta", weight=3]; 3435[label="map toEnum (takeWhile (flip (<=) yx26) (numericEnumFrom (Pos (Succ yx5700))))",fontsize=16,color="black",shape="box"];3435 -> 3510[label="",style="solid", color="black", weight=3]; 3436[label="yx26",fontsize=16,color="green",shape="box"];3231[label="map toEnum (takeWhile (flip (<=) yx18) (enforceWHNF (WHNF (Pos Zero)) (numericEnumFrom (Pos Zero))))",fontsize=16,color="black",shape="triangle"];3231 -> 3434[label="",style="solid", color="black", weight=3]; 3579 -> 576[label="",style="dashed", color="red", weight=0]; 3579[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3579 -> 3603[label="",style="dashed", color="magenta", weight=3]; 3580[label="yx26",fontsize=16,color="green",shape="box"];3581[label="yx570",fontsize=16,color="green",shape="box"];3510 -> 3609[label="",style="dashed", color="red", weight=0]; 3510[label="map toEnum (takeWhile (flip (<=) yx26) (Pos (Succ yx5700) : (numericEnumFrom $! Pos (Succ yx5700) + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3510 -> 3611[label="",style="dashed", color="magenta", weight=3]; 3510 -> 3612[label="",style="dashed", color="magenta", weight=3]; 3510 -> 3613[label="",style="dashed", color="magenta", weight=3]; 3434[label="map toEnum (takeWhile (flip (<=) yx18) (numericEnumFrom (Pos Zero)))",fontsize=16,color="black",shape="box"];3434 -> 3509[label="",style="solid", color="black", weight=3]; 3603[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3611 -> 576[label="",style="dashed", color="red", weight=0]; 3611[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3611 -> 3640[label="",style="dashed", color="magenta", weight=3]; 3612[label="yx26",fontsize=16,color="green",shape="box"];3613[label="yx5700",fontsize=16,color="green",shape="box"];3509 -> 3614[label="",style="dashed", color="red", weight=0]; 3509[label="map toEnum (takeWhile (flip (<=) yx18) (Pos Zero : (numericEnumFrom $! Pos Zero + fromInt (Pos (Succ Zero)))))",fontsize=16,color="magenta"];3509 -> 3616[label="",style="dashed", color="magenta", weight=3]; 3509 -> 3617[label="",style="dashed", color="magenta", weight=3]; 3640[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];3616[label="yx18",fontsize=16,color="green",shape="box"];3617 -> 576[label="",style="dashed", color="red", weight=0]; 3617[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];3617 -> 3639[label="",style="dashed", color="magenta", weight=3]; 3639[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];} ---------------------------------------- (14) Complex Obligation (AND) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) new_primDivNatS00(yx245, yx246, Succ(yx2470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) new_primDivNatS01(yx245, yx246) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS0(Zero, Zero) -> new_primDivNatS(new_primMinusNatS1, Zero) new_primDivNatS00(yx245, yx246, Succ(yx2470), Succ(yx2480)) -> new_primDivNatS00(yx245, yx246, yx2470, yx2480) new_primDivNatS00(yx245, yx246, Zero, Zero) -> new_primDivNatS01(yx245, yx246) The TRS R consists of the following rules: new_primMinusNatS1 -> Zero new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero new_primMinusNatS0(yx30000) -> Succ(yx30000) The set Q consists of the following terms: new_primMinusNatS0(x0) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS1 new_primMinusNatS2(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS00(yx245, yx246, Succ(yx2470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) new_primDivNatS00(yx245, yx246, Succ(yx2470), Succ(yx2480)) -> new_primDivNatS00(yx245, yx246, yx2470, yx2480) new_primDivNatS00(yx245, yx246, Zero, Zero) -> new_primDivNatS01(yx245, yx246) new_primDivNatS01(yx245, yx246) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) The TRS R consists of the following rules: new_primMinusNatS1 -> Zero new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero new_primMinusNatS0(yx30000) -> Succ(yx30000) The set Q consists of the following terms: new_primMinusNatS0(x0) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS1 new_primMinusNatS2(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_primDivNatS(Succ(yx4000), yx4100) -> new_primDivNatS0(yx4000, yx4100) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(new_primDivNatS(x_1, x_2)) = x_1 POL(new_primDivNatS0(x_1, x_2)) = x_1 POL(new_primDivNatS00(x_1, x_2, x_3, x_4)) = 1 + x_1 POL(new_primDivNatS01(x_1, x_2)) = 1 + x_1 POL(new_primMinusNatS0(x_1)) = 1 + x_1 POL(new_primMinusNatS2(x_1, x_2)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) new_primMinusNatS0(yx30000) -> Succ(yx30000) new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS00(yx245, yx246, Succ(yx2470), Zero) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS0(Succ(yx40000), Succ(yx41000)) -> new_primDivNatS00(yx40000, yx41000, yx40000, yx41000) new_primDivNatS00(yx245, yx246, Succ(yx2470), Succ(yx2480)) -> new_primDivNatS00(yx245, yx246, yx2470, yx2480) new_primDivNatS00(yx245, yx246, Zero, Zero) -> new_primDivNatS01(yx245, yx246) new_primDivNatS01(yx245, yx246) -> new_primDivNatS(new_primMinusNatS2(Succ(yx245), Succ(yx246)), Succ(yx246)) new_primDivNatS0(Succ(yx40000), Zero) -> new_primDivNatS(new_primMinusNatS0(yx40000), Zero) The TRS R consists of the following rules: new_primMinusNatS1 -> Zero new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero new_primMinusNatS0(yx30000) -> Succ(yx30000) The set Q consists of the following terms: new_primMinusNatS0(x0) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS1 new_primMinusNatS2(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS00(yx245, yx246, Succ(yx2470), Succ(yx2480)) -> new_primDivNatS00(yx245, yx246, yx2470, yx2480) The TRS R consists of the following rules: new_primMinusNatS1 -> Zero new_primMinusNatS2(Succ(yx2000), Zero) -> Succ(yx2000) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS2(yx2000, yx2010) new_primMinusNatS2(Zero, Succ(yx2010)) -> Zero new_primMinusNatS0(yx30000) -> Succ(yx30000) The set Q consists of the following terms: new_primMinusNatS0(x0) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS1 new_primMinusNatS2(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_primDivNatS00(yx245, yx246, Succ(yx2470), Succ(yx2480)) -> new_primDivNatS00(yx245, yx246, yx2470, yx2480) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) The TRS R consists of the following rules: new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_fromInt(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: (new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h),new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h)) ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) The TRS R consists of the following rules: new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_fromInt(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: (new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h),new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h)) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) The TRS R consists of the following rules: new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_fromInt(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) R is empty. The set Q consists of the following terms: new_fromInt(x0) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_fromInt(x0) ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map21(yx207, Zero, yx235, h) -> new_map20(yx207, yx235, yx207, h) we obtained the following new rules [LPAR04]: (new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2),new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2)) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(yx207, Neg(yx2120), Pos(Succ(yx21300)), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map20(yx207, yx212, Pos(Zero), h) -> new_map22(yx207, yx212, h) we obtained the following new rules [LPAR04]: (new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1),new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1)) ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map20(Pos(Zero), Pos(Succ(Zero)), Pos(Zero), z1) -> new_map22(Pos(Zero), Pos(Succ(Zero)), z1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map20(yx207, yx212, Neg(Zero), h) -> new_map23(yx207, yx212, h) we obtained the following new rules [LPAR04]: (new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1),new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1)) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map23(yx207, yx212, h) -> new_map22(yx207, yx212, h) we obtained the following new rules [LPAR04]: (new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0),new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0)) ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: new_map21(z0, Zero, Pos(Succ(Zero)), z2) -> new_map20(z0, Pos(Succ(Zero)), z0, z2) new_map22(yx207, Neg(yx2120), h) -> new_map21(yx207, yx2120, Pos(Succ(Zero)), h) new_map20(Neg(Zero), Pos(Succ(Zero)), Neg(Zero), z1) -> new_map23(Neg(Zero), Pos(Succ(Zero)), z1) new_map23(Neg(Zero), Pos(Succ(Zero)), z0) -> new_map22(Neg(Zero), Pos(Succ(Zero)), z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes. ---------------------------------------- (46) TRUE ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: new_map24(yx207, yx208, yx209, Succ(yx2100), Succ(yx2110), yx212, yx213, h) -> new_map24(yx207, yx208, yx209, yx2100, yx2110, yx212, yx213, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_map24(yx207, yx208, yx209, Succ(yx2100), Succ(yx2110), yx212, yx213, h) -> new_map24(yx207, yx208, yx209, yx2100, yx2110, yx212, yx213, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5, 6 >= 6, 7 >= 7, 8 >= 8 ---------------------------------------- (49) YES ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map11(yx207, yx212000, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map18(yx207, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) ---------------------------------------- (54) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (55) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: (new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, new_fromInt(Pos(Succ(Zero))), h) at position [2] we obtained the following new rules [LPAR04]: (new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h),new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h)) ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map6(yx255, yx254, ba) -> new_map19(yx254, new_fromInt(Pos(Succ(Zero))), ba) at position [1] we obtained the following new rules [LPAR04]: (new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba),new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba)) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map5(yx253, yx252, ba) -> new_map7(yx252, new_fromInt(Pos(Succ(Zero))), ba) at position [1] we obtained the following new rules [LPAR04]: (new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba),new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba)) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: (new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: (new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: (new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: (new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, new_fromInt(Pos(Succ(Zero))), bb) at position [3] we obtained the following new rules [LPAR04]: (new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb)) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb),new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb)) ---------------------------------------- (84) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (85) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, new_fromInt(Pos(Succ(Zero))), bb) at position [1] we obtained the following new rules [LPAR04]: (new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb)) ---------------------------------------- (86) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (87) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, new_fromInt(Pos(Succ(Zero))), bb) at position [2] we obtained the following new rules [LPAR04]: (new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb),new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb)) ---------------------------------------- (88) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero new_fromInt(yx8) -> yx8 The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (89) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (90) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_fromInt(x0) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (91) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_fromInt(x0) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map12(yx207, yx218, yx234, bb) -> new_map13(yx207, yx218, yx218, yx234, yx218, yx207, bb) we obtained the following new rules [LPAR04]: (new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2),new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2)) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Zero)), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4),new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4)) (new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3)) (new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) ---------------------------------------- (96) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (97) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map8(yx207, yx212000, yx238, bb) -> new_map4(yx207, Succ(yx212000), Succ(yx212000), yx238, Succ(yx212000), bb) we obtained the following new rules [LPAR04]: (new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3),new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3)) ---------------------------------------- (98) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (99) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Succ(yx2510), ba) -> new_map(yx19900, yx249, yx250, yx237, yx2510, yx19900, ba) we obtained the following new rules [LPAR04]: (new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3),new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3)) (new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2),new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2)) ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map10(yx207, yx239, bb) -> new_map4(yx207, Zero, Zero, yx239, Zero, bb) we obtained the following new rules [LPAR04]: (new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1),new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1)) ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (103) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Succ(yx19900)), yx249, yx250, yx237, Zero, ba) -> new_map(yx19900, yx249, yx250, yx237, Zero, Succ(yx19900), ba) we obtained the following new rules [LPAR04]: (new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3),new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3)) (new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1),new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1)) ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map3(yx272, yx278, yx280, h) -> new_map4(Pos(Succ(yx272)), yx278, yx278, yx280, yx278, h) we obtained the following new rules [LPAR04]: (new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3),new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3)) ---------------------------------------- (106) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (107) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Pos(Zero), yx250, yx2370, ba) we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2),new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2)) ---------------------------------------- (108) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (109) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Neg(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map6(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2),new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2)) (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (110) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (111) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1)) ---------------------------------------- (114) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (115) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map19(yx254, yx259, ba) -> new_map4(Neg(Zero), yx254, yx254, yx259, yx254, ba) we obtained the following new rules [LPAR04]: (new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2),new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2)) ---------------------------------------- (116) Obligation: Q DP problem: The TRS P consists of the following rules: new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map4(Neg(Zero), yx249, yx250, Neg(yx2370), Zero, ba) -> new_map1(Neg(Zero), yx250, yx2370, ba) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (117) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (118) Complex Obligation (AND) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (121) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (122) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map6(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (123) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (124) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (125) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (126) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (127) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (128) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: (new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1),new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1)) ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (131) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) R is empty. The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (132) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) ---------------------------------------- (133) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (134) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map6(yx255, yx254, ba) -> new_map19(yx254, Pos(Succ(Zero)), ba) we obtained the following new rules [LPAR04]: (new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0),new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0)) ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map19(z1, Pos(Succ(Zero)), z2) -> new_map4(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) we obtained the following new rules [LPAR04]: (new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0),new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0)) ---------------------------------------- (137) Obligation: Q DP problem: The TRS P consists of the following rules: new_map4(Neg(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map6(Succ(Zero), Succ(Zero), z1) new_map6(Succ(Zero), Succ(Zero), z0) -> new_map19(Succ(Zero), Pos(Succ(Zero)), z0) new_map19(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Neg(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (138) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (139) TRUE ---------------------------------------- (140) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (141) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map9(yx207, yx2180, bb) we obtained the following new rules [LPAR04]: (new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2),new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2)) (new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4),new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4)) ---------------------------------------- (142) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (143) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Succ(yx2190), Neg(Succ(Succ(yx213000))), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) (new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4),new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4)) (new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3)) ---------------------------------------- (144) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (145) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Succ(z0)), Succ(x4), Succ(x4), z2, Succ(x4), z3) -> new_map(z0, Succ(x4), Succ(x4), z2, x4, z0, z3) we obtained the following new rules [LPAR04]: (new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2),new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2)) ---------------------------------------- (146) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (147) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Zero), yx249, yx250, Pos(yx2370), Zero, ba) -> new_map5(new_primPlusNat0(yx250, yx2370), new_primPlusNat0(yx250, yx2370), ba) we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) (new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2),new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2)) ---------------------------------------- (148) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (149) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (150) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (151) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1)) ---------------------------------------- (152) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (153) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map7(yx252, yx258, ba) -> new_map4(Pos(Zero), yx252, yx252, yx258, yx252, ba) we obtained the following new rules [LPAR04]: (new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2),new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2)) ---------------------------------------- (154) Obligation: Q DP problem: The TRS P consists of the following rules: new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Neg(x2), Zero, z2) -> new_map1(Pos(Zero), Zero, x2, z2) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (155) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (156) Complex Obligation (AND) ---------------------------------------- (157) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (158) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (159) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (160) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(x2), Zero, z2) -> new_map5(new_primPlusNat0(Zero, x2), new_primPlusNat0(Zero, x2), z2) we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (161) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (162) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (163) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (164) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(new_primPlusNat0(Zero, Succ(Zero)), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [0] we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1)) ---------------------------------------- (165) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (166) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), new_primPlusNat0(Zero, Succ(Zero)), z1) at position [1] we obtained the following new rules [LPAR04]: (new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1),new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1)) ---------------------------------------- (167) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) The TRS R consists of the following rules: new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (168) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (169) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) R is empty. The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (170) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) ---------------------------------------- (171) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (172) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map5(yx253, yx252, ba) -> new_map7(yx252, Pos(Succ(Zero)), ba) we obtained the following new rules [LPAR04]: (new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0),new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0)) ---------------------------------------- (173) Obligation: Q DP problem: The TRS P consists of the following rules: new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (174) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map7(z1, Pos(Succ(Zero)), z2) -> new_map4(Pos(Zero), z1, z1, Pos(Succ(Zero)), z1, z2) we obtained the following new rules [LPAR04]: (new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0),new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0)) ---------------------------------------- (175) Obligation: Q DP problem: The TRS P consists of the following rules: new_map4(Pos(Zero), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map5(Succ(Zero), Succ(Zero), z1) new_map5(Succ(Zero), Succ(Zero), z0) -> new_map7(Succ(Zero), Pos(Succ(Zero)), z0) new_map7(Succ(Zero), Pos(Succ(Zero)), z0) -> new_map4(Pos(Zero), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (176) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (177) TRUE ---------------------------------------- (178) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (179) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, yx219, Neg(Zero), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: (new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2),new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2)) (new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4),new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4)) (new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3),new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3)) ---------------------------------------- (180) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (181) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map4(Pos(Succ(z0)), Zero, Zero, z2, Zero, z3) -> new_map(z0, Zero, Zero, z2, Zero, Succ(z0), z3) we obtained the following new rules [LPAR04]: (new_map4(Pos(Succ(z0)), Zero, Zero, Pos(Succ(Zero)), Zero, z2) -> new_map(z0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(z0), z2),new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1)) ---------------------------------------- (182) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (183) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map14(yx207, yx218, Succ(yx21200), yx232, bb) -> new_map13(yx207, Succ(new_primPlusNat0(yx218, yx21200)), Succ(new_primPlusNat0(yx218, yx21200)), yx232, Succ(new_primPlusNat0(yx218, yx21200)), yx207, bb) we obtained the following new rules [LPAR04]: (new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4),new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4)) ---------------------------------------- (184) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (185) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map1(yx207, yx212000, yx2180, bb) we obtained the following new rules [LPAR04]: (new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3),new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3)) ---------------------------------------- (186) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (187) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, Pos(Zero), yx219, Pos(yx2130), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: (new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3),new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3)) ---------------------------------------- (188) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (189) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, Neg(yx2120), yx219, Pos(yx2130), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: (new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3),new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3)) ---------------------------------------- (190) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (191) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map14(yx207, yx218, Zero, yx232, bb) -> new_map13(yx207, yx218, yx218, yx232, yx218, yx207, bb) we obtained the following new rules [LPAR04]: (new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4),new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4)) (new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3),new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3)) ---------------------------------------- (192) Obligation: Q DP problem: The TRS P consists of the following rules: new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map13(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), yx219, Pos(yx2130), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Pos(x5), Succ(x2), Succ(x2), Pos(Succ(Succ(x3))), Succ(x2), Pos(x5), z3) -> new_map1(Pos(x5), x3, x2, z3) new_map13(Pos(x4), z1, z1, Pos(Zero), z1, Pos(x4), z3) -> new_map12(Pos(x4), z1, Pos(Succ(Zero)), z3) new_map13(Pos(x5), z1, z1, Neg(x3), z1, Pos(x5), z3) -> new_map14(Pos(x5), z1, x3, Pos(Succ(Zero)), z3) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (193) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. ---------------------------------------- (194) Obligation: Q DP problem: The TRS P consists of the following rules: new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (195) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, yx218, yx212, Zero, Neg(Succ(Zero)), bb) -> new_map17(yx207, yx217, yx218, yx212, bb) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2)) ---------------------------------------- (196) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (197) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(yx207, yx217, Zero, Pos(Succ(Zero)), yx219, Pos(yx2130), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) we obtained the following new rules [LPAR04]: (new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2),new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2)) ---------------------------------------- (198) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (199) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Zero)), z4) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), z3, z4) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) (new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3),new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3)) ---------------------------------------- (200) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (201) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), z2, z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) ---------------------------------------- (202) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (203) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Succ(Succ(x5))), z4) -> new_map15(Neg(Succ(Succ(x5))), Succ(y_0), Succ(y_1), z3, x5, y_2, z4) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) (new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3),new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3)) ---------------------------------------- (204) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (205) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, Succ(x4), Neg(Succ(Succ(x5))), z3) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), z2, x5, x4, z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) ---------------------------------------- (206) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (207) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Zero), Succ(y_0), Succ(y_1), z3, Succ(y_2), Neg(Zero), z4) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), z3, z4) we obtained the following new rules [LPAR04]: (new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) (new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3),new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3)) ---------------------------------------- (208) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (209) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Zero), z1, z1, z2, z1, Neg(Zero), z3) -> new_map16(Neg(Zero), z1, z1, z2, z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2),new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2)) (new_map13(Neg(Zero), Succ(y_0), Succ(y_0), Pos(Succ(Zero)), Succ(y_0), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_0), Pos(Succ(Zero)), z3),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) ---------------------------------------- (210) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (211) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_map(yx272, yx273, yx274, Neg(yx2750), Zero, Succ(yx2770), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map17(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 1 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 0 POL(Zero) = 0 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map0(x_1, x_2, x_3, x_4)) = 0 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 POL(new_map10(x_1, x_2, x_3)) = 0 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(new_map14(x_1, x_2, x_3, x_4, x_5)) = 0 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = 0 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = 0 POL(new_map3(x_1, x_2, x_3, x_4)) = x_3 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(new_map8(x_1, x_2, x_3, x_4)) = 0 POL(new_map9(x_1, x_2, x_3)) = 0 POL(new_primPlusNat0(x_1, x_2)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (212) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (213) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_map2(yx272, yx273, yx274, Neg(yx2750), h) -> new_map1(Pos(Succ(yx272)), yx274, yx2750, h) new_map16(yx207, yx217, yx218, Neg(yx2120), bb) -> new_map14(yx207, yx218, yx2120, Pos(Succ(Zero)), bb) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 1 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 0 POL(Zero) = 0 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map0(x_1, x_2, x_3, x_4)) = 0 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 POL(new_map10(x_1, x_2, x_3)) = 0 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(new_map14(x_1, x_2, x_3, x_4, x_5)) = 0 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = 0 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map3(x_1, x_2, x_3, x_4)) = x_3 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = 0 POL(new_map8(x_1, x_2, x_3, x_4)) = 0 POL(new_map9(x_1, x_2, x_3)) = 0 POL(new_primPlusNat0(x_1, x_2)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (214) Obligation: Q DP problem: The TRS P consists of the following rules: new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map10(z0, Pos(Succ(Zero)), z1) -> new_map4(z0, Zero, Zero, Pos(Succ(Zero)), Zero, z1) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Zero, bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map8(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(z0, Succ(z2), Succ(z2), Pos(Succ(Zero)), Succ(z2), z3) new_map1(yx207, Zero, Zero, bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map14(z0, z2, Succ(x2), Pos(Succ(Zero)), z4) -> new_map13(z0, Succ(new_primPlusNat0(z2, x2)), Succ(new_primPlusNat0(z2, x2)), Pos(Succ(Zero)), Succ(new_primPlusNat0(z2, x2)), z0, z4) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map16(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, Zero, Pos(Succ(Succ(yx212000))), bb) -> new_map8(yx207, yx212000, Pos(Succ(Zero)), bb) new_map14(z0, z2, Zero, Pos(Succ(Zero)), z4) -> new_map13(z0, z2, z2, Pos(Succ(Zero)), z2, z0, z4) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map14(Pos(z0), z1, Zero, Pos(Succ(Zero)), z3) -> new_map13(Pos(z0), z1, z1, Pos(Succ(Zero)), z1, Pos(z0), z3) new_map17(yx207, yx217, Zero, Pos(Succ(Zero)), bb) -> new_map10(yx207, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), Zero, Neg(Succ(Zero)), z2) -> new_map17(Neg(Succ(Zero)), Zero, Zero, Pos(Succ(Zero)), z2) new_map13(Pos(x3), Zero, Zero, Pos(Succ(Zero)), Zero, Pos(x3), z2) -> new_map10(Pos(x3), Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (215) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 13 less nodes. ---------------------------------------- (216) Complex Obligation (AND) ---------------------------------------- (217) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (218) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_map4(Pos(Succ(x0)), Zero, Zero, Pos(Succ(Zero)), Zero, z1) -> new_map(x0, Zero, Zero, Pos(Succ(Zero)), Zero, Succ(x0), z1) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Pos(x_1)) = x_1 POL(Succ(x_1)) = 0 POL(Zero) = 1 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map0(x_1, x_2, x_3, x_4)) = x_3 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map3(x_1, x_2, x_3, x_4)) = x_2 + x_3 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = x_5 POL(new_primPlusNat0(x_1, x_2)) = x_2 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero ---------------------------------------- (219) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (220) QDPPairToRuleProof (EQUIVALENT) The dependency pair new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) was transformed to the following new rules: anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) the following new pairs maintain the fan-in: new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following new pairs maintain the fan-out: H(yx272, yx273, yx274, Pos(yx2750), h, cons_new_map(Zero, Succ(yx2770))) -> new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) ---------------------------------------- (221) Complex Obligation (AND) ---------------------------------------- (222) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, Pos(yx2750), h, cons_new_map(Zero, Succ(yx2770))) -> new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (223) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule H(yx272, yx273, yx274, Pos(yx2750), h, cons_new_map(Zero, Succ(yx2770))) -> new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) we obtained the following new rules [LPAR04]: (H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2),H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2)) ---------------------------------------- (224) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (225) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map(yx272, yx273, yx274, Pos(yx2750), Zero, Succ(yx2770), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) we obtained the following new rules [LPAR04]: (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) ---------------------------------------- (226) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (227) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) ---------------------------------------- (228) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (229) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) ---------------------------------------- (230) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (231) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [1,0,0] we obtained the following new rules [LPAR04]: (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2)) ---------------------------------------- (232) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (233) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2)) ---------------------------------------- (234) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (235) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2)) ---------------------------------------- (236) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (237) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z2) at position [2,0,0] we obtained the following new rules [LPAR04]: (new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2),new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2)) ---------------------------------------- (238) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (239) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule H(yx272, yx273, yx274, yx275, h, cons_new_map(Zero, Zero)) -> new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) we obtained the following new rules [LPAR04]: (H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2),H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2)) ---------------------------------------- (240) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (241) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map(yx272, yx273, yx274, yx275, Zero, Zero, h) -> new_map2(yx272, yx273, yx274, yx275, h) we obtained the following new rules [LPAR04]: (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2),new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2)) (new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2),new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2)) ---------------------------------------- (242) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (243) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map2(yx272, yx273, yx274, Pos(yx2750), h) -> new_map0(yx272, new_primPlusNat0(yx274, yx2750), new_primPlusNat0(yx274, yx2750), h) we obtained the following new rules [LPAR04]: (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) ---------------------------------------- (244) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (245) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, new_primPlusNat0(Succ(Zero), Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [1] we obtained the following new rules [LPAR04]: (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) ---------------------------------------- (246) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (247) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, new_primPlusNat0(Succ(z1), Succ(Zero)), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [1] we obtained the following new rules [LPAR04]: (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2)) ---------------------------------------- (248) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (249) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(new_primPlusNat0(Zero, Zero))), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [1,0,0] we obtained the following new rules [LPAR04]: (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0)) ---------------------------------------- (250) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (251) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), new_primPlusNat0(Succ(z1), Succ(Zero)), z2) at position [2] we obtained the following new rules [LPAR04]: (new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2),new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2)) ---------------------------------------- (252) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (253) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), new_primPlusNat0(Succ(Zero), Succ(Zero)), z0) at position [2] we obtained the following new rules [LPAR04]: (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0)) ---------------------------------------- (254) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (255) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (256) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (257) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(new_primPlusNat0(Zero, Zero))), z0) at position [2,0,0] we obtained the following new rules [LPAR04]: (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0),new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0)) ---------------------------------------- (258) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (259) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) the following chains were created: *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: *We consider the chain new_map3(x57, x58, Pos(Succ(Zero)), x59) -> new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59), new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) -> new_map(x60, Succ(x61), Succ(x61), Pos(Succ(Zero)), x61, x60, x62) which results in the following constraint: (1) (new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)=new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) ==> new_map3(x57, x58, Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) the following chains were created: *We consider the chain new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110) -> new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110), new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) -> new_map0(x111, Succ(Succ(new_primPlusNat0(x112, Zero))), Succ(Succ(new_primPlusNat0(x112, Zero))), x114) which results in the following constraint: (1) (new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)=new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) ==> new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110)_>=_new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) *We consider the chain new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117) -> new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117), new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) -> new_map0(Succ(x118), Succ(Succ(Zero)), Succ(Succ(Zero)), x119) which results in the following constraint: (1) (new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)=new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) ==> new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117)_>=_new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) *We consider the chain new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125) -> new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x126) which results in the following constraint: (1) (new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) ==> new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125)_>=_new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) *We consider the chain new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129) -> new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129), new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) -> new_map2(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), x132) which results in the following constraint: (1) (new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)=new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) ==> new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129)_>=_new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *We consider the chain H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213))) -> new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212), new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) -> new_map0(Succ(x214), Succ(Succ(Zero)), Succ(Succ(Zero)), x215) which results in the following constraint: (1) (new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)=new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) ==> H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) For Pair new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289) -> new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289), new_map0(x290, x291, x292, x293) -> new_map3(x290, x292, Pos(Succ(Zero)), x293) which results in the following constraint: (1) (new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)=new_map0(x290, x291, x292, x293) ==> new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: *We consider the chain H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero)) -> new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x343) which results in the following constraint: (1) (new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) ==> H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) For Pair new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366), new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) -> new_map0(x367, Succ(Succ(new_primPlusNat0(x368, Zero))), Succ(Succ(new_primPlusNat0(x368, Zero))), x369) which results in the following constraint: (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)=new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x371) which results in the following constraint: (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *We consider the chain new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410) -> new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x411) which results in the following constraint: (1) (new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) ==> new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) For Pair new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) the following chains were created: *We consider the chain new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452), new_map0(x453, x454, x455, x456) -> new_map3(x453, x455, Pos(Succ(Zero)), x456) which results in the following constraint: (1) (new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)=new_map0(x453, x454, x455, x456) ==> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) *(new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) *(new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) *(new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *(H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) *new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) *(new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) *(H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) *new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) *new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) *(new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 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. ---------------------------------------- (260) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (261) NonInfProof (EQUIVALENT) The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) the following chains were created: *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: *We consider the chain new_map3(x57, x58, Pos(Succ(Zero)), x59) -> new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59), new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) -> new_map(x60, Succ(x61), Succ(x61), Pos(Succ(Zero)), x61, x60, x62) which results in the following constraint: (1) (new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)=new_map4(Pos(Succ(x60)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x62) ==> new_map3(x57, x58, Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), x58, x58, Pos(Succ(Zero)), x58, x59)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) the following chains were created: *We consider the chain new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110) -> new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110), new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) -> new_map0(x111, Succ(Succ(new_primPlusNat0(x112, Zero))), Succ(Succ(new_primPlusNat0(x112, Zero))), x114) which results in the following constraint: (1) (new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)=new_map(x111, Succ(x112), Succ(x112), Pos(Succ(Zero)), Zero, Succ(x113), x114) ==> new_map4(Pos(Succ(x108)), Succ(x109), Succ(x109), Pos(Succ(Zero)), Succ(x109), x110)_>=_new_map(x108, Succ(x109), Succ(x109), Pos(Succ(Zero)), x109, x108, x110)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) *We consider the chain new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117) -> new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117), new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) -> new_map0(Succ(x118), Succ(Succ(Zero)), Succ(Succ(Zero)), x119) which results in the following constraint: (1) (new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)=new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x119) ==> new_map4(Pos(Succ(x115)), Succ(x116), Succ(x116), Pos(Succ(Zero)), Succ(x116), x117)_>=_new_map(x115, Succ(x116), Succ(x116), Pos(Succ(Zero)), x116, x115, x117)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) *We consider the chain new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125) -> new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x126) which results in the following constraint: (1) (new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x126) ==> new_map4(Pos(Succ(x123)), Succ(x124), Succ(x124), Pos(Succ(Zero)), Succ(x124), x125)_>=_new_map(x123, Succ(x124), Succ(x124), Pos(Succ(Zero)), x124, x123, x125)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) *We consider the chain new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129) -> new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129), new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) -> new_map2(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), x132) which results in the following constraint: (1) (new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)=new_map(x130, Succ(x131), Succ(x131), Pos(Succ(Zero)), Zero, Zero, x132) ==> new_map4(Pos(Succ(x127)), Succ(x128), Succ(x128), Pos(Succ(Zero)), Succ(x128), x129)_>=_new_map(x127, Succ(x128), Succ(x128), Pos(Succ(Zero)), x128, x127, x129)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *We consider the chain H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213))) -> new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212), new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) -> new_map0(Succ(x214), Succ(Succ(Zero)), Succ(Succ(Zero)), x215) which results in the following constraint: (1) (new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)=new_map(Succ(x214), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x214), x215) ==> H(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(x210, Succ(x211), Succ(x211), Pos(Succ(Zero)), Zero, Succ(x213), x212)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) For Pair new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289) -> new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289), new_map0(x290, x291, x292, x293) -> new_map3(x290, x292, Pos(Succ(Zero)), x293) which results in the following constraint: (1) (new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)=new_map0(x290, x291, x292, x293) ==> new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: *We consider the chain H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero)) -> new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342), new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x343) which results in the following constraint: (1) (new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)=new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x343) ==> H(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(x340, Succ(x341), Succ(x341), Pos(Succ(Zero)), Zero, Zero, x342)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) For Pair new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366), new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) -> new_map0(x367, Succ(Succ(new_primPlusNat0(x368, Zero))), Succ(Succ(new_primPlusNat0(x368, Zero))), x369) which results in the following constraint: (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)=new_map2(x367, Succ(x368), Succ(x368), Pos(Succ(Zero)), x369) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) *We consider the chain new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x371) which results in the following constraint: (1) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x371) ==> new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *We consider the chain new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410) -> new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410), new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x411) which results in the following constraint: (1) (new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)=new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x411) ==> new_map(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(x408, Succ(x409), Succ(x409), Pos(Succ(Zero)), x410)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) For Pair new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) the following chains were created: *We consider the chain new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452), new_map0(x453, x454, x455, x456) -> new_map3(x453, x455, Pos(Succ(Zero)), x456) which results in the following constraint: (1) (new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)=new_map0(x453, x454, x455, x456) ==> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) *(new_map3(x57, Succ(x61), Pos(Succ(Zero)), x59)_>=_new_map4(Pos(Succ(x57)), Succ(x61), Succ(x61), Pos(Succ(Zero)), Succ(x61), x59)) *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) *(new_map4(Pos(Succ(Succ(x113))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x110)_>=_new_map(Succ(x113), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x113), x110)) *(new_map4(Pos(Succ(Succ(x118))), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x117)_>=_new_map(Succ(x118), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x118), x117)) *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x125)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x125)) *(new_map4(Pos(Succ(Zero)), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Succ(Zero), x129)_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x129)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *(H(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x212, cons_new_map(Zero, Succ(x213)))_>=_new_map(Succ(x213), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x213), x212)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) *new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) *(new_map(Succ(x288), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x288), x289)_>=_new_map0(Succ(x288), Succ(Succ(Zero)), Succ(Succ(Zero)), x289)) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) *(H(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x342, cons_new_map(Zero, Zero))_>=_new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x342)) *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) *new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x366)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x366)) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x370)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x370)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *(new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, x410)_>=_new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x410)) *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) *new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) *(new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), x452)_>=_new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), x452)) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation [NONINF]: POL(H(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 - x_2 - x_4 - x_6 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(anew_new_map(x_1, x_2)) = 0 POL(c) = -3 POL(cons_new_map(x_1, x_2)) = 0 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 - x_2 - x_4 + x_5 POL(new_map0(x_1, x_2, x_3, x_4)) = -1 - x_3 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = -1 - x_3 - x_4 POL(new_map3(x_1, x_2, x_3, x_4)) = -1 - x_2 - x_3 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 - x_1 - x_2 + x_3 - x_4 - x_5 POL(new_new_map(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = x_1 The following pairs are in P_>: new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) The following pairs are in P_bound: new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map(Succ(x4), Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Succ(x4), z2) -> new_map0(Succ(x4), Succ(Succ(Zero)), Succ(Succ(Zero)), z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z0) -> new_map0(Zero, Succ(Succ(Zero)), Succ(Succ(Zero)), z0) The following rules are usable: new_new_map(yx2760, yx2770) -> anew_new_map(Succ(yx2760), Succ(yx2770)) Succ(yx2180) -> new_primPlusNat0(Succ(yx2180), Zero) Zero -> new_primPlusNat0(Zero, Zero) new_new_map(yx2760, yx2770) -> new_new_map(Succ(yx2760), Succ(yx2770)) cons_new_map(Zero, Succ(yx2770)) -> new_new_map(Zero, Succ(yx2770)) cons_new_map(Zero, Zero) -> new_new_map(Zero, Zero) ---------------------------------------- (262) Complex Obligation (AND) ---------------------------------------- (263) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> new_map(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z1, x0, z2) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(Zero, Succ(Zero), Succ(Zero), Pos(Succ(Zero)), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (264) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 8 less nodes. ---------------------------------------- (265) TRUE ---------------------------------------- (266) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (267) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) the following chains were created: *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 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. ---------------------------------------- (268) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (269) NonInfProof (EQUIVALENT) The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) the following chains were created: *We consider the chain new_map0(x4, x5, x6, x7) -> new_map3(x4, x6, Pos(Succ(Zero)), x7), new_map3(x8, x9, Pos(Succ(Zero)), x10) -> new_map4(Pos(Succ(x8)), x9, x9, Pos(Succ(Zero)), x9, x10) which results in the following constraint: (1) (new_map3(x4, x6, Pos(Succ(Zero)), x7)=new_map3(x8, x9, Pos(Succ(Zero)), x10) ==> new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) For Pair new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) the following chains were created: *We consider the chain new_map3(x63, x64, Pos(Succ(Zero)), x65) -> new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65), new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) -> H(x66, Succ(x67), Succ(x67), Pos(Succ(Zero)), x68, anew_new_map(x67, x66)) which results in the following constraint: (1) (new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)=new_map4(Pos(Succ(x66)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x68) ==> new_map3(x63, x64, Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), x64, x64, Pos(Succ(Zero)), x64, x65)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) For Pair new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) the following chains were created: *We consider the chain new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153) -> H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151)), H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) -> new_map(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), Zero, Succ(x157), x156) which results in the following constraint: (1) (H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))=H(x154, Succ(x155), Succ(x155), Pos(Succ(Zero)), x156, cons_new_map(Zero, Succ(x157))) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(x151)), Succ(x152), Succ(x152), Pos(Succ(Zero)), Succ(x152), x153)_>=_H(x151, Succ(x152), Succ(x152), Pos(Succ(Zero)), x153, anew_new_map(x152, x151))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x152, x151)=cons_new_map(Zero, Succ(x157)) which results in the following new constraint: (3) (new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(x468))), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), Succ(Succ(x469)), x153)_>=_H(Succ(x468), Succ(Succ(x469)), Succ(Succ(x469)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x469), Succ(x468)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x469, x468)=cons_new_map(Zero, Succ(x157)) which results in the following new constraints: (4) (new_new_map(x471, x470)=cons_new_map(Zero, Succ(x157)) & (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) (5) (cons_new_map(Zero, Succ(x474))=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Succ(x157)) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x472,x473:new_new_map(x471, x470)=cons_new_map(Zero, Succ(x472)) ==> new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x473)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x473, anew_new_map(Succ(x471), Succ(x470)))) with sigma = [x472 / x157, x473 / x153] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166) -> H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164)), H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) -> new_map(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), Zero, Zero, x169) which results in the following constraint: (1) (H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))=H(x167, Succ(x168), Succ(x168), Pos(Succ(Zero)), x169, cons_new_map(Zero, Zero)) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_map(x165, x164)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(x164)), Succ(x165), Succ(x165), Pos(Succ(Zero)), Succ(x165), x166)_>=_H(x164, Succ(x165), Succ(x165), Pos(Succ(Zero)), x166, anew_new_map(x165, x164))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_map(x165, x164)=cons_new_map(Zero, Zero) which results in the following new constraint: (3) (new_new_map(x476, x475)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x475))), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), Succ(Succ(x476)), x166)_>=_H(Succ(x475), Succ(Succ(x476)), Succ(Succ(x476)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x476), Succ(x475)))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_map(x476, x475)=cons_new_map(Zero, Zero) which results in the following new constraints: (4) (new_new_map(x478, x477)=cons_new_map(Zero, Zero) & (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) (5) (cons_new_map(Zero, Succ(x480))=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Succ(x480)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Succ(x480)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Succ(x480))))) (6) (cons_new_map(Zero, Zero)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x479:new_new_map(x478, x477)=cons_new_map(Zero, Zero) ==> new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x479)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x479, anew_new_map(Succ(x478), Succ(x477)))) with sigma = [x479 / x166] which results in the following new constraint: (7) (new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) the following chains were created: *We consider the chain H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205))) -> new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204), new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) -> new_map0(x206, Succ(Succ(new_primPlusNat0(x207, Zero))), Succ(Succ(new_primPlusNat0(x207, Zero))), x209) which results in the following constraint: (1) (new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)=new_map(x206, Succ(x207), Succ(x207), Pos(Succ(Zero)), Zero, Succ(x208), x209) ==> H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239) -> new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239), new_map0(x240, x241, x242, x243) -> new_map3(x240, x242, Pos(Succ(Zero)), x243) which results in the following constraint: (1) (new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)=new_map0(x240, x241, x242, x243) ==> new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) For Pair H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) the following chains were created: *We consider the chain H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero)) -> new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346), new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) -> new_map2(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), x349) which results in the following constraint: (1) (new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)=new_map(x347, Succ(x348), Succ(x348), Pos(Succ(Zero)), Zero, Zero, x349) ==> H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) For Pair new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) the following chains were created: *We consider the chain new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404) -> new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404), new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) -> new_map0(x405, Succ(Succ(new_primPlusNat0(x406, Zero))), Succ(Succ(new_primPlusNat0(x406, Zero))), x407) which results in the following constraint: (1) (new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)=new_map2(x405, Succ(x406), Succ(x406), Pos(Succ(Zero)), x407) ==> new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) For Pair new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) the following chains were created: *We consider the chain new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414) -> new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414), new_map0(x415, x416, x417, x418) -> new_map3(x415, x417, Pos(Succ(Zero)), x418) which results in the following constraint: (1) (new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)=new_map0(x415, x416, x417, x418) ==> new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) *(new_map0(x4, x5, x6, x7)_>=_new_map3(x4, x6, Pos(Succ(Zero)), x7)) *new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) *(new_map3(x63, Succ(x67), Pos(Succ(Zero)), x65)_>=_new_map4(Pos(Succ(x63)), Succ(x67), Succ(x67), Pos(Succ(Zero)), Succ(x67), x65)) *new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) *(new_map4(Pos(Succ(Succ(x470))), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), Succ(Succ(x471)), x153)_>=_H(Succ(x470), Succ(Succ(x471)), Succ(Succ(x471)), Pos(Succ(Zero)), x153, anew_new_map(Succ(x471), Succ(x470))) ==> new_map4(Pos(Succ(Succ(Succ(x470)))), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), Succ(Succ(Succ(x471))), x153)_>=_H(Succ(Succ(x470)), Succ(Succ(Succ(x471))), Succ(Succ(Succ(x471))), Pos(Succ(Zero)), x153, anew_new_map(Succ(Succ(x471)), Succ(Succ(x470))))) *(new_map4(Pos(Succ(Succ(Succ(x474)))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x153)_>=_H(Succ(Succ(x474)), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x153, anew_new_map(Succ(Zero), Succ(Succ(x474))))) *(new_map4(Pos(Succ(Succ(x477))), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), Succ(Succ(x478)), x166)_>=_H(Succ(x477), Succ(Succ(x478)), Succ(Succ(x478)), Pos(Succ(Zero)), x166, anew_new_map(Succ(x478), Succ(x477))) ==> new_map4(Pos(Succ(Succ(Succ(x477)))), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), Succ(Succ(Succ(x478))), x166)_>=_H(Succ(Succ(x477)), Succ(Succ(Succ(x478))), Succ(Succ(Succ(x478))), Pos(Succ(Zero)), x166, anew_new_map(Succ(Succ(x478)), Succ(Succ(x477))))) *(new_map4(Pos(Succ(Succ(Zero))), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), Succ(Succ(Zero)), x166)_>=_H(Succ(Zero), Succ(Succ(Zero)), Succ(Succ(Zero)), Pos(Succ(Zero)), x166, anew_new_map(Succ(Zero), Succ(Zero)))) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) *(H(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), x204, cons_new_map(Zero, Succ(x205)))_>=_new_map(x202, Succ(x203), Succ(x203), Pos(Succ(Zero)), Zero, Succ(x205), x204)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map(x236, Succ(x237), Succ(x237), Pos(Succ(Zero)), Zero, Succ(x238), x239)_>=_new_map0(x236, Succ(Succ(new_primPlusNat0(x237, Zero))), Succ(Succ(new_primPlusNat0(x237, Zero))), x239)) *H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) *(H(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), x346, cons_new_map(Zero, Zero))_>=_new_map(x344, Succ(x345), Succ(x345), Pos(Succ(Zero)), Zero, Zero, x346)) *new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) *(new_map(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), Zero, Zero, x404)_>=_new_map2(x402, Succ(x403), Succ(x403), Pos(Succ(Zero)), x404)) *new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) *(new_map2(x412, Succ(x413), Succ(x413), Pos(Succ(Zero)), x414)_>=_new_map0(x412, Succ(Succ(new_primPlusNat0(x413, Zero))), Succ(Succ(new_primPlusNat0(x413, Zero))), x414)) 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. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation [NONINF]: POL(H(x_1, x_2, x_3, x_4, x_5, x_6)) = x_1 - x_3 - x_4 POL(Pos(x_1)) = x_1 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(anew_new_map(x_1, x_2)) = 0 POL(c) = -3 POL(cons_new_map(x_1, x_2)) = x_2 POL(new_map(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = -1 + x_1 - x_2 - x_4 + x_5 POL(new_map0(x_1, x_2, x_3, x_4)) = -1 + x_1 - x_3 POL(new_map2(x_1, x_2, x_3, x_4, x_5)) = -1 + x_1 - x_3 - x_4 POL(new_map3(x_1, x_2, x_3, x_4)) = x_1 - x_2 - x_3 POL(new_map4(x_1, x_2, x_3, x_4, x_5, x_6)) = -1 + x_1 + x_2 - x_3 - x_4 - x_5 POL(new_new_map(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = x_1 The following pairs are in P_>: H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) The following pairs are in P_bound: new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) The following rules are usable: Succ(yx2180) -> new_primPlusNat0(Succ(yx2180), Zero) Zero -> new_primPlusNat0(Zero, Zero) ---------------------------------------- (270) Complex Obligation (AND) ---------------------------------------- (271) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) new_map4(Pos(Succ(x0)), Succ(z1), Succ(z1), Pos(Succ(Zero)), Succ(z1), z2) -> H(x0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, anew_new_map(z1, x0)) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (272) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 6 less nodes. ---------------------------------------- (273) TRUE ---------------------------------------- (274) Obligation: Q DP problem: The TRS P consists of the following rules: new_map0(yx272, yx279, yx278, h) -> new_map3(yx272, yx278, Pos(Succ(Zero)), h) new_map3(z0, z2, Pos(Succ(Zero)), z3) -> new_map4(Pos(Succ(z0)), z2, z2, Pos(Succ(Zero)), z2, z3) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Succ(x5))) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(x5), z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Succ(z3), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) H(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2, cons_new_map(Zero, Zero)) -> new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) new_map(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), Zero, Zero, z2) -> new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) new_map2(z0, Succ(z1), Succ(z1), Pos(Succ(Zero)), z2) -> new_map0(z0, Succ(Succ(new_primPlusNat0(z1, Zero))), Succ(Succ(new_primPlusNat0(z1, Zero))), z2) The TRS R consists of the following rules: new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) anew_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Succ(yx2760), Succ(yx2770)) -> new_new_map(yx2760, yx2770) new_new_map(Zero, Succ(yx2770)) -> cons_new_map(Zero, Succ(yx2770)) new_new_map(Zero, Zero) -> cons_new_map(Zero, Zero) The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) new_new_map(Succ(x0), Succ(x1)) anew_new_map(Succ(x0), Succ(x1)) new_new_map(Zero, Succ(x0)) new_new_map(Zero, Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (275) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 7 less nodes. ---------------------------------------- (276) TRUE ---------------------------------------- (277) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (278) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_map(yx272, yx273, yx274, yx275, Succ(yx2760), Succ(yx2770), h) -> new_map(yx272, yx273, yx274, yx275, yx2760, yx2770, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7 ---------------------------------------- (279) YES ---------------------------------------- (280) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (281) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Pos(x4), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Pos(x4), z4) -> new_map9(Pos(x4), y_1, z4) we obtained the following new rules [LPAR04]: (new_map13(Pos(x0), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Pos(x0), z2) -> new_map9(Pos(x0), x1, z2),new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2)) ---------------------------------------- (282) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) The TRS R consists of the following rules: new_primPlusNat0(Succ(yx2180), Zero) -> Succ(yx2180) new_primPlusNat0(Zero, Succ(yx212000)) -> Succ(yx212000) new_primPlusNat0(Succ(yx2180), Succ(yx212000)) -> Succ(Succ(new_primPlusNat0(yx2180, yx212000))) new_primPlusNat0(Zero, Zero) -> Zero The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (283) UsableRulesProof (EQUIVALENT) 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. ---------------------------------------- (284) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) R is empty. The set Q consists of the following terms: new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (285) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_primPlusNat0(Succ(x0), Zero) new_primPlusNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, Zero) new_primPlusNat0(Succ(x0), Succ(x1)) ---------------------------------------- (286) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (287) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Zero)), z3) -> new_map16(Neg(Succ(Zero)), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2)) ---------------------------------------- (288) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (289) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Succ(Succ(x0))), z3) -> new_map15(Neg(Succ(Succ(x0))), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), x0, y_2, z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), Succ(x1), Neg(Succ(Succ(x0))), z2) -> new_map15(Neg(Succ(Succ(x0))), Succ(x1), Succ(x1), Pos(Succ(Zero)), x0, x1, z2),new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2)) ---------------------------------------- (290) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (291) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_map13(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), Succ(y_2), Neg(Zero), z3) -> new_map16(Neg(Zero), Succ(y_0), Succ(y_1), Pos(Succ(Zero)), z3) we obtained the following new rules [LPAR04]: (new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2),new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2)) ---------------------------------------- (292) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (293) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_map16(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map17(yx207, yx217, yx218, Pos(Zero), bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 0 POL(Pos(x_1)) = x_1 POL(Succ(x_1)) = 0 POL(Zero) = 1 POL(new_map1(x_1, x_2, x_3, x_4)) = 0 POL(new_map12(x_1, x_2, x_3, x_4)) = x_3 POL(new_map13(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = 0 POL(new_map15(x_1, x_2, x_3, x_4, x_5, x_6, x_7)) = x_4 POL(new_map16(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map17(x_1, x_2, x_3, x_4, x_5)) = x_4 POL(new_map9(x_1, x_2, x_3)) = 0 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (294) Obligation: Q DP problem: The TRS P consists of the following rules: new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (295) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_map9(yx207, yx218, bb) -> new_map12(yx207, yx218, Pos(Succ(Zero)), bb) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4 *new_map12(z0, z1, Pos(Succ(Zero)), z2) -> new_map13(z0, z1, z1, Pos(Succ(Zero)), z1, z0, z2) The graph contains the following edges 1 >= 1, 2 >= 2, 2 >= 3, 3 >= 4, 2 >= 5, 1 >= 6, 4 >= 7 *new_map13(Pos(x4), Succ(x2), Succ(x2), Pos(Succ(Zero)), Succ(x2), Pos(x4), z2) -> new_map9(Pos(x4), x2, z2) The graph contains the following edges 1 >= 1, 6 >= 1, 2 > 2, 3 > 2, 5 > 2, 7 >= 3 *new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) The graph contains the following edges 1 >= 1, 3 > 2, 5 >= 3 *new_map13(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Succ(x5))), z2) -> new_map15(Neg(Succ(Succ(x5))), Succ(x4), Succ(x4), Pos(Succ(Zero)), x5, x4, z2) The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 1 > 5, 6 > 5, 2 > 6, 3 > 6, 5 > 6, 7 >= 7 *new_map15(yx207, yx217, yx218, yx212, Zero, Succ(yx2190), bb) -> new_map16(yx207, yx217, yx218, yx212, bb) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5 *new_map15(yx207, yx217, yx218, yx212, Succ(yx213000), Succ(yx2190), bb) -> new_map15(yx207, yx217, yx218, yx212, yx213000, yx2190, bb) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7 *new_map15(yx207, yx217, yx218, yx212, Zero, Zero, bb) -> new_map17(yx207, yx217, yx218, yx212, bb) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5 *new_map16(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) The graph contains the following edges 1 >= 1, 4 > 2, 3 > 3, 5 >= 4 *new_map1(yx207, Succ(yx212000), Succ(yx2180), bb) -> new_map1(yx207, yx212000, yx2180, bb) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4 *new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Succ(yx212000))), bb) -> new_map1(yx207, yx212000, yx2180, bb) The graph contains the following edges 1 >= 1, 4 > 2, 3 > 3, 5 >= 4 *new_map1(yx207, Zero, Succ(yx2180), bb) -> new_map9(yx207, yx2180, bb) The graph contains the following edges 1 >= 1, 3 > 2, 4 >= 3 *new_map17(yx207, yx217, Succ(yx2180), Pos(Succ(Zero)), bb) -> new_map9(yx207, yx2180, bb) The graph contains the following edges 1 >= 1, 3 > 2, 5 >= 3 *new_map13(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), Succ(x4), Neg(Succ(Zero)), z2) -> new_map16(Neg(Succ(Zero)), Succ(x4), Succ(x4), Pos(Succ(Zero)), z2) The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 *new_map13(Neg(Zero), z1, z1, Pos(Succ(Zero)), z1, Neg(Zero), z2) -> new_map16(Neg(Zero), z1, z1, Pos(Succ(Zero)), z2) The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 *new_map13(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), Succ(x0), Neg(Zero), z2) -> new_map16(Neg(Zero), Succ(x0), Succ(x0), Pos(Succ(Zero)), z2) The graph contains the following edges 1 >= 1, 6 >= 1, 2 >= 2, 3 >= 2, 5 >= 2, 2 >= 3, 3 >= 3, 5 >= 3, 4 >= 4, 7 >= 5 ---------------------------------------- (296) YES ---------------------------------------- (297) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS(yx2000, yx2010) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (298) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_primMinusNatS(Succ(yx2000), Succ(yx2010)) -> new_primMinusNatS(yx2000, yx2010) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (299) YES ---------------------------------------- (300) Obligation: Q DP problem: The TRS P consists of the following rules: new_primPlusNat(Succ(yx2180), Succ(yx212000)) -> new_primPlusNat(yx2180, yx212000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (301) QDPSizeChangeProof (EQUIVALENT) 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. From the DPs we obtained the following set of size-change graphs: *new_primPlusNat(Succ(yx2180), Succ(yx212000)) -> new_primPlusNat(yx2180, yx212000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (302) YES