75.87/40.61 MAYBE 78.21/41.32 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 78.21/41.32 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 78.21/41.32 78.21/41.32 78.21/41.32 H-Termination with start terms of the given HASKELL could not be shown: 78.21/41.32 78.21/41.32 (0) HASKELL 78.21/41.32 (1) LR [EQUIVALENT, 0 ms] 78.21/41.32 (2) HASKELL 78.21/41.32 (3) IFR [EQUIVALENT, 0 ms] 78.21/41.32 (4) HASKELL 78.21/41.32 (5) BR [EQUIVALENT, 0 ms] 78.21/41.32 (6) HASKELL 78.21/41.32 (7) COR [EQUIVALENT, 0 ms] 78.21/41.32 (8) HASKELL 78.21/41.32 (9) LetRed [EQUIVALENT, 0 ms] 78.21/41.32 (10) HASKELL 78.21/41.32 (11) NumRed [SOUND, 0 ms] 78.21/41.32 (12) HASKELL 78.21/41.32 (13) Narrow [SOUND, 0 ms] 78.21/41.32 (14) AND 78.21/41.32 (15) QDP 78.21/41.32 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (17) QDP 78.21/41.32 (18) QDPOrderProof [EQUIVALENT, 0 ms] 78.21/41.32 (19) QDP 78.21/41.32 (20) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (21) QDP 78.21/41.32 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (23) YES 78.21/41.32 (24) QDP 78.21/41.32 (25) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (26) QDP 78.21/41.32 (27) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (28) QDP 78.21/41.32 (29) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (30) QDP 78.21/41.32 (31) QReductionProof [EQUIVALENT, 20 ms] 78.21/41.32 (32) QDP 78.21/41.32 (33) TransformationProof [EQUIVALENT, 94 ms] 78.21/41.32 (34) QDP 78.21/41.32 (35) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (36) QDP 78.21/41.32 (37) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (38) QDP 78.21/41.32 (39) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (40) QDP 78.21/41.32 (41) TransformationProof [EQUIVALENT, 123 ms] 78.21/41.32 (42) QDP 78.21/41.32 (43) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (44) QDP 78.21/41.32 (45) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (46) QDP 78.21/41.32 (47) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (48) QDP 78.21/41.32 (49) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (50) QDP 78.21/41.32 (51) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (52) QDP 78.21/41.32 (53) TransformationProof [EQUIVALENT, 56 ms] 78.21/41.32 (54) QDP 78.21/41.32 (55) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (56) QDP 78.21/41.32 (57) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (58) QDP 78.21/41.32 (59) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (60) QDP 78.21/41.32 (61) TransformationProof [EQUIVALENT, 49 ms] 78.21/41.32 (62) QDP 78.21/41.32 (63) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (64) QDP 78.21/41.32 (65) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (66) QDP 78.21/41.32 (67) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (68) QDP 78.21/41.32 (69) TransformationProof [EQUIVALENT, 52 ms] 78.21/41.32 (70) QDP 78.21/41.32 (71) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (72) QDP 78.21/41.32 (73) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (74) QDP 78.21/41.32 (75) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (76) QDP 78.21/41.32 (77) InductionCalculusProof [EQUIVALENT, 0 ms] 78.21/41.32 (78) QDP 78.21/41.32 (79) QDPPairToRuleProof [EQUIVALENT, 0 ms] 78.21/41.32 (80) AND 78.21/41.32 (81) QDP 78.21/41.32 (82) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (83) QDP 78.21/41.32 (84) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (85) QDP 78.21/41.32 (86) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (87) QDP 78.21/41.32 (88) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (89) QDP 78.21/41.32 (90) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (91) QDP 78.21/41.32 (92) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (93) QDP 78.21/41.32 (94) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (95) QDP 78.21/41.32 (96) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (97) QDP 78.21/41.32 (98) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (99) QDP 78.21/41.32 (100) InductionCalculusProof [EQUIVALENT, 0 ms] 78.21/41.32 (101) QDP 78.21/41.32 (102) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (103) QDP 78.21/41.32 (104) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (105) QDP 78.21/41.32 (106) QReductionProof [EQUIVALENT, 0 ms] 78.21/41.32 (107) QDP 78.21/41.32 (108) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (109) QDP 78.21/41.32 (110) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (111) QDP 78.21/41.32 (112) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (113) QDP 78.21/41.32 (114) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (115) QDP 78.21/41.32 (116) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (117) QDP 78.21/41.32 (118) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (119) QDP 78.21/41.32 (120) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (121) QDP 78.21/41.32 (122) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (123) QDP 78.21/41.32 (124) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (125) QDP 78.21/41.32 (126) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (127) QDP 78.21/41.32 (128) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (129) QDP 78.21/41.32 (130) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (131) QDP 78.21/41.32 (132) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (133) QDP 78.21/41.32 (134) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (135) QDP 78.21/41.32 (136) TransformationProof [EQUIVALENT, 2 ms] 78.21/41.32 (137) QDP 78.21/41.32 (138) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (139) QDP 78.21/41.32 (140) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (141) QDP 78.21/41.32 (142) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (143) QDP 78.21/41.32 (144) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (145) QDP 78.21/41.32 (146) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (147) QDP 78.21/41.32 (148) TransformationProof [EQUIVALENT, 0 ms] 78.21/41.32 (149) QDP 78.21/41.32 (150) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (151) QDP 78.21/41.32 (152) UsableRulesProof [EQUIVALENT, 0 ms] 78.21/41.32 (153) QDP 78.21/41.32 (154) InductionCalculusProof [EQUIVALENT, 0 ms] 78.21/41.32 (155) QDP 78.21/41.32 (156) QDP 78.21/41.32 (157) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (158) YES 78.21/41.32 (159) QDP 78.21/41.32 (160) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (161) QDP 78.21/41.32 (162) QDPOrderProof [EQUIVALENT, 0 ms] 78.21/41.32 (163) QDP 78.21/41.32 (164) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (165) QDP 78.21/41.32 (166) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (167) YES 78.21/41.32 (168) QDP 78.21/41.32 (169) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (170) AND 78.21/41.32 (171) QDP 78.21/41.32 (172) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (173) YES 78.21/41.32 (174) QDP 78.21/41.32 (175) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (176) YES 78.21/41.32 (177) QDP 78.21/41.32 (178) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (179) YES 78.21/41.32 (180) QDP 78.21/41.32 (181) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (182) YES 78.21/41.32 (183) QDP 78.21/41.32 (184) QDPSizeChangeProof [EQUIVALENT, 0 ms] 78.21/41.32 (185) YES 78.21/41.32 (186) Narrow [COMPLETE, 0 ms] 78.21/41.32 (187) QDP 78.21/41.32 (188) DependencyGraphProof [EQUIVALENT, 0 ms] 78.21/41.32 (189) QDP 78.21/41.32 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (0) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (1) LR (EQUIVALENT) 78.21/41.32 Lambda Reductions: 78.21/41.32 The following Lambda expression 78.21/41.32 "\(_,d)->d" 78.21/41.32 is transformed to 78.21/41.32 "d0 (_,d) = d; 78.21/41.32 " 78.21/41.32 The following Lambda expression 78.21/41.32 "\(n',_)->n'" 78.21/41.32 is transformed to 78.21/41.32 "n'0 (n',_) = n'; 78.21/41.32 " 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (2) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (3) IFR (EQUIVALENT) 78.21/41.32 If Reductions: 78.21/41.32 The following If expression 78.21/41.32 "if n' == 0 then r' else showInt n' r'" 78.21/41.32 is transformed to 78.21/41.32 "showInt0 True = r'; 78.21/41.32 showInt0 False = showInt n' r'; 78.21/41.32 " 78.21/41.32 The following If expression 78.21/41.32 "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" 78.21/41.32 is transformed to 78.21/41.32 "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); 78.21/41.32 primDivNatS0 x y False = Zero; 78.21/41.32 " 78.21/41.32 The following If expression 78.21/41.32 "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" 78.21/41.32 is transformed to 78.21/41.32 "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); 78.21/41.32 primModNatS0 x y False = Succ x; 78.21/41.32 " 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (4) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (5) BR (EQUIVALENT) 78.21/41.32 Replaced joker patterns by fresh variables and removed binding patterns. 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (6) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (7) COR (EQUIVALENT) 78.21/41.32 Cond Reductions: 78.21/41.32 The following Function with conditions 78.21/41.32 "showInt n r|n < 0error []|otherwiselet { 78.21/41.32 d = d0 vu76; 78.21/41.32 ; 78.21/41.32 d0 (vx,d) = d; 78.21/41.32 ; 78.21/41.32 n' = n'0 vu76; 78.21/41.32 ; 78.21/41.32 n'0 (n',vw) = n'; 78.21/41.32 ; 78.21/41.32 r' = toEnum (fromEnum '0' + fromIntegral d) : r; 78.21/41.32 ; 78.21/41.32 showInt0 True = r'; 78.21/41.32 showInt0 False = showInt n' r'; 78.21/41.32 ; 78.21/41.32 vu76 = quotRem n 10; 78.21/41.32 } in showInt0 (n' == 0); 78.21/41.32 " 78.21/41.32 is transformed to 78.21/41.32 "showInt n r = showInt3 n r; 78.21/41.32 " 78.21/41.32 "showInt1 n r True = let { 78.21/41.32 d = d0 vu76; 78.21/41.32 ; 78.21/41.32 d0 (vx,d) = d; 78.21/41.32 ; 78.21/41.32 n' = n'0 vu76; 78.21/41.32 ; 78.21/41.32 n'0 (n',vw) = n'; 78.21/41.32 ; 78.21/41.32 r' = toEnum (fromEnum '0' + fromIntegral d) : r; 78.21/41.32 ; 78.21/41.32 showInt0 True = r'; 78.21/41.32 showInt0 False = showInt n' r'; 78.21/41.32 ; 78.21/41.32 vu76 = quotRem n 10; 78.21/41.32 } in showInt0 (n' == 0); 78.21/41.32 " 78.21/41.32 "showInt2 n r True = error []; 78.21/41.32 showInt2 n r False = showInt1 n r otherwise; 78.21/41.32 " 78.21/41.32 "showInt3 n r = showInt2 n r (n < 0); 78.21/41.32 " 78.21/41.32 The following Function with conditions 78.21/41.32 "undefined |Falseundefined; 78.21/41.32 " 78.21/41.32 is transformed to 78.21/41.32 "undefined = undefined1; 78.21/41.32 " 78.21/41.32 "undefined0 True = undefined; 78.21/41.32 " 78.21/41.32 "undefined1 = undefined0 False; 78.21/41.32 " 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (8) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (9) LetRed (EQUIVALENT) 78.21/41.32 Let/Where Reductions: 78.21/41.32 The bindings of the following Let/Where expression 78.21/41.32 "let { 78.21/41.32 d = d0 vu76; 78.21/41.32 ; 78.21/41.32 d0 (vx,d) = d; 78.21/41.32 ; 78.21/41.32 n' = n'0 vu76; 78.21/41.32 ; 78.21/41.32 n'0 (n',vw) = n'; 78.21/41.32 ; 78.21/41.32 r' = toEnum (fromEnum '0' + fromIntegral d) : r; 78.21/41.32 ; 78.21/41.32 showInt0 True = r'; 78.21/41.32 showInt0 False = showInt n' r'; 78.21/41.32 ; 78.21/41.32 vu76 = quotRem n 10; 78.21/41.32 } in showInt0 (n' == 0)" 78.21/41.32 are unpacked to the following functions on top level 78.21/41.32 "showInt1ShowInt0 wz xu True = showInt1R' wz xu; 78.21/41.32 showInt1ShowInt0 wz xu False = showInt (showInt1N' wz xu) (showInt1R' wz xu); 78.21/41.32 " 78.21/41.32 "showInt1R' wz xu = toEnum (fromEnum '0' + fromIntegral (showInt1D wz xu)) : wz; 78.21/41.32 " 78.21/41.32 "showInt1N'0 wz xu (n',vw) = n'; 78.21/41.32 " 78.21/41.32 "showInt1N' wz xu = showInt1N'0 wz xu (showInt1Vu76 wz xu); 78.21/41.32 " 78.21/41.32 "showInt1Vu76 wz xu = quotRem xu 10; 78.21/41.32 " 78.21/41.32 "showInt1D wz xu = showInt1D0 wz xu (showInt1Vu76 wz xu); 78.21/41.32 " 78.21/41.32 "showInt1D0 wz xu (vx,d) = d; 78.21/41.32 " 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (10) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (11) NumRed (SOUND) 78.21/41.32 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (12) 78.21/41.32 Obligation: 78.21/41.32 mainModule Main 78.21/41.32 module Main where { 78.21/41.32 import qualified Prelude; 78.21/41.32 } 78.21/41.32 78.21/41.32 ---------------------------------------- 78.21/41.32 78.21/41.32 (13) Narrow (SOUND) 78.21/41.32 Haskell To QDPs 78.21/41.32 78.21/41.32 digraph dp_graph { 78.21/41.32 node [outthreshold=100, inthreshold=100];1[label="showInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 78.21/41.32 3[label="showInt xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 78.21/41.32 4[label="showInt xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 78.21/41.32 5[label="showInt3 xv3 xv4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 78.21/41.32 6[label="showInt2 xv3 xv4 (xv3 < fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 78.21/41.32 7[label="showInt2 xv3 xv4 (compare xv3 (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 78.21/41.32 8[label="showInt2 xv3 xv4 (primCmpInt xv3 (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1445[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1445[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1445 -> 9[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1446[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1446[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1446 -> 10[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 9[label="showInt2 (Pos xv30) xv4 (primCmpInt (Pos xv30) (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1447[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];9 -> 1447[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1447 -> 11[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1448[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 1448[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1448 -> 12[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 10[label="showInt2 (Neg xv30) xv4 (primCmpInt (Neg xv30) (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1449[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];10 -> 1449[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1449 -> 13[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1450[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 1450[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1450 -> 14[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 11[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpInt (Pos (Succ xv300)) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 78.21/41.32 12[label="showInt2 (Pos Zero) xv4 (primCmpInt (Pos Zero) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 78.21/41.32 13[label="showInt2 (Neg (Succ xv300)) xv4 (primCmpInt (Neg (Succ xv300)) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 78.21/41.32 14[label="showInt2 (Neg Zero) xv4 (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 78.21/41.32 15[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpInt (Pos (Succ xv300)) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 78.21/41.32 16[label="showInt2 (Pos Zero) xv4 (primCmpInt (Pos Zero) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 78.21/41.32 17[label="showInt2 (Neg (Succ xv300)) xv4 (primCmpInt (Neg (Succ xv300)) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 78.21/41.32 18[label="showInt2 (Neg Zero) xv4 (primCmpInt (Neg Zero) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 78.21/41.32 19[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpNat (Succ xv300) Zero == LT)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 78.21/41.32 20[label="showInt2 (Pos Zero) xv4 (EQ == LT)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 78.21/41.32 21[label="showInt2 (Neg (Succ xv300)) xv4 (LT == LT)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 78.21/41.32 22[label="showInt2 (Neg Zero) xv4 (EQ == LT)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 78.21/41.32 23[label="showInt2 (Pos 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33[label="",style="solid", color="black", weight=3]; 78.21/41.32 30[label="showInt1 (Neg Zero) xv4 otherwise",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3]; 78.21/41.32 31[label="showInt1 (Pos (Succ xv300)) xv4 otherwise",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3]; 78.21/41.32 32[label="showInt1 (Pos Zero) xv4 True",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3]; 78.21/41.32 33[label="error []",fontsize=16,color="red",shape="box"];34[label="showInt1 (Neg Zero) xv4 True",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3]; 78.21/41.32 35[label="showInt1 (Pos (Succ xv300)) xv4 True",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3]; 78.21/41.32 36[label="showInt1ShowInt0 xv4 (Pos Zero) (showInt1N' xv4 (Pos Zero) == fromInt (Pos 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color="red", weight=0]; 78.21/41.32 45[label="showInt1ShowInt0 xv4 (Pos Zero) (primEqInt (showInt1N'0 xv4 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];45 -> 49[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 45 -> 50[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 46 -> 51[label="",style="dashed", color="red", weight=0]; 78.21/41.32 46[label="showInt1ShowInt0 xv4 (Neg Zero) (primEqInt (showInt1N'0 xv4 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];46 -> 52[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 46 -> 53[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 47 -> 54[label="",style="dashed", color="red", weight=0]; 78.21/41.32 47[label="showInt1ShowInt0 xv4 (Pos 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62[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (showInt1N'0 xv9 (Neg Zero) (primQuotInt (Neg Zero) (fromInt (Pos (Succ xv10))),primRemInt (Neg Zero) (fromInt (Pos (Succ xv10))))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];62 -> 65[label="",style="solid", color="black", weight=3]; 78.21/41.32 63[label="showInt1ShowInt0 xv12 (Pos (Succ xv13)) (primEqInt (showInt1N'0 xv12 (Pos (Succ xv13)) (primQuotInt (Pos (Succ xv13)) (fromInt (Pos (Succ xv14))),primRemInt (Pos (Succ xv13)) (fromInt (Pos (Succ xv14))))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];63 -> 66[label="",style="solid", color="black", weight=3]; 78.21/41.32 64[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (primQuotInt (Pos Zero) (fromInt (Pos (Succ xv7)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];64 -> 67[label="",style="solid", color="black", weight=3]; 78.21/41.32 65[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (primQuotInt (Neg Zero) (fromInt (Pos (Succ 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74[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (Neg Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3]; 78.21/41.32 75[label="showInt1ShowInt0 xv12 (Pos (Succ xv13)) (primEqInt (Pos (primDivNatS0 xv13 xv14 (primGEqNatS xv13 xv14))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1451[label="xv13/Succ xv130",fontsize=10,color="white",style="solid",shape="box"];75 -> 1451[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1451 -> 78[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1452[label="xv13/Zero",fontsize=10,color="white",style="solid",shape="box"];75 -> 1452[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1452 -> 79[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 76[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];76 -> 80[label="",style="solid", color="black", weight=3]; 78.21/41.32 77[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];77 -> 81[label="",style="solid", color="black", weight=3]; 78.21/41.32 78[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) xv14 (primGEqNatS (Succ xv130) xv14))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1453[label="xv14/Succ xv140",fontsize=10,color="white",style="solid",shape="box"];78 -> 1453[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1453 -> 82[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1454[label="xv14/Zero",fontsize=10,color="white",style="solid",shape="box"];78 -> 1454[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1454 -> 83[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 79[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero xv14 (primGEqNatS 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fromIntegral (showInt1D xv6 (Pos Zero))) : xv6",fontsize=16,color="green",shape="box"];92 -> 99[label="",style="dashed", color="green", weight=3]; 78.21/41.32 93[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv9 (Neg Zero))) : xv9",fontsize=16,color="green",shape="box"];93 -> 100[label="",style="dashed", color="green", weight=3]; 78.21/41.32 570[label="Succ 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xv980",fontsize=10,color="white",style="solid",shape="box"];620 -> 1459[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1459 -> 624[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1460[label="xv98/Zero",fontsize=10,color="white",style="solid",shape="box"];620 -> 1460[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1460 -> 625[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 621[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero xv98))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1461[label="xv98/Succ xv980",fontsize=10,color="white",style="solid",shape="box"];621 -> 1461[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1461 -> 626[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1462[label="xv98/Zero",fontsize=10,color="white",style="solid",shape="box"];621 -> 1462[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1462 -> 627[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 105[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS (Succ xv130) Zero) (Succ Zero)))) (Pos Zero))",fontsize=16,color="black",shape="box"];105 -> 118[label="",style="solid", color="black", weight=3]; 78.21/41.32 106[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];106 -> 119[label="",style="solid", color="black", weight=3]; 78.21/41.32 107[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (Pos Zero))",fontsize=16,color="black",shape="box"];107 -> 120[label="",style="solid", color="black", weight=3]; 78.21/41.32 147 -> 179[label="",style="dashed", color="red", weight=0]; 78.21/41.32 147[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ 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xv9 (Neg Zero)))",fontsize=16,color="magenta"];157 -> 183[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 157 -> 184[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 624[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS (Succ xv970) (Succ xv980)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];624 -> 631[label="",style="solid", color="black", weight=3]; 78.21/41.32 625[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS (Succ xv970) Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];625 -> 632[label="",style="solid", color="black", weight=3]; 78.21/41.32 626[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero (Succ xv980)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];626 -> 633[label="",style="solid", color="black", weight=3]; 78.21/41.32 627[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];627 -> 634[label="",style="solid", color="black", weight=3]; 78.21/41.32 118[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) False",fontsize=16,color="black",shape="triangle"];118 -> 144[label="",style="solid", color="black", weight=3]; 78.21/41.32 119[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) True",fontsize=16,color="black",shape="box"];119 -> 145[label="",style="solid", color="black", weight=3]; 78.21/41.32 120[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) False",fontsize=16,color="black",shape="box"];120 -> 146[label="",style="solid", color="black", weight=3]; 78.21/41.32 180[label="xv6",fontsize=16,color="green",shape="box"];181[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];179[label="primIntToChar (fromEnum (Char (Succ xv22)) + fromIntegral (showInt1D xv23 (Pos Zero)))",fontsize=16,color="black",shape="triangle"];179 -> 185[label="",style="solid", color="black", weight=3]; 78.21/41.32 183[label="xv9",fontsize=16,color="green",shape="box"];184[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];182[label="primIntToChar (fromEnum (Char (Succ xv25)) + fromIntegral (showInt1D xv26 (Neg Zero)))",fontsize=16,color="black",shape="triangle"];182 -> 186[label="",style="solid", color="black", weight=3]; 78.21/41.32 631 -> 569[label="",style="dashed", color="red", weight=0]; 78.21/41.32 631[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS xv970 xv980))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];631 -> 640[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 631 -> 641[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 632[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) True)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="triangle"];632 -> 642[label="",style="solid", color="black", weight=3]; 78.21/41.32 633[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) False)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];633 -> 643[label="",style="solid", color="black", weight=3]; 78.21/41.32 634 -> 632[label="",style="dashed", color="red", weight=0]; 78.21/41.32 634[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) True)) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];144 -> 4[label="",style="dashed", color="red", weight=0]; 78.21/41.32 144[label="showInt (showInt1N' xv12 (Pos (Succ (Succ xv130)))) (showInt1R' xv12 (Pos (Succ (Succ xv130))))",fontsize=16,color="magenta"];144 -> 174[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 144 -> 175[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 145[label="showInt1R' xv12 (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];145 -> 176[label="",style="solid", color="black", weight=3]; 78.21/41.32 146 -> 4[label="",style="dashed", color="red", weight=0]; 78.21/41.32 146[label="showInt (showInt1N' xv12 (Pos (Succ Zero))) (showInt1R' xv12 (Pos (Succ Zero)))",fontsize=16,color="magenta"];146 -> 177[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 146 -> 178[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 185[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv22))) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];185 -> 198[label="",style="solid", color="black", weight=3]; 78.21/41.32 186[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv25))) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];186 -> 199[label="",style="solid", color="black", weight=3]; 78.21/41.32 640[label="xv970",fontsize=16,color="green",shape="box"];641[label="xv980",fontsize=16,color="green",shape="box"];642[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS xv95 (Succ xv96)) (Succ (Succ xv96))))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];642 -> 649[label="",style="solid", color="black", weight=3]; 78.21/41.32 643[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];643 -> 650[label="",style="solid", color="black", weight=3]; 78.21/41.32 174[label="showInt1R' xv12 (Pos (Succ (Succ xv130)))",fontsize=16,color="black",shape="triangle"];174 -> 194[label="",style="solid", color="black", weight=3]; 78.21/41.32 175[label="showInt1N' xv12 (Pos (Succ (Succ xv130)))",fontsize=16,color="black",shape="box"];175 -> 195[label="",style="solid", color="black", weight=3]; 78.21/41.32 176[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero)))) : xv12",fontsize=16,color="green",shape="box"];176 -> 196[label="",style="dashed", color="green", weight=3]; 78.21/41.32 177 -> 145[label="",style="dashed", color="red", weight=0]; 78.21/41.32 177[label="showInt1R' xv12 (Pos (Succ Zero))",fontsize=16,color="magenta"];178[label="showInt1N' xv12 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];178 -> 197[label="",style="solid", color="black", weight=3]; 78.21/41.32 198[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv22))) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];198 -> 213[label="",style="solid", color="black", weight=3]; 78.21/41.32 199[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv25))) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];199 -> 214[label="",style="solid", color="black", weight=3]; 78.21/41.32 649[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS xv95 (Succ xv96)) (Succ (Succ xv96))))) (Pos Zero))",fontsize=16,color="black",shape="box"];649 -> 657[label="",style="solid", color="black", weight=3]; 78.21/41.32 650[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];650 -> 658[label="",style="solid", color="black", weight=3]; 78.21/41.32 194[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130))))) : xv12",fontsize=16,color="green",shape="box"];194 -> 208[label="",style="dashed", color="green", weight=3]; 78.21/41.32 195[label="showInt1N'0 xv12 (Pos (Succ (Succ xv130))) (showInt1Vu76 xv12 (Pos (Succ (Succ xv130))))",fontsize=16,color="black",shape="box"];195 -> 209[label="",style="solid", color="black", weight=3]; 78.21/41.32 196[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];196 -> 246[label="",style="solid", color="black", weight=3]; 78.21/41.32 197[label="showInt1N'0 xv12 (Pos (Succ Zero)) (showInt1Vu76 xv12 (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];197 -> 215[label="",style="solid", color="black", weight=3]; 78.21/41.32 213[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];213 -> 228[label="",style="solid", color="black", weight=3]; 78.21/41.32 214[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];214 -> 229[label="",style="solid", color="black", weight=3]; 78.21/41.32 657[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) False",fontsize=16,color="black",shape="box"];657 -> 666[label="",style="solid", color="black", weight=3]; 78.21/41.32 658[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) True",fontsize=16,color="black",shape="box"];658 -> 667[label="",style="solid", color="black", weight=3]; 78.21/41.32 208[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130)))))",fontsize=16,color="black",shape="box"];208 -> 270[label="",style="solid", color="black", weight=3]; 78.21/41.32 209 -> 474[label="",style="dashed", color="red", weight=0]; 78.21/41.32 209[label="showInt1N'0 xv12 (Pos (Succ (Succ xv130))) (quotRem (Pos (Succ (Succ xv130))) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];209 -> 475[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 209 -> 476[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 209 -> 477[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 246 -> 533[label="",style="dashed", color="red", weight=0]; 78.21/41.32 246[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero))))",fontsize=16,color="magenta"];246 -> 534[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 246 -> 535[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 246 -> 536[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 215 -> 474[label="",style="dashed", color="red", weight=0]; 78.21/41.32 215[label="showInt1N'0 xv12 (Pos (Succ Zero)) (quotRem (Pos (Succ Zero)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];215 -> 478[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 215 -> 479[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 215 -> 480[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 228[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];228 -> 247[label="",style="solid", color="black", weight=3]; 78.21/41.32 229[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];229 -> 248[label="",style="solid", color="black", weight=3]; 78.21/41.32 666 -> 4[label="",style="dashed", color="red", weight=0]; 78.21/41.32 666[label="showInt (showInt1N' xv94 (Pos (Succ xv95))) (showInt1R' xv94 (Pos (Succ xv95)))",fontsize=16,color="magenta"];666 -> 673[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 666 -> 674[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 667[label="showInt1R' xv94 (Pos (Succ xv95))",fontsize=16,color="black",shape="triangle"];667 -> 675[label="",style="solid", color="black", weight=3]; 78.21/41.32 270 -> 533[label="",style="dashed", color="red", weight=0]; 78.21/41.32 270[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130)))))",fontsize=16,color="magenta"];270 -> 537[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 270 -> 538[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 270 -> 539[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 475[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];476[label="xv12",fontsize=16,color="green",shape="box"];477[label="Succ xv130",fontsize=16,color="green",shape="box"];474[label="showInt1N'0 xv82 (Pos (Succ xv83)) (quotRem (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="triangle"];474 -> 490[label="",style="solid", color="black", weight=3]; 78.21/41.32 534[label="Zero",fontsize=16,color="green",shape="box"];535[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];536[label="xv12",fontsize=16,color="green",shape="box"];533[label="primIntToChar (fromEnum (Char (Succ xv86)) + fromIntegral (showInt1D xv87 (Pos (Succ xv88))))",fontsize=16,color="black",shape="triangle"];533 -> 549[label="",style="solid", color="black", weight=3]; 78.21/41.32 478[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];479[label="xv12",fontsize=16,color="green",shape="box"];480[label="Zero",fontsize=16,color="green",shape="box"];247[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger (toInteger (showInt1D xv23 (Pos Zero)))))",fontsize=16,color="black",shape="box"];247 -> 272[label="",style="solid", color="black", weight=3]; 78.21/41.32 248[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger (toInteger (showInt1D xv26 (Neg Zero)))))",fontsize=16,color="black",shape="box"];248 -> 273[label="",style="solid", color="black", weight=3]; 78.21/41.32 673 -> 667[label="",style="dashed", color="red", weight=0]; 78.21/41.32 673[label="showInt1R' xv94 (Pos (Succ xv95))",fontsize=16,color="magenta"];674[label="showInt1N' xv94 (Pos (Succ xv95))",fontsize=16,color="black",shape="box"];674 -> 683[label="",style="solid", color="black", weight=3]; 78.21/41.32 675[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95)))) : xv94",fontsize=16,color="green",shape="box"];675 -> 684[label="",style="dashed", color="green", weight=3]; 78.21/41.32 537[label="Succ xv130",fontsize=16,color="green",shape="box"];538[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];539[label="xv12",fontsize=16,color="green",shape="box"];490[label="showInt1N'0 xv82 (Pos (Succ xv83)) (primQrmInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="box"];490 -> 509[label="",style="solid", color="black", weight=3]; 78.21/41.32 549[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv86))) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];549 -> 560[label="",style="solid", color="black", weight=3]; 78.21/41.32 272[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger (Integer (showInt1D xv23 (Pos Zero)))))",fontsize=16,color="black",shape="box"];272 -> 290[label="",style="solid", color="black", weight=3]; 78.21/41.32 273[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger (Integer (showInt1D xv26 (Neg Zero)))))",fontsize=16,color="black",shape="box"];273 -> 291[label="",style="solid", color="black", weight=3]; 78.21/41.32 683[label="showInt1N'0 xv94 (Pos (Succ xv95)) (showInt1Vu76 xv94 (Pos (Succ xv95)))",fontsize=16,color="black",shape="box"];683 -> 694[label="",style="solid", color="black", weight=3]; 78.21/41.32 684[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95))))",fontsize=16,color="black",shape="box"];684 -> 713[label="",style="solid", color="black", weight=3]; 78.21/41.32 509[label="showInt1N'0 xv82 (Pos (Succ xv83)) (primQuotInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))),primRemInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="box"];509 -> 532[label="",style="solid", color="black", weight=3]; 78.21/41.32 560[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv86))) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];560 -> 567[label="",style="solid", color="black", weight=3]; 78.21/41.32 290[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D xv23 (Pos Zero)))",fontsize=16,color="black",shape="box"];290 -> 304[label="",style="solid", color="black", weight=3]; 78.21/41.32 291[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D xv26 (Neg Zero)))",fontsize=16,color="black",shape="box"];291 -> 305[label="",style="solid", color="black", weight=3]; 78.21/41.32 694 -> 474[label="",style="dashed", color="red", weight=0]; 78.21/41.32 694[label="showInt1N'0 xv94 (Pos (Succ xv95)) (quotRem (Pos (Succ xv95)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];694 -> 699[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 694 -> 700[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 694 -> 701[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 713 -> 533[label="",style="dashed", color="red", weight=0]; 78.21/41.32 713[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95))))",fontsize=16,color="magenta"];713 -> 723[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 713 -> 724[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 713 -> 725[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 532[label="primQuotInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84)))",fontsize=16,color="black",shape="triangle"];532 -> 550[label="",style="solid", color="black", weight=3]; 78.21/41.32 567[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];567 -> 622[label="",style="solid", color="black", weight=3]; 78.21/41.32 304[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D0 xv23 (Pos Zero) (showInt1Vu76 xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];304 -> 317[label="",style="solid", color="black", weight=3]; 78.21/41.32 305[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D0 xv26 (Neg Zero) (showInt1Vu76 xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];305 -> 318[label="",style="solid", color="black", weight=3]; 78.21/41.32 699[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];700[label="xv94",fontsize=16,color="green",shape="box"];701[label="xv95",fontsize=16,color="green",shape="box"];723[label="xv95",fontsize=16,color="green",shape="box"];724[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];725[label="xv94",fontsize=16,color="green",shape="box"];550[label="primQuotInt (Pos (Succ xv83)) (Pos (Succ xv84))",fontsize=16,color="black",shape="box"];550 -> 561[label="",style="solid", color="black", weight=3]; 78.21/41.32 622[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];622 -> 628[label="",style="solid", color="black", weight=3]; 78.21/41.32 317 -> 337[label="",style="dashed", color="red", weight=0]; 78.21/41.32 317[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D0 xv23 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];317 -> 338[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 317 -> 339[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 317 -> 340[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 318 -> 341[label="",style="dashed", color="red", weight=0]; 78.21/41.32 318[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D0 xv26 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];318 -> 342[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 318 -> 343[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 318 -> 344[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 561[label="Pos (primDivNatS (Succ xv83) (Succ xv84))",fontsize=16,color="green",shape="box"];561 -> 568[label="",style="dashed", color="green", weight=3]; 78.21/41.32 628[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger (toInteger (showInt1D xv87 (Pos (Succ xv88))))))",fontsize=16,color="black",shape="box"];628 -> 635[label="",style="solid", color="black", weight=3]; 78.21/41.32 338[label="xv22",fontsize=16,color="green",shape="box"];339[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];340[label="xv23",fontsize=16,color="green",shape="box"];337[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="triangle"];337 -> 359[label="",style="solid", color="black", weight=3]; 78.21/41.32 342[label="xv25",fontsize=16,color="green",shape="box"];343[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];344[label="xv26",fontsize=16,color="green",shape="box"];341[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="triangle"];341 -> 360[label="",style="solid", color="black", weight=3]; 78.21/41.32 568[label="primDivNatS (Succ xv83) (Succ xv84)",fontsize=16,color="black",shape="triangle"];568 -> 623[label="",style="solid", color="black", weight=3]; 78.21/41.32 635[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger (Integer (showInt1D xv87 (Pos (Succ xv88))))))",fontsize=16,color="black",shape="box"];635 -> 644[label="",style="solid", color="black", weight=3]; 78.21/41.32 359[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (primQrmInt (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="box"];359 -> 371[label="",style="solid", color="black", weight=3]; 78.21/41.32 360[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (primQrmInt (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="box"];360 -> 372[label="",style="solid", color="black", weight=3]; 78.21/41.32 623[label="primDivNatS0 xv83 xv84 (primGEqNatS xv83 xv84)",fontsize=16,color="burlywood",shape="box"];1463[label="xv83/Succ xv830",fontsize=10,color="white",style="solid",shape="box"];623 -> 1463[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1463 -> 629[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1464[label="xv83/Zero",fontsize=10,color="white",style="solid",shape="box"];623 -> 1464[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1464 -> 630[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 644[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D xv87 (Pos (Succ xv88))))",fontsize=16,color="black",shape="box"];644 -> 651[label="",style="solid", color="black", weight=3]; 78.21/41.32 371[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (primQuotInt (Pos Zero) (fromInt (Pos (Succ xv60))),primRemInt (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="box"];371 -> 385[label="",style="solid", color="black", weight=3]; 78.21/41.32 372[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (primQuotInt (Neg Zero) (fromInt (Pos (Succ xv64))),primRemInt (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="box"];372 -> 386[label="",style="solid", color="black", weight=3]; 78.21/41.32 629[label="primDivNatS0 (Succ xv830) xv84 (primGEqNatS (Succ xv830) xv84)",fontsize=16,color="burlywood",shape="box"];1465[label="xv84/Succ xv840",fontsize=10,color="white",style="solid",shape="box"];629 -> 1465[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1465 -> 636[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1466[label="xv84/Zero",fontsize=10,color="white",style="solid",shape="box"];629 -> 1466[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1466 -> 637[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 630[label="primDivNatS0 Zero xv84 (primGEqNatS Zero xv84)",fontsize=16,color="burlywood",shape="box"];1467[label="xv84/Succ xv840",fontsize=10,color="white",style="solid",shape="box"];630 -> 1467[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1467 -> 638[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1468[label="xv84/Zero",fontsize=10,color="white",style="solid",shape="box"];630 -> 1468[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1468 -> 639[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 651[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D0 xv87 (Pos (Succ xv88)) (showInt1Vu76 xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];651 -> 659[label="",style="solid", color="black", weight=3]; 78.21/41.32 385[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (primRemInt (Pos Zero) (fromInt (Pos (Succ xv60)))))",fontsize=16,color="black",shape="box"];385 -> 401[label="",style="solid", color="black", weight=3]; 78.21/41.32 386[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (primRemInt (Neg Zero) (fromInt (Pos (Succ xv64)))))",fontsize=16,color="black",shape="box"];386 -> 402[label="",style="solid", color="black", weight=3]; 78.21/41.32 636[label="primDivNatS0 (Succ xv830) (Succ xv840) (primGEqNatS (Succ xv830) (Succ xv840))",fontsize=16,color="black",shape="box"];636 -> 645[label="",style="solid", color="black", weight=3]; 78.21/41.32 637[label="primDivNatS0 (Succ xv830) Zero (primGEqNatS (Succ xv830) Zero)",fontsize=16,color="black",shape="box"];637 -> 646[label="",style="solid", color="black", weight=3]; 78.21/41.32 638[label="primDivNatS0 Zero (Succ xv840) (primGEqNatS Zero (Succ xv840))",fontsize=16,color="black",shape="box"];638 -> 647[label="",style="solid", color="black", weight=3]; 78.21/41.32 639[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];639 -> 648[label="",style="solid", color="black", weight=3]; 78.21/41.32 659 -> 668[label="",style="dashed", color="red", weight=0]; 78.21/41.32 659[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D0 xv87 (Pos (Succ xv88)) (quotRem (Pos (Succ xv88)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];659 -> 669[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 659 -> 670[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 659 -> 671[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 659 -> 672[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 401[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (primRemInt (Pos Zero) (Pos (Succ xv60))))",fontsize=16,color="black",shape="box"];401 -> 421[label="",style="solid", color="black", weight=3]; 78.21/41.32 402[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (primRemInt (Neg Zero) (Pos (Succ xv64))))",fontsize=16,color="black",shape="box"];402 -> 422[label="",style="solid", color="black", weight=3]; 78.21/41.32 645 -> 917[label="",style="dashed", color="red", weight=0]; 78.21/41.32 645[label="primDivNatS0 (Succ xv830) (Succ xv840) (primGEqNatS xv830 xv840)",fontsize=16,color="magenta"];645 -> 918[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 645 -> 919[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 645 -> 920[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 645 -> 921[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 646[label="primDivNatS0 (Succ xv830) Zero True",fontsize=16,color="black",shape="box"];646 -> 654[label="",style="solid", color="black", weight=3]; 78.21/41.32 647[label="primDivNatS0 Zero (Succ xv840) False",fontsize=16,color="black",shape="box"];647 -> 655[label="",style="solid", color="black", weight=3]; 78.21/41.32 648[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];648 -> 656[label="",style="solid", color="black", weight=3]; 78.21/41.32 669[label="xv87",fontsize=16,color="green",shape="box"];670[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];671[label="xv88",fontsize=16,color="green",shape="box"];672[label="xv86",fontsize=16,color="green",shape="box"];668[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (quotRem (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="triangle"];668 -> 676[label="",style="solid", color="black", weight=3]; 78.21/41.32 421[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (Pos (primModNatS Zero (Succ xv60))))",fontsize=16,color="black",shape="box"];421 -> 444[label="",style="solid", color="black", weight=3]; 78.21/41.32 422[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (Neg (primModNatS Zero (Succ xv64))))",fontsize=16,color="black",shape="box"];422 -> 445[label="",style="solid", color="black", weight=3]; 78.21/41.32 918[label="xv840",fontsize=16,color="green",shape="box"];919[label="xv840",fontsize=16,color="green",shape="box"];920[label="xv830",fontsize=16,color="green",shape="box"];921[label="xv830",fontsize=16,color="green",shape="box"];917[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS xv127 xv128)",fontsize=16,color="burlywood",shape="triangle"];1469[label="xv127/Succ xv1270",fontsize=10,color="white",style="solid",shape="box"];917 -> 1469[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1469 -> 950[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1470[label="xv127/Zero",fontsize=10,color="white",style="solid",shape="box"];917 -> 1470[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1470 -> 951[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 654[label="Succ (primDivNatS (primMinusNatS (Succ xv830) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];654 -> 664[label="",style="dashed", color="green", weight=3]; 78.21/41.32 655[label="Zero",fontsize=16,color="green",shape="box"];656[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];656 -> 665[label="",style="dashed", color="green", weight=3]; 78.21/41.32 676[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (primQrmInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="box"];676 -> 685[label="",style="solid", color="black", weight=3]; 78.21/41.32 444[label="primIntToChar (Pos (primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60))))",fontsize=16,color="black",shape="box"];444 -> 467[label="",style="solid", color="black", weight=3]; 78.21/41.32 445[label="primIntToChar (primMinusNat (Succ xv62) (primModNatS Zero (Succ xv64)))",fontsize=16,color="black",shape="box"];445 -> 468[label="",style="solid", color="black", weight=3]; 78.21/41.32 950[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) xv128)",fontsize=16,color="burlywood",shape="box"];1471[label="xv128/Succ xv1280",fontsize=10,color="white",style="solid",shape="box"];950 -> 1471[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1471 -> 963[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1472[label="xv128/Zero",fontsize=10,color="white",style="solid",shape="box"];950 -> 1472[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1472 -> 964[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 951[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero xv128)",fontsize=16,color="burlywood",shape="box"];1473[label="xv128/Succ xv1280",fontsize=10,color="white",style="solid",shape="box"];951 -> 1473[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1473 -> 965[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1474[label="xv128/Zero",fontsize=10,color="white",style="solid",shape="box"];951 -> 1474[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1474 -> 966[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 664 -> 1379[label="",style="dashed", color="red", weight=0]; 78.21/41.32 664[label="primDivNatS (primMinusNatS (Succ xv830) Zero) (Succ Zero)",fontsize=16,color="magenta"];664 -> 1380[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 664 -> 1381[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 664 -> 1382[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 665 -> 1379[label="",style="dashed", color="red", weight=0]; 78.21/41.32 665[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];665 -> 1383[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 665 -> 1384[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 665 -> 1385[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 685 -> 711[label="",style="dashed", color="red", weight=0]; 78.21/41.32 685[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (primQuotInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))),primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="magenta"];685 -> 712[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 467[label="Char (primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60)))",fontsize=16,color="green",shape="box"];467 -> 502[label="",style="dashed", color="green", weight=3]; 78.21/41.32 468[label="primIntToChar (primMinusNat (Succ xv62) Zero)",fontsize=16,color="black",shape="box"];468 -> 503[label="",style="solid", color="black", weight=3]; 78.21/41.32 963[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) (Succ xv1280))",fontsize=16,color="black",shape="box"];963 -> 976[label="",style="solid", color="black", weight=3]; 78.21/41.32 964[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) Zero)",fontsize=16,color="black",shape="box"];964 -> 977[label="",style="solid", color="black", weight=3]; 78.21/41.32 965[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero (Succ xv1280))",fontsize=16,color="black",shape="box"];965 -> 978[label="",style="solid", color="black", weight=3]; 78.21/41.32 966[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];966 -> 979[label="",style="solid", color="black", weight=3]; 78.21/41.32 1380[label="Zero",fontsize=16,color="green",shape="box"];1381[label="Succ xv830",fontsize=16,color="green",shape="box"];1382[label="Zero",fontsize=16,color="green",shape="box"];1379[label="primDivNatS (primMinusNatS xv168 xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="triangle"];1475[label="xv168/Succ xv1680",fontsize=10,color="white",style="solid",shape="box"];1379 -> 1475[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1475 -> 1413[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1476[label="xv168/Zero",fontsize=10,color="white",style="solid",shape="box"];1379 -> 1476[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1476 -> 1414[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1383[label="Zero",fontsize=16,color="green",shape="box"];1384[label="Zero",fontsize=16,color="green",shape="box"];1385[label="Zero",fontsize=16,color="green",shape="box"];712 -> 532[label="",style="dashed", color="red", weight=0]; 78.21/41.32 712[label="primQuotInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103)))",fontsize=16,color="magenta"];712 -> 714[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 712 -> 715[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 711[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (xv108,primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="triangle"];711 -> 716[label="",style="solid", color="black", weight=3]; 78.21/41.32 502[label="primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60))",fontsize=16,color="black",shape="triangle"];502 -> 522[label="",style="solid", color="black", weight=3]; 78.21/41.32 503[label="primIntToChar (Pos (Succ xv62))",fontsize=16,color="black",shape="box"];503 -> 523[label="",style="solid", color="black", weight=3]; 78.21/41.32 976 -> 917[label="",style="dashed", color="red", weight=0]; 78.21/41.32 976[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS xv1270 xv1280)",fontsize=16,color="magenta"];976 -> 988[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 976 -> 989[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 977[label="primDivNatS0 (Succ xv125) (Succ xv126) True",fontsize=16,color="black",shape="triangle"];977 -> 990[label="",style="solid", color="black", weight=3]; 78.21/41.32 978[label="primDivNatS0 (Succ xv125) (Succ xv126) False",fontsize=16,color="black",shape="box"];978 -> 991[label="",style="solid", color="black", weight=3]; 78.21/41.32 979 -> 977[label="",style="dashed", color="red", weight=0]; 78.21/41.32 979[label="primDivNatS0 (Succ xv125) (Succ xv126) True",fontsize=16,color="magenta"];1413[label="primDivNatS (primMinusNatS (Succ xv1680) xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="box"];1477[label="xv169/Succ xv1690",fontsize=10,color="white",style="solid",shape="box"];1413 -> 1477[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1477 -> 1417[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1478[label="xv169/Zero",fontsize=10,color="white",style="solid",shape="box"];1413 -> 1478[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1478 -> 1418[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1414[label="primDivNatS (primMinusNatS Zero xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="box"];1479[label="xv169/Succ xv1690",fontsize=10,color="white",style="solid",shape="box"];1414 -> 1479[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1479 -> 1419[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1480[label="xv169/Zero",fontsize=10,color="white",style="solid",shape="box"];1414 -> 1480[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1480 -> 1420[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 714[label="xv103",fontsize=16,color="green",shape="box"];715[label="xv102",fontsize=16,color="green",shape="box"];716[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103)))))",fontsize=16,color="black",shape="box"];716 -> 726[label="",style="solid", color="black", weight=3]; 78.21/41.32 522[label="primPlusNat (Succ xv58) Zero",fontsize=16,color="black",shape="box"];522 -> 551[label="",style="solid", color="black", weight=3]; 78.21/41.32 523[label="Char (Succ xv62)",fontsize=16,color="green",shape="box"];988[label="xv1280",fontsize=16,color="green",shape="box"];989[label="xv1270",fontsize=16,color="green",shape="box"];990[label="Succ (primDivNatS (primMinusNatS (Succ xv125) (Succ xv126)) (Succ (Succ xv126)))",fontsize=16,color="green",shape="box"];990 -> 1002[label="",style="dashed", color="green", weight=3]; 78.21/41.32 991[label="Zero",fontsize=16,color="green",shape="box"];1417[label="primDivNatS (primMinusNatS (Succ xv1680) (Succ xv1690)) (Succ xv170)",fontsize=16,color="black",shape="box"];1417 -> 1425[label="",style="solid", color="black", weight=3]; 78.21/41.32 1418[label="primDivNatS (primMinusNatS (Succ xv1680) Zero) (Succ xv170)",fontsize=16,color="black",shape="box"];1418 -> 1426[label="",style="solid", color="black", weight=3]; 78.21/41.32 1419[label="primDivNatS (primMinusNatS Zero (Succ xv1690)) (Succ xv170)",fontsize=16,color="black",shape="box"];1419 -> 1427[label="",style="solid", color="black", weight=3]; 78.21/41.32 1420[label="primDivNatS (primMinusNatS Zero Zero) (Succ xv170)",fontsize=16,color="black",shape="box"];1420 -> 1428[label="",style="solid", color="black", weight=3]; 78.21/41.32 726[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (primRemInt (Pos (Succ xv102)) (Pos (Succ xv103))))",fontsize=16,color="black",shape="box"];726 -> 733[label="",style="solid", color="black", weight=3]; 78.21/41.32 551[label="Succ xv58",fontsize=16,color="green",shape="box"];1002 -> 1379[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1002[label="primDivNatS (primMinusNatS (Succ xv125) (Succ xv126)) (Succ (Succ xv126))",fontsize=16,color="magenta"];1002 -> 1386[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1002 -> 1387[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1002 -> 1388[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1425 -> 1379[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1425[label="primDivNatS (primMinusNatS xv1680 xv1690) (Succ xv170)",fontsize=16,color="magenta"];1425 -> 1433[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1425 -> 1434[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1426 -> 568[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1426[label="primDivNatS (Succ xv1680) (Succ xv170)",fontsize=16,color="magenta"];1426 -> 1435[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1426 -> 1436[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1427[label="primDivNatS Zero (Succ xv170)",fontsize=16,color="black",shape="triangle"];1427 -> 1437[label="",style="solid", color="black", weight=3]; 78.21/41.32 1428 -> 1427[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1428[label="primDivNatS Zero (Succ xv170)",fontsize=16,color="magenta"];733[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (Pos (primModNatS (Succ xv102) (Succ xv103))))",fontsize=16,color="black",shape="box"];733 -> 741[label="",style="solid", color="black", weight=3]; 78.21/41.32 1386[label="Succ xv126",fontsize=16,color="green",shape="box"];1387[label="Succ xv125",fontsize=16,color="green",shape="box"];1388[label="Succ xv126",fontsize=16,color="green",shape="box"];1433[label="xv1680",fontsize=16,color="green",shape="box"];1434[label="xv1690",fontsize=16,color="green",shape="box"];1435[label="xv170",fontsize=16,color="green",shape="box"];1436[label="xv1680",fontsize=16,color="green",shape="box"];1437[label="Zero",fontsize=16,color="green",shape="box"];741[label="primIntToChar (Pos (primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103))))",fontsize=16,color="black",shape="box"];741 -> 751[label="",style="solid", color="black", weight=3]; 78.21/41.32 751[label="Char (primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103)))",fontsize=16,color="green",shape="box"];751 -> 758[label="",style="dashed", color="green", weight=3]; 78.21/41.32 758[label="primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103))",fontsize=16,color="black",shape="triangle"];758 -> 766[label="",style="solid", color="black", weight=3]; 78.21/41.32 766[label="primPlusNat (Succ xv100) (primModNatS0 xv102 xv103 (primGEqNatS xv102 xv103))",fontsize=16,color="burlywood",shape="box"];1481[label="xv102/Succ xv1020",fontsize=10,color="white",style="solid",shape="box"];766 -> 1481[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1481 -> 775[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1482[label="xv102/Zero",fontsize=10,color="white",style="solid",shape="box"];766 -> 1482[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1482 -> 776[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 775[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) xv103 (primGEqNatS (Succ xv1020) xv103))",fontsize=16,color="burlywood",shape="box"];1483[label="xv103/Succ xv1030",fontsize=10,color="white",style="solid",shape="box"];775 -> 1483[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1483 -> 786[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1484[label="xv103/Zero",fontsize=10,color="white",style="solid",shape="box"];775 -> 1484[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1484 -> 787[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 776[label="primPlusNat (Succ xv100) (primModNatS0 Zero xv103 (primGEqNatS Zero xv103))",fontsize=16,color="burlywood",shape="box"];1485[label="xv103/Succ xv1030",fontsize=10,color="white",style="solid",shape="box"];776 -> 1485[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1485 -> 788[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1486[label="xv103/Zero",fontsize=10,color="white",style="solid",shape="box"];776 -> 1486[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1486 -> 789[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 786[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) (Succ xv1030) (primGEqNatS (Succ xv1020) (Succ xv1030)))",fontsize=16,color="black",shape="box"];786 -> 797[label="",style="solid", color="black", weight=3]; 78.21/41.32 787[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) Zero (primGEqNatS (Succ xv1020) Zero))",fontsize=16,color="black",shape="box"];787 -> 798[label="",style="solid", color="black", weight=3]; 78.21/41.32 788[label="primPlusNat (Succ xv100) (primModNatS0 Zero (Succ xv1030) (primGEqNatS Zero (Succ xv1030)))",fontsize=16,color="black",shape="box"];788 -> 799[label="",style="solid", color="black", weight=3]; 78.21/41.32 789[label="primPlusNat (Succ xv100) (primModNatS0 Zero Zero (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];789 -> 800[label="",style="solid", color="black", weight=3]; 78.21/41.32 797[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) (Succ xv1030) (primGEqNatS xv1020 xv1030))",fontsize=16,color="burlywood",shape="box"];1487[label="xv1020/Succ xv10200",fontsize=10,color="white",style="solid",shape="box"];797 -> 1487[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1487 -> 809[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1488[label="xv1020/Zero",fontsize=10,color="white",style="solid",shape="box"];797 -> 1488[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1488 -> 810[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 798[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) Zero True)",fontsize=16,color="black",shape="box"];798 -> 811[label="",style="solid", color="black", weight=3]; 78.21/41.32 799[label="primPlusNat (Succ xv100) (primModNatS0 Zero (Succ xv1030) False)",fontsize=16,color="black",shape="box"];799 -> 812[label="",style="solid", color="black", weight=3]; 78.21/41.32 800[label="primPlusNat (Succ xv100) (primModNatS0 Zero Zero True)",fontsize=16,color="black",shape="box"];800 -> 813[label="",style="solid", color="black", weight=3]; 78.21/41.32 809[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ xv1030) (primGEqNatS (Succ xv10200) xv1030))",fontsize=16,color="burlywood",shape="box"];1489[label="xv1030/Succ xv10300",fontsize=10,color="white",style="solid",shape="box"];809 -> 1489[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1489 -> 822[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1490[label="xv1030/Zero",fontsize=10,color="white",style="solid",shape="box"];809 -> 1490[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1490 -> 823[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 810[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ xv1030) (primGEqNatS Zero xv1030))",fontsize=16,color="burlywood",shape="box"];1491[label="xv1030/Succ xv10300",fontsize=10,color="white",style="solid",shape="box"];810 -> 1491[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1491 -> 824[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1492[label="xv1030/Zero",fontsize=10,color="white",style="solid",shape="box"];810 -> 1492[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1492 -> 825[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 811[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv1020) Zero) (Succ Zero))",fontsize=16,color="black",shape="box"];811 -> 826[label="",style="solid", color="black", weight=3]; 78.21/41.32 812 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 812[label="primPlusNat (Succ xv100) (Succ Zero)",fontsize=16,color="magenta"];812 -> 1098[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 812 -> 1099[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 813[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="black",shape="box"];813 -> 828[label="",style="solid", color="black", weight=3]; 78.21/41.32 822[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ (Succ xv10300)) (primGEqNatS (Succ xv10200) (Succ xv10300)))",fontsize=16,color="black",shape="box"];822 -> 846[label="",style="solid", color="black", weight=3]; 78.21/41.32 823[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ Zero) (primGEqNatS (Succ xv10200) Zero))",fontsize=16,color="black",shape="box"];823 -> 847[label="",style="solid", color="black", weight=3]; 78.21/41.32 824[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ (Succ xv10300)) (primGEqNatS Zero (Succ xv10300)))",fontsize=16,color="black",shape="box"];824 -> 848[label="",style="solid", color="black", weight=3]; 78.21/41.32 825[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ Zero) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];825 -> 849[label="",style="solid", color="black", weight=3]; 78.21/41.32 826 -> 758[label="",style="dashed", color="red", weight=0]; 78.21/41.32 826[label="primPlusNat (Succ xv100) (primModNatS (Succ xv1020) (Succ Zero))",fontsize=16,color="magenta"];826 -> 850[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 826 -> 851[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1098[label="xv100",fontsize=16,color="green",shape="box"];1099[label="Zero",fontsize=16,color="green",shape="box"];1097[label="primPlusNat (Succ xv133) (Succ xv134)",fontsize=16,color="black",shape="triangle"];1097 -> 1114[label="",style="solid", color="black", weight=3]; 78.21/41.32 828 -> 502[label="",style="dashed", color="red", weight=0]; 78.21/41.32 828[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ Zero))",fontsize=16,color="magenta"];828 -> 853[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 828 -> 854[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 846[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ (Succ xv10300)) (primGEqNatS xv10200 xv10300))",fontsize=16,color="burlywood",shape="box"];1493[label="xv10200/Succ xv102000",fontsize=10,color="white",style="solid",shape="box"];846 -> 1493[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1493 -> 884[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1494[label="xv10200/Zero",fontsize=10,color="white",style="solid",shape="box"];846 -> 1494[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1494 -> 885[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 847[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ Zero) True)",fontsize=16,color="black",shape="box"];847 -> 886[label="",style="solid", color="black", weight=3]; 78.21/41.32 848[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ (Succ xv10300)) False)",fontsize=16,color="black",shape="box"];848 -> 887[label="",style="solid", color="black", weight=3]; 78.21/41.32 849[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ Zero) True)",fontsize=16,color="black",shape="box"];849 -> 888[label="",style="solid", color="black", weight=3]; 78.21/41.32 850[label="Zero",fontsize=16,color="green",shape="box"];851[label="xv1020",fontsize=16,color="green",shape="box"];1114[label="Succ (Succ (primPlusNat xv133 xv134))",fontsize=16,color="green",shape="box"];1114 -> 1128[label="",style="dashed", color="green", weight=3]; 78.21/41.32 853[label="xv100",fontsize=16,color="green",shape="box"];854[label="Zero",fontsize=16,color="green",shape="box"];884[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ xv10300)) (primGEqNatS (Succ xv102000) xv10300))",fontsize=16,color="burlywood",shape="box"];1495[label="xv10300/Succ xv103000",fontsize=10,color="white",style="solid",shape="box"];884 -> 1495[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1495 -> 896[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1496[label="xv10300/Zero",fontsize=10,color="white",style="solid",shape="box"];884 -> 1496[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1496 -> 897[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 885[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ xv10300)) (primGEqNatS Zero xv10300))",fontsize=16,color="burlywood",shape="box"];1497[label="xv10300/Succ xv103000",fontsize=10,color="white",style="solid",shape="box"];885 -> 1497[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1497 -> 898[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1498[label="xv10300/Zero",fontsize=10,color="white",style="solid",shape="box"];885 -> 1498[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1498 -> 899[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 886[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv10200)) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];886 -> 900[label="",style="solid", color="black", weight=3]; 78.21/41.32 887 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 887[label="primPlusNat (Succ xv100) (Succ (Succ Zero))",fontsize=16,color="magenta"];887 -> 1100[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 887 -> 1101[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 888[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];888 -> 902[label="",style="solid", color="black", weight=3]; 78.21/41.32 1128[label="primPlusNat xv133 xv134",fontsize=16,color="burlywood",shape="triangle"];1499[label="xv133/Succ xv1330",fontsize=10,color="white",style="solid",shape="box"];1128 -> 1499[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1499 -> 1144[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1500[label="xv133/Zero",fontsize=10,color="white",style="solid",shape="box"];1128 -> 1500[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1500 -> 1145[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 896[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ (Succ xv103000))) (primGEqNatS (Succ xv102000) (Succ xv103000)))",fontsize=16,color="black",shape="box"];896 -> 910[label="",style="solid", color="black", weight=3]; 78.21/41.32 897[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ Zero)) (primGEqNatS (Succ xv102000) Zero))",fontsize=16,color="black",shape="box"];897 -> 911[label="",style="solid", color="black", weight=3]; 78.21/41.32 898[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ (Succ xv103000))) (primGEqNatS Zero (Succ xv103000)))",fontsize=16,color="black",shape="box"];898 -> 912[label="",style="solid", color="black", weight=3]; 78.21/41.32 899[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ Zero)) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];899 -> 913[label="",style="solid", color="black", weight=3]; 78.21/41.32 900[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv10200) Zero) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];900 -> 914[label="",style="solid", color="black", weight=3]; 78.21/41.32 1100[label="xv100",fontsize=16,color="green",shape="box"];1101[label="Succ Zero",fontsize=16,color="green",shape="box"];902[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];902 -> 916[label="",style="solid", color="black", weight=3]; 78.21/41.32 1144[label="primPlusNat (Succ xv1330) xv134",fontsize=16,color="burlywood",shape="box"];1501[label="xv134/Succ xv1340",fontsize=10,color="white",style="solid",shape="box"];1144 -> 1501[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1501 -> 1164[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1502[label="xv134/Zero",fontsize=10,color="white",style="solid",shape="box"];1144 -> 1502[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1502 -> 1165[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1145[label="primPlusNat Zero xv134",fontsize=16,color="burlywood",shape="box"];1503[label="xv134/Succ xv1340",fontsize=10,color="white",style="solid",shape="box"];1145 -> 1503[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1503 -> 1166[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1504[label="xv134/Zero",fontsize=10,color="white",style="solid",shape="box"];1145 -> 1504[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1504 -> 1167[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 910[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ (Succ xv103000))) (primGEqNatS xv102000 xv103000))",fontsize=16,color="burlywood",shape="box"];1505[label="xv102000/Succ xv1020000",fontsize=10,color="white",style="solid",shape="box"];910 -> 1505[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1505 -> 952[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1506[label="xv102000/Zero",fontsize=10,color="white",style="solid",shape="box"];910 -> 1506[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1506 -> 953[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 911[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ Zero)) True)",fontsize=16,color="black",shape="box"];911 -> 954[label="",style="solid", color="black", weight=3]; 78.21/41.32 912[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ (Succ xv103000))) False)",fontsize=16,color="black",shape="box"];912 -> 955[label="",style="solid", color="black", weight=3]; 78.21/41.32 913[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ Zero)) True)",fontsize=16,color="black",shape="box"];913 -> 956[label="",style="solid", color="black", weight=3]; 78.21/41.32 914 -> 758[label="",style="dashed", color="red", weight=0]; 78.21/41.32 914[label="primPlusNat (Succ xv100) (primModNatS (Succ xv10200) (Succ (Succ Zero)))",fontsize=16,color="magenta"];914 -> 957[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 914 -> 958[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 916 -> 502[label="",style="dashed", color="red", weight=0]; 78.21/41.32 916[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ Zero)))",fontsize=16,color="magenta"];916 -> 961[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 916 -> 962[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1164[label="primPlusNat (Succ xv1330) (Succ xv1340)",fontsize=16,color="black",shape="box"];1164 -> 1179[label="",style="solid", color="black", weight=3]; 78.21/41.32 1165[label="primPlusNat (Succ xv1330) Zero",fontsize=16,color="black",shape="box"];1165 -> 1180[label="",style="solid", color="black", weight=3]; 78.21/41.32 1166[label="primPlusNat Zero (Succ xv1340)",fontsize=16,color="black",shape="box"];1166 -> 1181[label="",style="solid", color="black", weight=3]; 78.21/41.32 1167[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];1167 -> 1182[label="",style="solid", color="black", weight=3]; 78.21/41.32 952[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ xv103000))) (primGEqNatS (Succ xv1020000) xv103000))",fontsize=16,color="burlywood",shape="box"];1507[label="xv103000/Succ xv1030000",fontsize=10,color="white",style="solid",shape="box"];952 -> 1507[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1507 -> 967[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1508[label="xv103000/Zero",fontsize=10,color="white",style="solid",shape="box"];952 -> 1508[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1508 -> 968[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 953[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ xv103000))) (primGEqNatS Zero xv103000))",fontsize=16,color="burlywood",shape="box"];1509[label="xv103000/Succ xv1030000",fontsize=10,color="white",style="solid",shape="box"];953 -> 1509[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1509 -> 969[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1510[label="xv103000/Zero",fontsize=10,color="white",style="solid",shape="box"];953 -> 1510[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1510 -> 970[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 954[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv102000))) (Succ (Succ Zero))) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];954 -> 971[label="",style="solid", color="black", weight=3]; 78.21/41.32 955 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 955[label="primPlusNat (Succ xv100) (Succ (Succ (Succ Zero)))",fontsize=16,color="magenta"];955 -> 1102[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 955 -> 1103[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 956[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];956 -> 973[label="",style="solid", color="black", weight=3]; 78.21/41.32 957[label="Succ Zero",fontsize=16,color="green",shape="box"];958[label="xv10200",fontsize=16,color="green",shape="box"];961[label="xv100",fontsize=16,color="green",shape="box"];962[label="Succ Zero",fontsize=16,color="green",shape="box"];1179[label="Succ (Succ (primPlusNat xv1330 xv1340))",fontsize=16,color="green",shape="box"];1179 -> 1196[label="",style="dashed", color="green", weight=3]; 78.21/41.32 1180[label="Succ xv1330",fontsize=16,color="green",shape="box"];1181[label="Succ xv1340",fontsize=16,color="green",shape="box"];1182[label="Zero",fontsize=16,color="green",shape="box"];967[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS (Succ xv1020000) (Succ xv1030000)))",fontsize=16,color="black",shape="box"];967 -> 980[label="",style="solid", color="black", weight=3]; 78.21/41.32 968[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero))) (primGEqNatS (Succ xv1020000) Zero))",fontsize=16,color="black",shape="box"];968 -> 981[label="",style="solid", color="black", weight=3]; 78.21/41.32 969[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS Zero (Succ xv1030000)))",fontsize=16,color="black",shape="box"];969 -> 982[label="",style="solid", color="black", weight=3]; 78.21/41.32 970[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];970 -> 983[label="",style="solid", color="black", weight=3]; 78.21/41.32 971[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv102000)) (Succ Zero)) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];971 -> 984[label="",style="solid", color="black", weight=3]; 78.21/41.32 1102[label="xv100",fontsize=16,color="green",shape="box"];1103[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];973[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];973 -> 986[label="",style="solid", color="black", weight=3]; 78.21/41.32 1196 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1196[label="primPlusNat xv1330 xv1340",fontsize=16,color="magenta"];1196 -> 1212[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1196 -> 1213[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 980[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS xv1020000 xv1030000))",fontsize=16,color="burlywood",shape="box"];1511[label="xv1020000/Succ xv10200000",fontsize=10,color="white",style="solid",shape="box"];980 -> 1511[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1511 -> 992[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1512[label="xv1020000/Zero",fontsize=10,color="white",style="solid",shape="box"];980 -> 1512[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1512 -> 993[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 981[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero))) True)",fontsize=16,color="black",shape="box"];981 -> 994[label="",style="solid", color="black", weight=3]; 78.21/41.32 982[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ xv1030000)))) False)",fontsize=16,color="black",shape="box"];982 -> 995[label="",style="solid", color="black", weight=3]; 78.21/41.32 983[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero))) True)",fontsize=16,color="black",shape="box"];983 -> 996[label="",style="solid", color="black", weight=3]; 78.21/41.32 984[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv102000) Zero) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];984 -> 997[label="",style="solid", color="black", weight=3]; 78.21/41.32 986[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];986 -> 1000[label="",style="solid", color="black", weight=3]; 78.21/41.32 1212[label="xv1330",fontsize=16,color="green",shape="box"];1213[label="xv1340",fontsize=16,color="green",shape="box"];992[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS (Succ xv10200000) xv1030000))",fontsize=16,color="burlywood",shape="box"];1513[label="xv1030000/Succ xv10300000",fontsize=10,color="white",style="solid",shape="box"];992 -> 1513[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1513 -> 1003[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1514[label="xv1030000/Zero",fontsize=10,color="white",style="solid",shape="box"];992 -> 1514[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1514 -> 1004[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 993[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS Zero xv1030000))",fontsize=16,color="burlywood",shape="box"];1515[label="xv1030000/Succ xv10300000",fontsize=10,color="white",style="solid",shape="box"];993 -> 1515[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1515 -> 1005[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1516[label="xv1030000/Zero",fontsize=10,color="white",style="solid",shape="box"];993 -> 1516[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1516 -> 1006[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 994[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];994 -> 1007[label="",style="solid", color="black", weight=3]; 78.21/41.32 995 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 995[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];995 -> 1104[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 995 -> 1105[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 996[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];996 -> 1009[label="",style="solid", color="black", weight=3]; 78.21/41.32 997 -> 758[label="",style="dashed", color="red", weight=0]; 78.21/41.32 997[label="primPlusNat (Succ xv100) (primModNatS (Succ xv102000) (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];997 -> 1010[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 997 -> 1011[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1000 -> 502[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1000[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];1000 -> 1014[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1000 -> 1015[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1003[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS (Succ xv10200000) (Succ xv10300000)))",fontsize=16,color="black",shape="box"];1003 -> 1017[label="",style="solid", color="black", weight=3]; 78.21/41.32 1004[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero)))) (primGEqNatS (Succ xv10200000) Zero))",fontsize=16,color="black",shape="box"];1004 -> 1018[label="",style="solid", color="black", weight=3]; 78.21/41.32 1005[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS Zero (Succ xv10300000)))",fontsize=16,color="black",shape="box"];1005 -> 1019[label="",style="solid", color="black", weight=3]; 78.21/41.32 1006[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1006 -> 1020[label="",style="solid", color="black", weight=3]; 78.21/41.32 1007[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv1020000))) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1007 -> 1021[label="",style="solid", color="black", weight=3]; 78.21/41.32 1104[label="xv100",fontsize=16,color="green",shape="box"];1105[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1009[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1009 -> 1023[label="",style="solid", color="black", weight=3]; 78.21/41.32 1010[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];1011[label="xv102000",fontsize=16,color="green",shape="box"];1014[label="xv100",fontsize=16,color="green",shape="box"];1015[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];1017[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS xv10200000 xv10300000))",fontsize=16,color="burlywood",shape="box"];1517[label="xv10200000/Succ xv102000000",fontsize=10,color="white",style="solid",shape="box"];1017 -> 1517[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1517 -> 1027[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1518[label="xv10200000/Zero",fontsize=10,color="white",style="solid",shape="box"];1017 -> 1518[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1518 -> 1028[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1018[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero)))) True)",fontsize=16,color="black",shape="box"];1018 -> 1029[label="",style="solid", color="black", weight=3]; 78.21/41.32 1019[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) False)",fontsize=16,color="black",shape="box"];1019 -> 1030[label="",style="solid", color="black", weight=3]; 78.21/41.32 1020[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))) True)",fontsize=16,color="black",shape="box"];1020 -> 1031[label="",style="solid", color="black", weight=3]; 78.21/41.32 1021[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv1020000)) (Succ Zero)) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1021 -> 1032[label="",style="solid", color="black", weight=3]; 78.21/41.32 1023[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1023 -> 1035[label="",style="solid", color="black", weight=3]; 78.21/41.32 1027[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS (Succ xv102000000) xv10300000))",fontsize=16,color="burlywood",shape="box"];1519[label="xv10300000/Succ xv103000000",fontsize=10,color="white",style="solid",shape="box"];1027 -> 1519[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1519 -> 1041[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1520[label="xv10300000/Zero",fontsize=10,color="white",style="solid",shape="box"];1027 -> 1520[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1520 -> 1042[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1028[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS Zero xv10300000))",fontsize=16,color="burlywood",shape="box"];1521[label="xv10300000/Succ xv103000000",fontsize=10,color="white",style="solid",shape="box"];1028 -> 1521[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1521 -> 1043[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1522[label="xv10300000/Zero",fontsize=10,color="white",style="solid",shape="box"];1028 -> 1522[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1522 -> 1044[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1029[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1029 -> 1045[label="",style="solid", color="black", weight=3]; 78.21/41.32 1030 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1030[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1030 -> 1106[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1030 -> 1107[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1031[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1031 -> 1047[label="",style="solid", color="black", weight=3]; 78.21/41.32 1032[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv1020000) Zero) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1032 -> 1048[label="",style="solid", color="black", weight=3]; 78.21/41.32 1035[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1035 -> 1051[label="",style="solid", color="black", weight=3]; 78.21/41.32 1041[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS (Succ xv102000000) (Succ xv103000000)))",fontsize=16,color="black",shape="box"];1041 -> 1056[label="",style="solid", color="black", weight=3]; 78.21/41.32 1042[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ Zero))))) (primGEqNatS (Succ xv102000000) Zero))",fontsize=16,color="black",shape="box"];1042 -> 1057[label="",style="solid", color="black", weight=3]; 78.21/41.32 1043[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS Zero (Succ xv103000000)))",fontsize=16,color="black",shape="box"];1043 -> 1058[label="",style="solid", color="black", weight=3]; 78.21/41.32 1044[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1044 -> 1059[label="",style="solid", color="black", weight=3]; 78.21/41.32 1045[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ xv10200000)))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1045 -> 1060[label="",style="solid", color="black", weight=3]; 78.21/41.32 1106[label="xv100",fontsize=16,color="green",shape="box"];1107[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1047[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1047 -> 1062[label="",style="solid", color="black", weight=3]; 78.21/41.32 1048 -> 758[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1048[label="primPlusNat (Succ xv100) (primModNatS (Succ xv1020000) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1048 -> 1063[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1048 -> 1064[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1051 -> 502[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1051[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1051 -> 1066[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1051 -> 1067[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1056[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS xv102000000 xv103000000))",fontsize=16,color="burlywood",shape="box"];1523[label="xv102000000/Succ xv1020000000",fontsize=10,color="white",style="solid",shape="box"];1056 -> 1523[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1523 -> 1074[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1524[label="xv102000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1056 -> 1524[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1524 -> 1075[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1057 -> 1078[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1057[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="magenta"];1057 -> 1079[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1057 -> 1080[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1058[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) False)",fontsize=16,color="black",shape="box"];1058 -> 1077[label="",style="solid", color="black", weight=3]; 78.21/41.32 1059 -> 1078[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1059[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="magenta"];1059 -> 1081[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1059 -> 1082[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1060[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv10200000))) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1060 -> 1087[label="",style="solid", color="black", weight=3]; 78.21/41.32 1062[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1062 -> 1090[label="",style="solid", color="black", weight=3]; 78.21/41.32 1063[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1064[label="xv1020000",fontsize=16,color="green",shape="box"];1066[label="xv100",fontsize=16,color="green",shape="box"];1067[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1074[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS (Succ xv1020000000) xv103000000))",fontsize=16,color="burlywood",shape="box"];1525[label="xv103000000/Succ xv1030000000",fontsize=10,color="white",style="solid",shape="box"];1074 -> 1525[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1525 -> 1092[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1526[label="xv103000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1074 -> 1526[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1526 -> 1093[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1075[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS Zero xv103000000))",fontsize=16,color="burlywood",shape="box"];1527[label="xv103000000/Succ xv1030000000",fontsize=10,color="white",style="solid",shape="box"];1075 -> 1527[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1527 -> 1094[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1528[label="xv103000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1075 -> 1528[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1528 -> 1095[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1079[label="Succ (Succ (Succ (Succ (Succ xv102000000))))",fontsize=16,color="green",shape="box"];1080[label="xv100",fontsize=16,color="green",shape="box"];1078[label="primPlusNat (Succ xv130) (primModNatS0 (Succ xv131) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="black",shape="triangle"];1078 -> 1096[label="",style="solid", color="black", weight=3]; 78.21/41.32 1077 -> 1097[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1077[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1077 -> 1108[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1077 -> 1109[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1081[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1082[label="xv100",fontsize=16,color="green",shape="box"];1087[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv10200000)) (Succ Zero)) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1087 -> 1115[label="",style="solid", color="black", weight=3]; 78.21/41.32 1090[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1090 -> 1116[label="",style="solid", color="black", weight=3]; 78.21/41.32 1092[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS (Succ xv1020000000) (Succ xv1030000000)))",fontsize=16,color="black",shape="box"];1092 -> 1117[label="",style="solid", color="black", weight=3]; 78.21/41.32 1093[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (primGEqNatS (Succ xv1020000000) Zero))",fontsize=16,color="black",shape="box"];1093 -> 1118[label="",style="solid", color="black", weight=3]; 78.21/41.32 1094[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS Zero (Succ xv1030000000)))",fontsize=16,color="black",shape="box"];1094 -> 1119[label="",style="solid", color="black", weight=3]; 78.21/41.32 1095[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1095 -> 1120[label="",style="solid", color="black", weight=3]; 78.21/41.32 1096[label="primPlusNat (Succ xv130) (primModNatS (primMinusNatS (Succ xv131) (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="black",shape="box"];1096 -> 1121[label="",style="solid", color="black", weight=3]; 78.21/41.32 1108[label="xv100",fontsize=16,color="green",shape="box"];1109[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1115[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv10200000) Zero) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1115 -> 1129[label="",style="solid", color="black", weight=3]; 78.21/41.32 1116[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1116 -> 1130[label="",style="solid", color="black", weight=3]; 78.21/41.32 1117[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS xv1020000000 xv1030000000))",fontsize=16,color="burlywood",shape="box"];1529[label="xv1020000000/Succ xv10200000000",fontsize=10,color="white",style="solid",shape="box"];1117 -> 1529[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1529 -> 1131[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1530[label="xv1020000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1117 -> 1530[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1530 -> 1132[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1118 -> 1135[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1118[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1118 -> 1136[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1118 -> 1137[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1119[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) False)",fontsize=16,color="black",shape="box"];1119 -> 1134[label="",style="solid", color="black", weight=3]; 78.21/41.32 1120 -> 1135[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1120[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1120 -> 1138[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1120 -> 1139[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1121 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1121[label="primPlusNat (Succ xv130) (primModNatS (primMinusNatS xv131 (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1121 -> 1146[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1121 -> 1147[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1129 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1129[label="primPlusNat (Succ xv100) (primModNatS (Succ xv10200000) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1129 -> 1148[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1129 -> 1149[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1130 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1130[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1130 -> 1150[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1130 -> 1151[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1131[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS (Succ xv10200000000) xv1030000000))",fontsize=16,color="burlywood",shape="box"];1531[label="xv1030000000/Succ xv10300000000",fontsize=10,color="white",style="solid",shape="box"];1131 -> 1531[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1531 -> 1152[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1532[label="xv1030000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1131 -> 1532[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1532 -> 1153[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1132[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS Zero xv1030000000))",fontsize=16,color="burlywood",shape="box"];1533[label="xv1030000000/Succ xv10300000000",fontsize=10,color="white",style="solid",shape="box"];1132 -> 1533[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1533 -> 1154[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1534[label="xv1030000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1132 -> 1534[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1534 -> 1155[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1136[label="Succ (Succ (Succ (Succ (Succ (Succ xv1020000000)))))",fontsize=16,color="green",shape="box"];1137[label="xv100",fontsize=16,color="green",shape="box"];1135 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1135[label="primPlusNat (Succ xv136) (primModNatS0 (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1135 -> 1156[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1135 -> 1157[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1134 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1134[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1134 -> 1158[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1134 -> 1159[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1138[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1139[label="xv100",fontsize=16,color="green",shape="box"];1146[label="Succ xv130",fontsize=16,color="green",shape="box"];1147 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1147[label="primModNatS (primMinusNatS xv131 (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1147 -> 1283[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1147 -> 1284[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1147 -> 1285[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1148[label="Succ xv100",fontsize=16,color="green",shape="box"];1149 -> 1170[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1149[label="primModNatS (Succ xv10200000) (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1149 -> 1171[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1149 -> 1172[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1150[label="Succ xv100",fontsize=16,color="green",shape="box"];1151 -> 1216[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1151[label="primModNatS Zero (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1151 -> 1217[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1152[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS (Succ xv10200000000) (Succ xv10300000000)))",fontsize=16,color="black",shape="box"];1152 -> 1184[label="",style="solid", color="black", weight=3]; 78.21/41.32 1153[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (primGEqNatS (Succ xv10200000000) Zero))",fontsize=16,color="black",shape="box"];1153 -> 1185[label="",style="solid", color="black", weight=3]; 78.21/41.32 1154[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS Zero (Succ xv10300000000)))",fontsize=16,color="black",shape="box"];1154 -> 1186[label="",style="solid", color="black", weight=3]; 78.21/41.32 1155[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1155 -> 1187[label="",style="solid", color="black", weight=3]; 78.21/41.32 1156[label="Succ xv136",fontsize=16,color="green",shape="box"];1157[label="primModNatS0 (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True",fontsize=16,color="black",shape="box"];1157 -> 1188[label="",style="solid", color="black", weight=3]; 78.21/41.32 1158[label="Succ xv100",fontsize=16,color="green",shape="box"];1159[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1283[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1284[label="xv131",fontsize=16,color="green",shape="box"];1285[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1282[label="primModNatS (primMinusNatS xv151 xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="triangle"];1535[label="xv151/Succ xv1510",fontsize=10,color="white",style="solid",shape="box"];1282 -> 1535[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1535 -> 1308[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1536[label="xv151/Zero",fontsize=10,color="white",style="solid",shape="box"];1282 -> 1536[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1536 -> 1309[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1171[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1172[label="xv10200000",fontsize=16,color="green",shape="box"];1170[label="primModNatS (Succ xv139) (Succ xv140)",fontsize=16,color="black",shape="triangle"];1170 -> 1191[label="",style="solid", color="black", weight=3]; 78.21/41.32 1217[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1216[label="primModNatS Zero (Succ xv145)",fontsize=16,color="black",shape="triangle"];1216 -> 1225[label="",style="solid", color="black", weight=3]; 78.21/41.32 1184 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1184[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS xv10200000000 xv10300000000))",fontsize=16,color="magenta"];1184 -> 1197[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1184 -> 1198[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1185 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1185[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True)",fontsize=16,color="magenta"];1185 -> 1199[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1185 -> 1200[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1186 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1186[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) False)",fontsize=16,color="magenta"];1186 -> 1201[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1186 -> 1202[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1187 -> 1128[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1187[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True)",fontsize=16,color="magenta"];1187 -> 1203[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1187 -> 1204[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1188 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.32 1188[label="primModNatS (primMinusNatS (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1188 -> 1286[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1188 -> 1287[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1188 -> 1288[label="",style="dashed", color="magenta", weight=3]; 78.21/41.32 1308[label="primModNatS (primMinusNatS (Succ xv1510) xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="box"];1537[label="xv152/Succ xv1520",fontsize=10,color="white",style="solid",shape="box"];1308 -> 1537[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1537 -> 1325[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1538[label="xv152/Zero",fontsize=10,color="white",style="solid",shape="box"];1308 -> 1538[label="",style="solid", color="burlywood", weight=9]; 78.21/41.32 1538 -> 1326[label="",style="solid", color="burlywood", weight=3]; 78.21/41.32 1309[label="primModNatS (primMinusNatS Zero xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="box"];1539[label="xv152/Succ xv1520",fontsize=10,color="white",style="solid",shape="box"];1309 -> 1539[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1539 -> 1327[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1540[label="xv152/Zero",fontsize=10,color="white",style="solid",shape="box"];1309 -> 1540[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1540 -> 1328[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1191[label="primModNatS0 xv139 xv140 (primGEqNatS xv139 xv140)",fontsize=16,color="burlywood",shape="box"];1541[label="xv139/Succ xv1390",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1541[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1541 -> 1226[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1542[label="xv139/Zero",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1542[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1542 -> 1227[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1225[label="Zero",fontsize=16,color="green",shape="box"];1197[label="Succ xv100",fontsize=16,color="green",shape="box"];1198 -> 1362[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1198[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS xv10200000000 xv10300000000)",fontsize=16,color="magenta"];1198 -> 1363[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1198 -> 1364[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1198 -> 1365[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1198 -> 1366[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1199[label="Succ xv100",fontsize=16,color="green",shape="box"];1200 -> 1230[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1200[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="magenta"];1200 -> 1231[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1201[label="Succ xv100",fontsize=16,color="green",shape="box"];1202 -> 1237[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1202[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) False",fontsize=16,color="magenta"];1202 -> 1238[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1203[label="Succ xv100",fontsize=16,color="green",shape="box"];1204 -> 1230[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1204[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="magenta"];1204 -> 1232[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1286[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1287[label="Succ xv137",fontsize=16,color="green",shape="box"];1288[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1325[label="primModNatS (primMinusNatS (Succ xv1510) (Succ xv1520)) (Succ xv153)",fontsize=16,color="black",shape="box"];1325 -> 1336[label="",style="solid", color="black", weight=3]; 78.21/41.33 1326[label="primModNatS (primMinusNatS (Succ xv1510) Zero) (Succ xv153)",fontsize=16,color="black",shape="box"];1326 -> 1337[label="",style="solid", color="black", weight=3]; 78.21/41.33 1327[label="primModNatS (primMinusNatS Zero (Succ xv1520)) (Succ xv153)",fontsize=16,color="black",shape="box"];1327 -> 1338[label="",style="solid", color="black", weight=3]; 78.21/41.33 1328[label="primModNatS (primMinusNatS Zero Zero) (Succ xv153)",fontsize=16,color="black",shape="box"];1328 -> 1339[label="",style="solid", color="black", weight=3]; 78.21/41.33 1226[label="primModNatS0 (Succ xv1390) xv140 (primGEqNatS (Succ xv1390) xv140)",fontsize=16,color="burlywood",shape="box"];1543[label="xv140/Succ xv1400",fontsize=10,color="white",style="solid",shape="box"];1226 -> 1543[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1543 -> 1247[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1544[label="xv140/Zero",fontsize=10,color="white",style="solid",shape="box"];1226 -> 1544[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1544 -> 1248[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1227[label="primModNatS0 Zero xv140 (primGEqNatS Zero xv140)",fontsize=16,color="burlywood",shape="box"];1545[label="xv140/Succ xv1400",fontsize=10,color="white",style="solid",shape="box"];1227 -> 1545[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1545 -> 1249[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1546[label="xv140/Zero",fontsize=10,color="white",style="solid",shape="box"];1227 -> 1546[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1546 -> 1250[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1363[label="xv10300000000",fontsize=16,color="green",shape="box"];1364[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000))))))",fontsize=16,color="green",shape="box"];1365[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000))))))",fontsize=16,color="green",shape="box"];1366[label="xv10200000000",fontsize=16,color="green",shape="box"];1362[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS xv165 xv166)",fontsize=16,color="burlywood",shape="triangle"];1547[label="xv165/Succ xv1650",fontsize=10,color="white",style="solid",shape="box"];1362 -> 1547[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1547 -> 1415[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1548[label="xv165/Zero",fontsize=10,color="white",style="solid",shape="box"];1362 -> 1548[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1548 -> 1416[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1231[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000))))))",fontsize=16,color="green",shape="box"];1230[label="primModNatS0 (Succ xv147) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="black",shape="triangle"];1230 -> 1255[label="",style="solid", color="black", weight=3]; 78.21/41.33 1238[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000))))))",fontsize=16,color="green",shape="box"];1237[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ xv149) False",fontsize=16,color="black",shape="triangle"];1237 -> 1256[label="",style="solid", color="black", weight=3]; 78.21/41.33 1232[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1336 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1336[label="primModNatS (primMinusNatS xv1510 xv1520) (Succ xv153)",fontsize=16,color="magenta"];1336 -> 1346[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1336 -> 1347[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1337 -> 1170[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1337[label="primModNatS (Succ xv1510) (Succ xv153)",fontsize=16,color="magenta"];1337 -> 1348[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1337 -> 1349[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1338 -> 1216[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1338[label="primModNatS Zero (Succ xv153)",fontsize=16,color="magenta"];1338 -> 1350[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1339 -> 1216[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1339[label="primModNatS Zero (Succ xv153)",fontsize=16,color="magenta"];1339 -> 1351[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1247[label="primModNatS0 (Succ xv1390) (Succ xv1400) (primGEqNatS (Succ xv1390) (Succ xv1400))",fontsize=16,color="black",shape="box"];1247 -> 1268[label="",style="solid", color="black", weight=3]; 78.21/41.33 1248[label="primModNatS0 (Succ xv1390) Zero (primGEqNatS (Succ xv1390) Zero)",fontsize=16,color="black",shape="box"];1248 -> 1269[label="",style="solid", color="black", weight=3]; 78.21/41.33 1249[label="primModNatS0 Zero (Succ xv1400) (primGEqNatS Zero (Succ xv1400))",fontsize=16,color="black",shape="box"];1249 -> 1270[label="",style="solid", color="black", weight=3]; 78.21/41.33 1250[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1250 -> 1271[label="",style="solid", color="black", weight=3]; 78.21/41.33 1415[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) xv166)",fontsize=16,color="burlywood",shape="box"];1549[label="xv166/Succ xv1660",fontsize=10,color="white",style="solid",shape="box"];1415 -> 1549[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1549 -> 1421[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1550[label="xv166/Zero",fontsize=10,color="white",style="solid",shape="box"];1415 -> 1550[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1550 -> 1422[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1416[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero xv166)",fontsize=16,color="burlywood",shape="box"];1551[label="xv166/Succ xv1660",fontsize=10,color="white",style="solid",shape="box"];1416 -> 1551[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1551 -> 1423[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1552[label="xv166/Zero",fontsize=10,color="white",style="solid",shape="box"];1416 -> 1552[label="",style="solid", color="burlywood", weight=9]; 78.21/41.33 1552 -> 1424[label="",style="solid", color="burlywood", weight=3]; 78.21/41.33 1255 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1255[label="primModNatS (primMinusNatS (Succ xv147) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="magenta"];1255 -> 1298[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1255 -> 1299[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1255 -> 1300[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1256[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="green",shape="box"];1346[label="xv1520",fontsize=16,color="green",shape="box"];1347[label="xv1510",fontsize=16,color="green",shape="box"];1348[label="xv153",fontsize=16,color="green",shape="box"];1349[label="xv1510",fontsize=16,color="green",shape="box"];1350[label="xv153",fontsize=16,color="green",shape="box"];1351[label="xv153",fontsize=16,color="green",shape="box"];1268 -> 1362[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1268[label="primModNatS0 (Succ xv1390) (Succ xv1400) (primGEqNatS xv1390 xv1400)",fontsize=16,color="magenta"];1268 -> 1367[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1268 -> 1368[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1268 -> 1369[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1268 -> 1370[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1269[label="primModNatS0 (Succ xv1390) Zero True",fontsize=16,color="black",shape="box"];1269 -> 1312[label="",style="solid", color="black", weight=3]; 78.21/41.33 1270[label="primModNatS0 Zero (Succ xv1400) False",fontsize=16,color="black",shape="box"];1270 -> 1313[label="",style="solid", color="black", weight=3]; 78.21/41.33 1271[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];1271 -> 1314[label="",style="solid", color="black", weight=3]; 78.21/41.33 1421[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) (Succ xv1660))",fontsize=16,color="black",shape="box"];1421 -> 1429[label="",style="solid", color="black", weight=3]; 78.21/41.33 1422[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) Zero)",fontsize=16,color="black",shape="box"];1422 -> 1430[label="",style="solid", color="black", weight=3]; 78.21/41.33 1423[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero (Succ xv1660))",fontsize=16,color="black",shape="box"];1423 -> 1431[label="",style="solid", color="black", weight=3]; 78.21/41.33 1424[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1424 -> 1432[label="",style="solid", color="black", weight=3]; 78.21/41.33 1298[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1299[label="Succ xv147",fontsize=16,color="green",shape="box"];1300[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1367[label="xv1400",fontsize=16,color="green",shape="box"];1368[label="xv1390",fontsize=16,color="green",shape="box"];1369[label="xv1400",fontsize=16,color="green",shape="box"];1370[label="xv1390",fontsize=16,color="green",shape="box"];1312 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1312[label="primModNatS (primMinusNatS (Succ xv1390) Zero) (Succ Zero)",fontsize=16,color="magenta"];1312 -> 1356[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1312 -> 1357[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1312 -> 1358[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1313[label="Succ Zero",fontsize=16,color="green",shape="box"];1314 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1314[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];1314 -> 1359[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1314 -> 1360[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1314 -> 1361[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1429 -> 1362[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1429[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS xv1650 xv1660)",fontsize=16,color="magenta"];1429 -> 1438[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1429 -> 1439[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1430[label="primModNatS0 (Succ xv163) (Succ xv164) True",fontsize=16,color="black",shape="triangle"];1430 -> 1440[label="",style="solid", color="black", weight=3]; 78.21/41.33 1431[label="primModNatS0 (Succ xv163) (Succ xv164) False",fontsize=16,color="black",shape="box"];1431 -> 1441[label="",style="solid", color="black", weight=3]; 78.21/41.33 1432 -> 1430[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1432[label="primModNatS0 (Succ xv163) (Succ xv164) True",fontsize=16,color="magenta"];1356[label="Zero",fontsize=16,color="green",shape="box"];1357[label="Succ xv1390",fontsize=16,color="green",shape="box"];1358[label="Zero",fontsize=16,color="green",shape="box"];1359[label="Zero",fontsize=16,color="green",shape="box"];1360[label="Zero",fontsize=16,color="green",shape="box"];1361[label="Zero",fontsize=16,color="green",shape="box"];1438[label="xv1660",fontsize=16,color="green",shape="box"];1439[label="xv1650",fontsize=16,color="green",shape="box"];1440 -> 1282[label="",style="dashed", color="red", weight=0]; 78.21/41.33 1440[label="primModNatS (primMinusNatS (Succ xv163) (Succ xv164)) (Succ (Succ xv164))",fontsize=16,color="magenta"];1440 -> 1442[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1440 -> 1443[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1440 -> 1444[label="",style="dashed", color="magenta", weight=3]; 78.21/41.33 1441[label="Succ (Succ xv163)",fontsize=16,color="green",shape="box"];1442[label="Succ xv164",fontsize=16,color="green",shape="box"];1443[label="Succ xv163",fontsize=16,color="green",shape="box"];1444[label="Succ xv164",fontsize=16,color="green",shape="box"];} 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (14) 78.21/41.33 Complex Obligation (AND) 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (15) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_primDivNatS0(xv125, xv126, Zero, Zero) -> new_primDivNatS00(xv125, xv126) 78.21/41.33 new_primDivNatS00(xv125, xv126) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 new_primDivNatS(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS1(Succ(xv830), Zero) -> new_primDivNatS(Succ(xv830), Zero, Zero) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS0(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 new_primDivNatS1(Succ(xv830), Succ(xv840)) -> new_primDivNatS0(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS1(Zero, Zero) -> new_primDivNatS(Zero, Zero, Zero) 78.21/41.33 new_primDivNatS(Succ(xv1680), Zero, xv170) -> new_primDivNatS1(xv1680, xv170) 78.21/41.33 78.21/41.33 R is empty. 78.21/41.33 Q is empty. 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (16) DependencyGraphProof (EQUIVALENT) 78.21/41.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (17) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_primDivNatS00(xv125, xv126) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 new_primDivNatS(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS(Succ(xv1680), Zero, xv170) -> new_primDivNatS1(xv1680, xv170) 78.21/41.33 new_primDivNatS1(Succ(xv830), Zero) -> new_primDivNatS(Succ(xv830), Zero, Zero) 78.21/41.33 new_primDivNatS1(Succ(xv830), Succ(xv840)) -> new_primDivNatS0(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS0(xv125, xv126, Zero, Zero) -> new_primDivNatS00(xv125, xv126) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS0(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 78.21/41.33 R is empty. 78.21/41.33 Q is empty. 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (18) QDPOrderProof (EQUIVALENT) 78.21/41.33 We use the reduction pair processor [LPAR04,JAR06]. 78.21/41.33 78.21/41.33 78.21/41.33 The following pairs can be oriented strictly and are deleted. 78.21/41.33 78.21/41.33 new_primDivNatS(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS1(Succ(xv830), Zero) -> new_primDivNatS(Succ(xv830), Zero, Zero) 78.21/41.33 new_primDivNatS1(Succ(xv830), Succ(xv840)) -> new_primDivNatS0(xv830, xv840, xv830, xv840) 78.21/41.33 The remaining pairs can at least be oriented weakly. 78.21/41.33 Used ordering: Polynomial interpretation [POLO]: 78.21/41.33 78.21/41.33 POL(Succ(x_1)) = 1 + x_1 78.21/41.33 POL(Zero) = 0 78.21/41.33 POL(new_primDivNatS(x_1, x_2, x_3)) = x_1 78.21/41.33 POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1 78.21/41.33 POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1 78.21/41.33 POL(new_primDivNatS1(x_1, x_2)) = 1 + x_1 78.21/41.33 78.21/41.33 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 78.21/41.33 none 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (19) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_primDivNatS00(xv125, xv126) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 new_primDivNatS(Succ(xv1680), Zero, xv170) -> new_primDivNatS1(xv1680, xv170) 78.21/41.33 new_primDivNatS0(xv125, xv126, Zero, Zero) -> new_primDivNatS00(xv125, xv126) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS0(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS(Succ(xv125), Succ(xv126), Succ(xv126)) 78.21/41.33 78.21/41.33 R is empty. 78.21/41.33 Q is empty. 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (20) DependencyGraphProof (EQUIVALENT) 78.21/41.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (21) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_primDivNatS0(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS0(xv125, xv126, xv1270, xv1280) 78.21/41.33 78.21/41.33 R is empty. 78.21/41.33 Q is empty. 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (22) QDPSizeChangeProof (EQUIVALENT) 78.21/41.33 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. 78.21/41.33 78.21/41.33 From the DPs we obtained the following set of size-change graphs: 78.21/41.33 *new_primDivNatS0(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS0(xv125, xv126, xv1270, xv1280) 78.21/41.33 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (23) 78.21/41.33 YES 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (24) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt0(xv12, Zero, Zero) -> new_showInt(new_showInt1N'0(xv12, Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'0(xv12)) 78.21/41.33 new_showInt1ShowInt0(xv12, Succ(xv130), Succ(xv140)) -> new_showInt1ShowInt00(xv12, Succ(xv130), xv140, xv130, xv140) 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt02(xv12, xv130) -> new_showInt(new_showInt1N'0(xv12, Succ(xv130), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'(xv12, xv130)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt0(xv12, Succ(xv130), Zero) -> new_showInt(new_showInt1N'0(xv12, Succ(xv130), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'(xv12, xv130)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_showInt1R'(xv12, xv130) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Succ(xv130)), xv12) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'0(xv12) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Zero), xv12) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_showInt1R'0(x0) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_showInt1R'(x0, x1) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (25) DependencyGraphProof (EQUIVALENT) 78.21/41.33 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (26) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt0(xv12, Succ(xv130), Succ(xv140)) -> new_showInt1ShowInt00(xv12, Succ(xv130), xv140, xv130, xv140) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_showInt1R'(xv12, xv130) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Succ(xv130)), xv12) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'0(xv12) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Zero), xv12) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_showInt1R'0(x0) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_showInt1R'(x0, x1) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (27) TransformationProof (EQUIVALENT) 78.21/41.33 By instantiating [LPAR04] the rule new_showInt1ShowInt0(xv12, Succ(xv130), Succ(xv140)) -> new_showInt1ShowInt00(xv12, Succ(xv130), xv140, xv130, xv140) we obtained the following new rules [LPAR04]: 78.21/41.33 78.21/41.33 (new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (28) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_showInt1R'(xv12, xv130) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Succ(xv130)), xv12) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'0(xv12) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Zero), xv12) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_showInt1R'0(x0) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_showInt1R'(x0, x1) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (29) UsableRulesProof (EQUIVALENT) 78.21/41.33 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. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (30) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_showInt1R'0(x0) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_showInt1R'(x0, x1) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (31) QReductionProof (EQUIVALENT) 78.21/41.33 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.21/41.33 78.21/41.33 new_showInt1R'0(x0) 78.21/41.33 new_showInt1R'(x0, x1) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (32) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (33) TransformationProof (EQUIVALENT) 78.21/41.33 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) at position [0] we obtained the following new rules [LPAR04]: 78.21/41.33 78.21/41.33 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95))) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (34) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (35) TransformationProof (EQUIVALENT) 78.21/41.33 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) at position [0] we obtained the following new rules [LPAR04]: 78.21/41.33 78.21/41.33 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95))) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (36) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt1N'0(xv82, xv83, xv84) -> new_primQuotInt(xv83, xv84) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (37) UsableRulesProof (EQUIVALENT) 78.21/41.33 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. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (38) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (39) QReductionProof (EQUIVALENT) 78.21/41.33 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.21/41.33 78.21/41.33 new_showInt1N'0(x0, x1, x2) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (40) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (41) TransformationProof (EQUIVALENT) 78.21/41.33 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) at position [0] we obtained the following new rules [LPAR04]: 78.21/41.33 78.21/41.33 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95))) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (42) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.33 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.33 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.33 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.33 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.33 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.33 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.33 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.33 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.33 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.33 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.33 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.33 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.33 new_primModNatS4(xv145) -> Zero 78.21/41.33 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.33 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.33 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.33 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.33 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.33 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.33 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.33 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.33 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.33 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.33 new_primDivNatS4(xv170) -> Zero 78.21/41.33 78.21/41.33 The set Q consists of the following terms: 78.21/41.33 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.33 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primDivNatS2(Zero, Zero) 78.21/41.33 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS3(Zero, Zero, x0) 78.21/41.33 new_primModNatS3(Zero, Succ(x0)) 78.21/41.33 new_primPlusNat4(x0, x1) 78.21/41.33 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat2(Succ(x0), Zero) 78.21/41.33 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.33 new_primQuotInt(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Zero) 78.21/41.33 new_primModNatS03(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.33 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.33 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.33 new_primPlusNat3(x0, Zero, Zero) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.33 new_primDivNatS2(Succ(x0), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.33 new_primModNatS3(Zero, Zero) 78.21/41.33 new_primDivNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.33 new_primModNatS01(x0) 78.21/41.33 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.33 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.33 new_primModNatS2(Zero, Zero, x0) 78.21/41.33 new_primModNatS4(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.33 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.33 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.33 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.33 new_primModNatS04(x0) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primIntToChar(x0, x1, x2) 78.21/41.33 new_primPlusNat1(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.33 new_primDivNatS02(x0, x1) 78.21/41.33 new_primPlusNat5(x0, x1) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.33 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.33 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.33 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.33 new_primModNatS3(Succ(x0), Zero) 78.21/41.33 new_primPlusNat6(x0, x1) 78.21/41.33 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.33 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.33 new_showInt1R'1(x0, x1) 78.21/41.33 78.21/41.33 We have to consider all minimal (P,Q,R)-chains. 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (43) TransformationProof (EQUIVALENT) 78.21/41.33 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1R'1(xv94, xv95)) at position [0] we obtained the following new rules [LPAR04]: 78.21/41.33 78.21/41.33 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95))) 78.21/41.33 78.21/41.33 78.21/41.33 ---------------------------------------- 78.21/41.33 78.21/41.33 (44) 78.21/41.33 Obligation: 78.21/41.33 Q DP problem: 78.21/41.33 The TRS P consists of the following rules: 78.21/41.33 78.21/41.33 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.33 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.33 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.33 78.21/41.33 The TRS R consists of the following rules: 78.21/41.33 78.21/41.33 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.33 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.33 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.33 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.33 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.33 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.33 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.33 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS4(xv145) -> Zero 78.21/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS4(xv170) -> Zero 78.21/41.34 78.21/41.34 The set Q consists of the following terms: 78.21/41.34 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primDivNatS2(Zero, Zero) 78.21/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.34 new_primDivNatS3(Zero, Zero, x0) 78.21/41.34 new_primModNatS3(Zero, Succ(x0)) 78.21/41.34 new_primPlusNat4(x0, x1) 78.21/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat2(Succ(x0), Zero) 78.21/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.34 new_primQuotInt(x0, x1) 78.21/41.34 new_primPlusNat2(Zero, Zero) 78.21/41.34 new_primModNatS03(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.34 new_primPlusNat3(x0, Zero, Zero) 78.21/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS2(Succ(x0), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.34 new_primModNatS3(Zero, Zero) 78.21/41.34 new_primDivNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primModNatS01(x0) 78.21/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.34 new_primModNatS2(Zero, Zero, x0) 78.21/41.34 new_primModNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.34 new_primModNatS04(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primIntToChar(x0, x1, x2) 78.21/41.34 new_primPlusNat1(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.34 new_primDivNatS02(x0, x1) 78.21/41.34 new_primPlusNat5(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.34 new_primModNatS3(Succ(x0), Zero) 78.21/41.34 new_primPlusNat6(x0, x1) 78.21/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.34 new_showInt1R'1(x0, x1) 78.21/41.34 78.21/41.34 We have to consider all minimal (P,Q,R)-chains. 78.21/41.34 ---------------------------------------- 78.21/41.34 78.21/41.34 (45) TransformationProof (EQUIVALENT) 78.21/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) at position [1] we obtained the following new rules [LPAR04]: 78.21/41.34 78.21/41.34 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94))) 78.21/41.34 78.21/41.34 78.21/41.34 ---------------------------------------- 78.21/41.34 78.21/41.34 (46) 78.21/41.34 Obligation: 78.21/41.34 Q DP problem: 78.21/41.34 The TRS P consists of the following rules: 78.21/41.34 78.21/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) 78.21/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.21/41.34 78.21/41.34 The TRS R consists of the following rules: 78.21/41.34 78.21/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.34 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS4(xv145) -> Zero 78.21/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS4(xv170) -> Zero 78.21/41.34 78.21/41.34 The set Q consists of the following terms: 78.21/41.34 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primDivNatS2(Zero, Zero) 78.21/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.34 new_primDivNatS3(Zero, Zero, x0) 78.21/41.34 new_primModNatS3(Zero, Succ(x0)) 78.21/41.34 new_primPlusNat4(x0, x1) 78.21/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat2(Succ(x0), Zero) 78.21/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.34 new_primQuotInt(x0, x1) 78.21/41.34 new_primPlusNat2(Zero, Zero) 78.21/41.34 new_primModNatS03(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.34 new_primPlusNat3(x0, Zero, Zero) 78.21/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS2(Succ(x0), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.34 new_primModNatS3(Zero, Zero) 78.21/41.34 new_primDivNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primModNatS01(x0) 78.21/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.34 new_primModNatS2(Zero, Zero, x0) 78.21/41.34 new_primModNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.21/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.21/41.34 new_primModNatS04(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primIntToChar(x0, x1, x2) 78.21/41.34 new_primPlusNat1(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.21/41.34 new_primDivNatS02(x0, x1) 78.21/41.34 new_primPlusNat5(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.21/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.21/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.21/41.34 new_primModNatS3(Succ(x0), Zero) 78.21/41.34 new_primPlusNat6(x0, x1) 78.21/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.21/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.21/41.34 new_showInt1R'1(x0, x1) 78.21/41.34 78.21/41.34 We have to consider all minimal (P,Q,R)-chains. 78.21/41.34 ---------------------------------------- 78.21/41.34 78.21/41.34 (47) TransformationProof (EQUIVALENT) 78.21/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), new_showInt1R'1(xv94, xv95)) at position [1] we obtained the following new rules [LPAR04]: 78.21/41.34 78.21/41.34 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94))) 78.21/41.34 78.21/41.34 78.21/41.34 ---------------------------------------- 78.21/41.34 78.21/41.34 (48) 78.21/41.34 Obligation: 78.21/41.34 Q DP problem: 78.21/41.34 The TRS P consists of the following rules: 78.21/41.34 78.21/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.21/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.21/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.21/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.21/41.34 78.21/41.34 The TRS R consists of the following rules: 78.21/41.34 78.21/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.21/41.34 new_showInt1R'1(xv94, xv95) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94) 78.21/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.21/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.21/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.21/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.21/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.21/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.21/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.21/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.21/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.21/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.21/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.21/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.21/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.21/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.21/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.21/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.21/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.21/41.34 new_primModNatS4(xv145) -> Zero 78.21/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.21/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.21/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.21/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.21/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.21/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.21/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.21/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.21/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.21/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.21/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.21/41.34 new_primDivNatS4(xv170) -> Zero 78.21/41.34 78.21/41.34 The set Q consists of the following terms: 78.21/41.34 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.21/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.21/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primDivNatS2(Zero, Zero) 78.21/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.21/41.34 new_primDivNatS3(Zero, Zero, x0) 78.21/41.34 new_primModNatS3(Zero, Succ(x0)) 78.21/41.34 new_primPlusNat4(x0, x1) 78.21/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.21/41.34 new_primPlusNat2(Succ(x0), Zero) 78.21/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.21/41.34 new_primQuotInt(x0, x1) 78.21/41.34 new_primPlusNat2(Zero, Zero) 78.21/41.34 new_primModNatS03(x0, x1) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.21/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.21/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.21/41.34 new_primPlusNat3(x0, Zero, Zero) 78.21/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.21/41.34 new_primDivNatS2(Succ(x0), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.21/41.34 new_primModNatS3(Zero, Zero) 78.21/41.34 new_primDivNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.21/41.34 new_primModNatS01(x0) 78.21/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.21/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.21/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.21/41.34 new_primModNatS2(Zero, Zero, x0) 78.21/41.34 new_primModNatS4(x0) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.21/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.21/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.21/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.21/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 new_showInt1R'1(x0, x1) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (49) UsableRulesProof (EQUIVALENT) 78.58/41.34 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. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (50) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 new_showInt1R'1(x0, x1) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (51) QReductionProof (EQUIVALENT) 78.58/41.34 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.58/41.34 78.58/41.34 new_showInt1R'1(x0, x1) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (52) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (53) TransformationProof (EQUIVALENT) 78.58/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.34 78.58/41.34 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94))) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (54) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (55) TransformationProof (EQUIVALENT) 78.58/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.34 78.58/41.34 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94))) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (56) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar(xv86, xv87, xv88) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (57) UsableRulesProof (EQUIVALENT) 78.58/41.34 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. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (58) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (59) QReductionProof (EQUIVALENT) 78.58/41.34 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.58/41.34 78.58/41.34 new_primIntToChar(x0, x1, x2) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (60) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (61) TransformationProof (EQUIVALENT) 78.58/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.34 78.58/41.34 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94))) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (62) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (63) TransformationProof (EQUIVALENT) 78.58/41.34 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar1(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.34 78.58/41.34 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94))) 78.58/41.34 78.58/41.34 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (64) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.34 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.34 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.34 78.58/41.34 The TRS R consists of the following rules: 78.58/41.34 78.58/41.34 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.34 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.34 new_primIntToChar1(xv100, xv101, xv102, xv103) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103), xv103) 78.58/41.34 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.34 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.34 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.34 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.34 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.34 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.34 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.34 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.34 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.34 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.34 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.34 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.34 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.34 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.34 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.34 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.34 new_primModNatS4(xv145) -> Zero 78.58/41.34 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.34 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.34 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.34 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.34 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.34 new_primDivNatS4(xv170) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.34 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.34 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.34 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.34 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.34 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.34 78.58/41.34 The set Q consists of the following terms: 78.58/41.34 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.34 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primDivNatS2(Zero, Zero) 78.58/41.34 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS3(Zero, Zero, x0) 78.58/41.34 new_primModNatS3(Zero, Succ(x0)) 78.58/41.34 new_primPlusNat4(x0, x1) 78.58/41.34 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat2(Succ(x0), Zero) 78.58/41.34 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.34 new_primQuotInt(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Zero) 78.58/41.34 new_primModNatS03(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.34 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.34 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.34 new_primPlusNat3(x0, Zero, Zero) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.34 new_primDivNatS2(Succ(x0), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.34 new_primModNatS3(Zero, Zero) 78.58/41.34 new_primDivNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.34 new_primModNatS01(x0) 78.58/41.34 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.34 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.34 new_primModNatS2(Zero, Zero, x0) 78.58/41.34 new_primModNatS4(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.34 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.34 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.34 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.34 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.34 new_primModNatS04(x0) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.34 new_primPlusNat1(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.34 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.34 new_primDivNatS02(x0, x1) 78.58/41.34 new_primPlusNat5(x0, x1) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.34 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.34 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.34 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.34 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.34 new_primModNatS3(Succ(x0), Zero) 78.58/41.34 new_primPlusNat6(x0, x1) 78.58/41.34 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.34 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.34 78.58/41.34 We have to consider all minimal (P,Q,R)-chains. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (65) UsableRulesProof (EQUIVALENT) 78.58/41.34 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. 78.58/41.34 ---------------------------------------- 78.58/41.34 78.58/41.34 (66) 78.58/41.34 Obligation: 78.58/41.34 Q DP problem: 78.58/41.34 The TRS P consists of the following rules: 78.58/41.34 78.58/41.34 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.35 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (67) QReductionProof (EQUIVALENT) 78.58/41.35 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.58/41.35 78.58/41.35 new_primIntToChar1(x0, x1, x2, x3) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (68) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.35 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (69) TransformationProof (EQUIVALENT) 78.58/41.35 By rewriting [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.35 78.58/41.35 (new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)),new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94))) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (70) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.35 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (71) TransformationProof (EQUIVALENT) 78.58/41.35 By rewriting [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(new_primIntToChar0(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, new_primQuotInt(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), xv94)) at position [1,0] we obtained the following new rules [LPAR04]: 78.58/41.35 78.58/41.35 (new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)),new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94))) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (72) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primQuotInt(xv83, xv84) -> Pos(new_primDivNatS2(xv83, xv84)) 78.58/41.35 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103) -> Char(new_primPlusNat3(xv100, xv102, xv103)) 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(xv1020), Zero) -> new_primPlusNat3(xv100, xv1020, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero))) -> new_primPlusNat6(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Zero, Zero) -> new_primPlusNat6(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero)) -> new_primPlusNat6(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat3(xv100, xv10200, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primModNatS01(xv147) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primPlusNat6(xv58, xv60) -> Succ(xv58) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat1(xv136, xv137) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat4(xv130, xv131) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (73) UsableRulesProof (EQUIVALENT) 78.58/41.35 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. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (74) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (75) QReductionProof (EQUIVALENT) 78.58/41.35 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.58/41.35 78.58/41.35 new_primPlusNat4(x0, x1) 78.58/41.35 new_primQuotInt(x0, x1) 78.58/41.35 new_primModNatS01(x0) 78.58/41.35 new_primIntToChar0(x0, x1, x2, x3, x4) 78.58/41.35 new_primPlusNat1(x0, x1) 78.58/41.35 new_primPlusNat6(x0, x1) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (76) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (77) InductionCalculusProof (EQUIVALENT) 78.58/41.35 Note that final constraints are written in bold face. 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) the following chains were created: 78.58/41.35 *We consider the chain new_showInt(Pos(Succ(x6)), x7) -> new_showInt1ShowInt0(x7, x6, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1ShowInt0(x8, Succ(x9), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(x8, Succ(x9), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x9, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt0(x7, x6, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))=new_showInt1ShowInt0(x8, Succ(x9), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) ==> new_showInt(Pos(Succ(x6)), x7)_>=_new_showInt1ShowInt0(x7, x6, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt(Pos(Succ(Succ(x9))), x7)_>=_new_showInt1ShowInt0(x7, Succ(x9), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) the following chains were created: 78.58/41.35 *We consider the chain new_showInt1ShowInt00(x26, x27, x28, Zero, Zero) -> new_showInt1ShowInt01(x26, x27, x28), new_showInt1ShowInt01(x29, x30, x31) -> new_showInt(Pos(new_primDivNatS2(x30, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x30, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x29)) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt01(x26, x27, x28)=new_showInt1ShowInt01(x29, x30, x31) ==> new_showInt1ShowInt00(x26, x27, x28, Zero, Zero)_>=_new_showInt1ShowInt01(x26, x27, x28)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt1ShowInt00(x26, x27, x28, Zero, Zero)_>=_new_showInt1ShowInt01(x26, x27, x28)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) the following chains were created: 78.58/41.35 *We consider the chain new_showInt1ShowInt00(x40, x41, x42, Succ(x43), Succ(x44)) -> new_showInt1ShowInt00(x40, x41, x42, x43, x44), new_showInt1ShowInt00(x45, x46, x47, Zero, Zero) -> new_showInt1ShowInt01(x45, x46, x47) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt00(x40, x41, x42, x43, x44)=new_showInt1ShowInt00(x45, x46, x47, Zero, Zero) ==> new_showInt1ShowInt00(x40, x41, x42, Succ(x43), Succ(x44))_>=_new_showInt1ShowInt00(x40, x41, x42, x43, x44)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt1ShowInt00(x40, x41, x42, Succ(Zero), Succ(Zero))_>=_new_showInt1ShowInt00(x40, x41, x42, Zero, Zero)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *We consider the chain new_showInt1ShowInt00(x48, x49, x50, Succ(x51), Succ(x52)) -> new_showInt1ShowInt00(x48, x49, x50, x51, x52), new_showInt1ShowInt00(x53, x54, x55, Succ(x56), Succ(x57)) -> new_showInt1ShowInt00(x53, x54, x55, x56, x57) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt00(x48, x49, x50, x51, x52)=new_showInt1ShowInt00(x53, x54, x55, Succ(x56), Succ(x57)) ==> new_showInt1ShowInt00(x48, x49, x50, Succ(x51), Succ(x52))_>=_new_showInt1ShowInt00(x48, x49, x50, x51, x52)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt1ShowInt00(x48, x49, x50, Succ(Succ(x56)), Succ(Succ(x57)))_>=_new_showInt1ShowInt00(x48, x49, x50, Succ(x56), Succ(x57))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *We consider the chain new_showInt1ShowInt00(x68, x69, x70, Succ(x71), Succ(x72)) -> new_showInt1ShowInt00(x68, x69, x70, x71, x72), new_showInt1ShowInt00(x73, x74, x75, Succ(x76), Zero) -> new_showInt(Pos(new_primDivNatS2(x74, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x74, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x73)) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt00(x68, x69, x70, x71, x72)=new_showInt1ShowInt00(x73, x74, x75, Succ(x76), Zero) ==> new_showInt1ShowInt00(x68, x69, x70, Succ(x71), Succ(x72))_>=_new_showInt1ShowInt00(x68, x69, x70, x71, x72)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt1ShowInt00(x68, x69, x70, Succ(Succ(x76)), Succ(Zero))_>=_new_showInt1ShowInt00(x68, x69, x70, Succ(x76), Zero)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) the following chains were created: 78.58/41.35 *We consider the chain new_showInt1ShowInt0(x81, Succ(x82), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(x81, Succ(x82), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x82, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), new_showInt1ShowInt00(x83, x84, x85, Succ(x86), Succ(x87)) -> new_showInt1ShowInt00(x83, x84, x85, x86, x87) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt1ShowInt00(x81, Succ(x82), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x82, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=new_showInt1ShowInt00(x83, x84, x85, Succ(x86), Succ(x87)) ==> new_showInt1ShowInt0(x81, Succ(x82), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_new_showInt1ShowInt00(x81, Succ(x82), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x82, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (new_showInt1ShowInt0(x81, Succ(Succ(x86)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_new_showInt1ShowInt00(x81, Succ(Succ(x86)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x86), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) the following chains were created: 78.58/41.35 *We consider the chain new_showInt1ShowInt01(x94, x95, x96) -> new_showInt(Pos(new_primDivNatS2(x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94)), new_showInt(Pos(Succ(x97)), x98) -> new_showInt1ShowInt0(x98, x97, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt(Pos(new_primDivNatS2(x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))=new_showInt(Pos(Succ(x97)), x98) ==> new_showInt1ShowInt01(x94, x95, x96)_>=_new_showInt(Pos(new_primDivNatS2(x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=x140 & new_primDivNatS2(x95, x140)=Succ(x97) ==> new_showInt1ShowInt01(x94, x95, x96)_>=_new_showInt(Pos(new_primDivNatS2(x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS2(x95, x140)=Succ(x97) which results in the following new constraints: 78.58/41.35 78.58/41.35 (3) (new_primDivNatS01(x142, x141, x142, x141)=Succ(x97) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Succ(x141) ==> new_showInt1ShowInt01(x94, Succ(x142), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x142), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x142), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 (4) (Succ(new_primDivNatS3(Zero, Zero, Zero))=Succ(x97) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_showInt1ShowInt01(x94, Zero, x96)_>=_new_showInt(Pos(new_primDivNatS2(Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 (5) (Succ(new_primDivNatS3(Succ(x144), Zero, Zero))=Succ(x97) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_showInt1ShowInt01(x94, Succ(x144), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x144), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x144), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (6) (x142=x145 & x141=x146 & new_primDivNatS01(x142, x141, x145, x146)=Succ(x97) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x141 ==> new_showInt1ShowInt01(x94, Succ(x142), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x142), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x142), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x142, x141, x145, x146)=Succ(x97) which results in the following new constraints: 78.58/41.35 78.58/41.35 (7) (new_primDivNatS02(x149, x148)=Succ(x97) & x149=Succ(x147) & x148=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x148 ==> new_showInt1ShowInt01(x94, Succ(x149), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x149), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x149), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 (8) (new_primDivNatS02(x154, x153)=Succ(x97) & x154=Zero & x153=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x153 ==> new_showInt1ShowInt01(x94, Succ(x154), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x154), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x154), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 (9) (new_primDivNatS01(x158, x157, x156, x155)=Succ(x97) & x158=Succ(x156) & x157=Succ(x155) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x157 & (\/x159,x160,x161:new_primDivNatS01(x158, x157, x156, x155)=Succ(x159) & x158=x156 & x157=x155 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x157 ==> new_showInt1ShowInt01(x160, Succ(x158), x161)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x158), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x158), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x160))) ==> new_showInt1ShowInt01(x94, Succ(x158), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x158), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x158), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We solved constraint (7) using rules (I), (II), (III).We solved constraint (8) using rules (I), (II), (III).We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (10) (new_showInt1ShowInt01(x94, Succ(Succ(x156)), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(Succ(x156)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x156)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 For Pair new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) the following chains were created: 78.58/41.35 *We consider the chain new_showInt1ShowInt00(x114, x115, x116, Succ(x117), Zero) -> new_showInt(Pos(new_primDivNatS2(x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114)), new_showInt(Pos(Succ(x118)), x119) -> new_showInt1ShowInt0(x119, x118, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.35 78.58/41.35 (1) (new_showInt(Pos(new_primDivNatS2(x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))=new_showInt(Pos(Succ(x118)), x119) ==> new_showInt1ShowInt00(x114, x115, x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=x164 & new_primDivNatS2(x115, x164)=Succ(x118) ==> new_showInt1ShowInt00(x114, x115, x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), x115, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS2(x115, x164)=Succ(x118) which results in the following new constraints: 78.58/41.35 78.58/41.35 (3) (new_primDivNatS01(x166, x165, x166, x165)=Succ(x118) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Succ(x165) ==> new_showInt1ShowInt00(x114, Succ(x166), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x166), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x166), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 (4) (Succ(new_primDivNatS3(Zero, Zero, Zero))=Succ(x118) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_showInt1ShowInt00(x114, Zero, x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 (5) (Succ(new_primDivNatS3(Succ(x168), Zero, Zero))=Succ(x118) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_showInt1ShowInt00(x114, Succ(x168), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x168), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x168), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (6) (x166=x169 & x165=x170 & new_primDivNatS01(x166, x165, x169, x170)=Succ(x118) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x165 ==> new_showInt1ShowInt00(x114, Succ(x166), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x166), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x166), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x166, x165, x169, x170)=Succ(x118) which results in the following new constraints: 78.58/41.35 78.58/41.35 (7) (new_primDivNatS02(x173, x172)=Succ(x118) & x173=Succ(x171) & x172=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x172 ==> new_showInt1ShowInt00(x114, Succ(x173), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x173), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x173), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 (8) (new_primDivNatS02(x178, x177)=Succ(x118) & x178=Zero & x177=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x177 ==> new_showInt1ShowInt00(x114, Succ(x178), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x178), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x178), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 (9) (new_primDivNatS01(x182, x181, x180, x179)=Succ(x118) & x182=Succ(x180) & x181=Succ(x179) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x181 & (\/x183,x184,x185,x186:new_primDivNatS01(x182, x181, x180, x179)=Succ(x183) & x182=x180 & x181=x179 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x181 ==> new_showInt1ShowInt00(x184, Succ(x182), x185, Succ(x186), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x184))) ==> new_showInt1ShowInt00(x114, Succ(x182), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 We solved constraint (7) using rules (I), (II), (III).We solved constraint (8) using rules (I), (II), (III).We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: 78.58/41.35 78.58/41.35 (10) (new_showInt1ShowInt00(x114, Succ(Succ(x180)), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(Succ(x180)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x180)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 To summarize, we get the following constraints P__>=_ for the following pairs. 78.58/41.35 78.58/41.35 *new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 *(new_showInt(Pos(Succ(Succ(x9))), x7)_>=_new_showInt1ShowInt0(x7, Succ(x9), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt00(x26, x27, x28, Zero, Zero)_>=_new_showInt1ShowInt01(x26, x27, x28)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt00(x40, x41, x42, Succ(Zero), Succ(Zero))_>=_new_showInt1ShowInt00(x40, x41, x42, Zero, Zero)) 78.58/41.35 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt00(x48, x49, x50, Succ(Succ(x56)), Succ(Succ(x57)))_>=_new_showInt1ShowInt00(x48, x49, x50, Succ(x56), Succ(x57))) 78.58/41.35 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt00(x68, x69, x70, Succ(Succ(x76)), Succ(Zero))_>=_new_showInt1ShowInt00(x68, x69, x70, Succ(x76), Zero)) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt0(x81, Succ(Succ(x86)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_new_showInt1ShowInt00(x81, Succ(Succ(x86)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x86), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt01(x94, Succ(Succ(x156)), x96)_>=_new_showInt(Pos(new_primDivNatS2(Succ(Succ(x156)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x156)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x94))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 *new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 *(new_showInt1ShowInt00(x114, Succ(Succ(x180)), x116, Succ(x117), Zero)_>=_new_showInt(Pos(new_primDivNatS2(Succ(Succ(x180)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x180)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x114))) 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 78.58/41.35 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. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (78) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (79) QDPPairToRuleProof (EQUIVALENT) 78.58/41.35 The dependency pair new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) was transformed to the following new rules: 78.58/41.35 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.35 78.58/41.35 the following new pairs maintain the fan-in: 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 the following new pairs maintain the fan-out: 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (80) 78.58/41.35 Complex Obligation (AND) 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (81) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> new_showInt1ShowInt00(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (82) DependencyGraphProof (EQUIVALENT) 78.58/41.35 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (83) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (84) TransformationProof (EQUIVALENT) 78.58/41.35 By instantiating [LPAR04] the rule H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) we obtained the following new rules [LPAR04]: 78.58/41.35 78.58/41.35 (H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero),H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (85) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (86) TransformationProof (EQUIVALENT) 78.58/41.35 By instantiating [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero) -> new_showInt1ShowInt01(xv94, xv95, xv96) we obtained the following new rules [LPAR04]: 78.58/41.35 78.58/41.35 (new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (87) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.35 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.35 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.35 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.35 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.35 new_primDivNatS4(xv170) -> Zero 78.58/41.35 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.35 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.35 78.58/41.35 The set Q consists of the following terms: 78.58/41.35 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.35 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primDivNatS2(Zero, Zero) 78.58/41.35 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS3(Zero, Zero, x0) 78.58/41.35 new_primModNatS3(Zero, Succ(x0)) 78.58/41.35 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat2(Succ(x0), Zero) 78.58/41.35 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.35 new_primPlusNat2(Zero, Zero) 78.58/41.35 new_primModNatS03(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.35 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.35 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.35 new_primPlusNat3(x0, Zero, Zero) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.35 new_primDivNatS2(Succ(x0), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.35 new_primModNatS3(Zero, Zero) 78.58/41.35 new_primDivNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.35 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.35 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.35 new_primModNatS2(Zero, Zero, x0) 78.58/41.35 new_primModNatS4(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.35 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.35 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.35 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.35 new_primModNatS04(x0) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.35 new_primDivNatS02(x0, x1) 78.58/41.35 new_primPlusNat5(x0, x1) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.35 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.35 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.35 new_primModNatS3(Succ(x0), Zero) 78.58/41.35 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.35 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.35 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.35 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.35 78.58/41.35 We have to consider all minimal (P,Q,R)-chains. 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (88) TransformationProof (EQUIVALENT) 78.58/41.35 By instantiating [LPAR04] the rule new_showInt1ShowInt01(xv94, xv95, xv96) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) we obtained the following new rules [LPAR04]: 78.58/41.35 78.58/41.35 (new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)),new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0))) 78.58/41.35 78.58/41.35 78.58/41.35 ---------------------------------------- 78.58/41.35 78.58/41.35 (89) 78.58/41.35 Obligation: 78.58/41.35 Q DP problem: 78.58/41.35 The TRS P consists of the following rules: 78.58/41.35 78.58/41.35 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.35 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.35 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.35 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.35 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.35 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.35 78.58/41.35 The TRS R consists of the following rules: 78.58/41.35 78.58/41.35 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.35 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.35 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.35 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.35 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.35 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.35 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.35 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.35 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.35 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.35 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.35 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.35 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.35 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.35 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.35 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.35 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.35 new_primModNatS4(xv145) -> Zero 78.58/41.35 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.35 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.35 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (90) TransformationProof (EQUIVALENT) 78.58/41.36 By rewriting [LPAR04] the rule new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)),new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0))) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (91) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) 78.58/41.36 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (92) TransformationProof (EQUIVALENT) 78.58/41.36 By instantiating [LPAR04] the rule H(xv94, xv95, xv96, cons_new_showInt1ShowInt00(Succ(xv970), Zero)) -> new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero),H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (93) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (94) TransformationProof (EQUIVALENT) 78.58/41.36 By instantiating [LPAR04] the rule new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero) -> new_showInt(Pos(new_primDivNatS2(xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), xv94)) we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)),new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0))) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (95) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primPlusNat3(xv100, Zero, Succ(xv1030)) -> new_primPlusNat5(xv100, Zero) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (96) UsableRulesProof (EQUIVALENT) 78.58/41.36 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. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (97) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (98) TransformationProof (EQUIVALENT) 78.58/41.36 By rewriting [LPAR04] the rule new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS2(Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)),new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0))) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (99) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (100) InductionCalculusProof (EQUIVALENT) 78.58/41.36 Note that final constraints are written in bold face. 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) the following chains were created: 78.58/41.36 *We consider the chain new_showInt1ShowInt0(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x5, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), H(x6, Succ(x7), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(x6, Succ(x7), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x5, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))=H(x6, Succ(x7), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) ==> new_showInt1ShowInt0(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x5, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x130 & anew_new_showInt1ShowInt00(x5, x130)=cons_new_showInt1ShowInt00(Zero, Zero) ==> new_showInt1ShowInt0(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x5, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_showInt1ShowInt00(x5, x130)=cons_new_showInt1ShowInt00(Zero, Zero) which results in the following new constraint: 78.58/41.36 78.58/41.36 (3) (new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Succ(x131) ==> new_showInt1ShowInt0(x4, Succ(Succ(x132)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(x132)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x132), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (3) using rules (I), (II) which results in the following new constraint: 78.58/41.36 78.58/41.36 (4) (new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x131 ==> new_showInt1ShowInt0(x4, Succ(Succ(x132)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(x132)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x132), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) which results in the following new constraints: 78.58/41.36 78.58/41.36 (5) (new_new_showInt1ShowInt00(x134, x133)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Succ(x133) & (\/x135:new_new_showInt1ShowInt00(x134, x133)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x133 ==> new_showInt1ShowInt0(x135, Succ(Succ(x134)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x135, Succ(Succ(x134)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x134), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) ==> new_showInt1ShowInt0(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x134)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 (6) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt0(x4, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 (7) (cons_new_showInt1ShowInt00(Succ(x136), Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt0(x4, Succ(Succ(Succ(x136))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(Succ(x136))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x136)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (5) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (8) (new_showInt1ShowInt0(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x134)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (6) using rules (I), (II).We solved constraint (7) using rules (I), (II). 78.58/41.36 *We consider the chain new_showInt1ShowInt0(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), H(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x16), Zero)) -> new_showInt1ShowInt00(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x16), Zero) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (H(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))=H(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x16), Zero)) ==> new_showInt1ShowInt0(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x137 & anew_new_showInt1ShowInt00(x13, x137)=cons_new_showInt1ShowInt00(Succ(x16), Zero) ==> new_showInt1ShowInt0(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(x13), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_showInt1ShowInt00(x13, x137)=cons_new_showInt1ShowInt00(Succ(x16), Zero) which results in the following new constraint: 78.58/41.36 78.58/41.36 (3) (new_new_showInt1ShowInt00(x139, x138)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Succ(x138) ==> new_showInt1ShowInt0(x12, Succ(Succ(x139)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(x139)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x139), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (3) using rules (I), (II) which results in the following new constraint: 78.58/41.36 78.58/41.36 (4) (new_new_showInt1ShowInt00(x139, x138)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x138 ==> new_showInt1ShowInt0(x12, Succ(Succ(x139)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(x139)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x139), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x139, x138)=cons_new_showInt1ShowInt00(Succ(x16), Zero) which results in the following new constraints: 78.58/41.36 78.58/41.36 (5) (new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Succ(x140) & (\/x142,x143:new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x142), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x140 ==> new_showInt1ShowInt0(x143, Succ(Succ(x141)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x143, Succ(Succ(x141)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(x141), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x141)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 (6) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 (7) (cons_new_showInt1ShowInt00(Succ(x144), Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(x144))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(x144))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x144)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (5) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (8) (new_showInt1ShowInt0(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x141)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (6) using rules (I), (II).We solved constraint (7) using rules (I), (II). 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) the following chains were created: 78.58/41.36 *We consider the chain new_showInt(Pos(Succ(x19)), x20) -> new_showInt1ShowInt0(x20, x19, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1ShowInt0(x21, Succ(x22), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x21, Succ(x22), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x22, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt1ShowInt0(x20, x19, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))=new_showInt1ShowInt0(x21, Succ(x22), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) ==> new_showInt(Pos(Succ(x19)), x20)_>=_new_showInt1ShowInt0(x20, x19, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (new_showInt(Pos(Succ(Succ(x22))), x20)_>=_new_showInt1ShowInt0(x20, Succ(x22), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) the following chains were created: 78.58/41.36 *We consider the chain H(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero), new_showInt1ShowInt00(x43, Succ(x44), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(x43, Succ(x44), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)=new_showInt1ShowInt00(x43, Succ(x44), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) ==> H(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (H(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) the following chains were created: 78.58/41.36 *We consider the chain new_showInt1ShowInt00(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), new_showInt1ShowInt01(x61, Succ(x62), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(x62, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x62, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x62), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x61)) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt1ShowInt01(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=new_showInt1ShowInt01(x61, Succ(x62), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) ==> new_showInt1ShowInt00(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (new_showInt1ShowInt00(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) the following chains were created: 78.58/41.36 *We consider the chain new_showInt1ShowInt01(x69, Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69)), new_showInt(Pos(Succ(x71)), x72) -> new_showInt1ShowInt0(x72, x71, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt(Pos(new_primDivNatS01(x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))=new_showInt(Pos(Succ(x71)), x72) ==> new_showInt1ShowInt01(x69, Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x145 & x70=x146 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x147 & new_primDivNatS01(x70, x145, x146, x147)=Succ(x71) ==> new_showInt1ShowInt01(x69, Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x70, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x70), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x70, x145, x146, x147)=Succ(x71) which results in the following new constraints: 78.58/41.36 78.58/41.36 (3) (new_primDivNatS02(x150, x149)=Succ(x71) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x149 & x150=Succ(x148) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Zero ==> new_showInt1ShowInt01(x69, Succ(x150), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x150, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x150, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x150), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 (4) (new_primDivNatS02(x155, x154)=Succ(x71) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x154 & x155=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Zero ==> new_showInt1ShowInt01(x69, Succ(x155), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x155, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x155, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x155), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 (5) (new_primDivNatS01(x159, x158, x157, x156)=Succ(x71) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & x159=Succ(x157) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Succ(x156) & (\/x160,x161:new_primDivNatS01(x159, x158, x157, x156)=Succ(x160) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & x159=x157 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x156 ==> new_showInt1ShowInt01(x161, Succ(x159), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x159, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x159, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x159), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x161))) ==> new_showInt1ShowInt01(x69, Succ(x159), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(x159, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x159, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x159), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (3) using rules (I), (II).We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (6) (Succ(x157)=x162 & new_primDivNatS01(x162, x158, x157, x156)=Succ(x71) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x156 ==> new_showInt1ShowInt01(x69, Succ(Succ(x157)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(x157), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x157), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x157)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x162, x158, x157, x156)=Succ(x71) which results in the following new constraints: 78.58/41.36 78.58/41.36 (7) (new_primDivNatS02(x165, x164)=Succ(x71) & Succ(Succ(x163))=x165 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x164 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt01(x69, Succ(Succ(Succ(x163))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x163)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x163)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x163))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 (8) (new_primDivNatS02(x170, x169)=Succ(x71) & Succ(Zero)=x170 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x169 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt01(x69, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 (9) (new_primDivNatS01(x174, x173, x172, x171)=Succ(x71) & Succ(Succ(x172))=x174 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x173 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Succ(x171) & (\/x175,x176:new_primDivNatS01(x174, x173, x172, x171)=Succ(x175) & Succ(x172)=x174 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x173 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x171 ==> new_showInt1ShowInt01(x176, Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(x172), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x172), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x176))) ==> new_showInt1ShowInt01(x69, Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (7) using rules (I), (II).We solved constraint (8) using rules (I), (II).We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (10) (new_showInt1ShowInt01(x69, Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) the following chains were created: 78.58/41.36 *We consider the chain H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero)) -> new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero), new_showInt1ShowInt00(x104, Succ(x105), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x106), Zero) -> new_showInt(Pos(new_primDivNatS01(x105, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x105, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x105), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x104)) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)=new_showInt1ShowInt00(x104, Succ(x105), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x106), Zero) ==> H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 For Pair new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) the following chains were created: 78.58/41.36 *We consider the chain new_showInt1ShowInt00(x110, Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero) -> new_showInt(Pos(new_primDivNatS01(x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110)), new_showInt(Pos(Succ(x113)), x114) -> new_showInt1ShowInt0(x114, x113, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.36 78.58/41.36 (1) (new_showInt(Pos(new_primDivNatS01(x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))=new_showInt(Pos(Succ(x113)), x114) ==> new_showInt1ShowInt00(x110, Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x177 & x111=x178 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x179 & new_primDivNatS01(x111, x177, x178, x179)=Succ(x113) ==> new_showInt1ShowInt00(x110, Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x111, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x111), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x111, x177, x178, x179)=Succ(x113) which results in the following new constraints: 78.58/41.36 78.58/41.36 (3) (new_primDivNatS02(x182, x181)=Succ(x113) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x181 & x182=Succ(x180) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Zero ==> new_showInt1ShowInt00(x110, Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x182, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x182, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x182), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 (4) (new_primDivNatS02(x187, x186)=Succ(x113) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x186 & x187=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Zero ==> new_showInt1ShowInt00(x110, Succ(x187), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x187, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x187, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x187), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 (5) (new_primDivNatS01(x191, x190, x189, x188)=Succ(x113) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x190 & x191=Succ(x189) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=Succ(x188) & (\/x192,x193,x194:new_primDivNatS01(x191, x190, x189, x188)=Succ(x192) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x190 & x191=x189 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x188 ==> new_showInt1ShowInt00(x193, Succ(x191), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x194), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x191, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x191, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x191), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x193))) ==> new_showInt1ShowInt00(x110, Succ(x191), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(x191, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x191, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(x191), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (3) using rules (I), (II).We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: 78.58/41.36 78.58/41.36 (6) (Succ(x189)=x195 & new_primDivNatS01(x195, x190, x189, x188)=Succ(x113) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x190 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x188 ==> new_showInt1ShowInt00(x110, Succ(Succ(x189)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(x189), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x189), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x189)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x195, x190, x189, x188)=Succ(x113) which results in the following new constraints: 78.58/41.36 78.58/41.36 (7) (new_primDivNatS02(x198, x197)=Succ(x113) & Succ(Succ(x196))=x198 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x197 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt00(x110, Succ(Succ(Succ(x196))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x196)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x196)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x196))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 (8) (new_primDivNatS02(x203, x202)=Succ(x113) & Succ(Zero)=x203 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x202 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Zero ==> new_showInt1ShowInt00(x110, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 (9) (new_primDivNatS01(x207, x206, x205, x204)=Succ(x113) & Succ(Succ(x205))=x207 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x206 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=Succ(x204) & (\/x208,x209,x210:new_primDivNatS01(x207, x206, x205, x204)=Succ(x208) & Succ(x205)=x207 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x206 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))=x204 ==> new_showInt1ShowInt00(x209, Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x210), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(x205), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x205), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x209))) ==> new_showInt1ShowInt00(x110, Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 We solved constraint (7) using rules (I), (II).We solved constraint (8) using rules (I), (II).We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.36 78.58/41.36 (10) (new_showInt1ShowInt00(x110, Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 To summarize, we get the following constraints P__>=_ for the following pairs. 78.58/41.36 78.58/41.36 *new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 *(new_showInt1ShowInt0(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x4, Succ(Succ(Succ(x134))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x134)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 *(new_showInt1ShowInt0(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(x141))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(Succ(Succ(x141)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 *(new_showInt(Pos(Succ(Succ(x22))), x20)_>=_new_showInt1ShowInt0(x20, Succ(x22), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 78.58/41.36 *(H(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 78.58/41.36 *(new_showInt1ShowInt00(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x59, Succ(x60), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 *(new_showInt1ShowInt01(x69, Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x172)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x172))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x69))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 78.58/41.36 *(H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 *new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 *(new_showInt1ShowInt00(x110, Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x112), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x205)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x205))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x110))) 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 78.58/41.36 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. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (101) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (102) TransformationProof (EQUIVALENT) 78.58/41.36 By narrowing [LPAR04] the rule new_showInt1ShowInt0(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(z1, Succ(x1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), anew_new_showInt1ShowInt00(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) at position [3] we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (103) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (104) UsableRulesProof (EQUIVALENT) 78.58/41.36 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. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (105) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (106) QReductionProof (EQUIVALENT) 78.58/41.36 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 78.58/41.36 78.58/41.36 anew_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (107) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (108) TransformationProof (EQUIVALENT) 78.58/41.36 By narrowing [LPAR04] the rule new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.36 78.58/41.36 (new_showInt1ShowInt01(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.36 (new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.36 78.58/41.36 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (109) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.36 new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.36 new_primModNatS04(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primDivNatS02(x0, x1) 78.58/41.36 new_primPlusNat5(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.36 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.36 new_primModNatS3(Succ(x0), Zero) 78.58/41.36 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.36 78.58/41.36 We have to consider all minimal (P,Q,R)-chains. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (110) DependencyGraphProof (EQUIVALENT) 78.58/41.36 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.36 ---------------------------------------- 78.58/41.36 78.58/41.36 (111) 78.58/41.36 Obligation: 78.58/41.36 Q DP problem: 78.58/41.36 The TRS P consists of the following rules: 78.58/41.36 78.58/41.36 new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.36 new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.36 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.36 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.36 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.36 78.58/41.36 The TRS R consists of the following rules: 78.58/41.36 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.36 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.36 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.36 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.36 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.36 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.36 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.36 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.36 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.36 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.36 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.36 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.36 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.36 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.36 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.36 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.36 new_primModNatS4(xv145) -> Zero 78.58/41.36 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.36 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.36 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.36 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.36 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.36 new_primDivNatS4(xv170) -> Zero 78.58/41.36 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.36 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.36 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.36 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.36 78.58/41.36 The set Q consists of the following terms: 78.58/41.36 78.58/41.36 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.36 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.36 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primDivNatS2(Zero, Zero) 78.58/41.36 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.36 new_primDivNatS3(Zero, Zero, x0) 78.58/41.36 new_primModNatS3(Zero, Succ(x0)) 78.58/41.36 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.36 new_primPlusNat2(Succ(x0), Zero) 78.58/41.36 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.36 new_primPlusNat2(Zero, Zero) 78.58/41.36 new_primModNatS03(x0, x1) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.36 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.36 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.36 new_primPlusNat3(x0, Zero, Zero) 78.58/41.36 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.36 new_primDivNatS2(Succ(x0), Zero) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.36 new_primModNatS3(Zero, Zero) 78.58/41.36 new_primDivNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.36 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.36 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.36 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.36 new_primModNatS2(Zero, Zero, x0) 78.58/41.36 new_primModNatS4(x0) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.36 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (112) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt0(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(x0)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) at position [3] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))),new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (113) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (114) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt01(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (115) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (116) DependencyGraphProof (EQUIVALENT) 78.58/41.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (117) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (118) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt01(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (119) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (120) DependencyGraphProof (EQUIVALENT) 78.58/41.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (121) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (122) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 (new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (123) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (124) DependencyGraphProof (EQUIVALENT) 78.58/41.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (125) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (126) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt0(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) at position [3] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))),new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (127) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (128) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(z2), Zero) -> new_showInt(Pos(new_primDivNatS01(z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), z1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), z0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt00(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 (new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (129) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (130) DependencyGraphProof (EQUIVALENT) 78.58/41.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (131) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300))) -> new_primPlusNat5(xv100, Succ(Zero)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (132) UsableRulesProof (EQUIVALENT) 78.58/41.37 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. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (133) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (134) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(x0)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Succ(Zero))))))) at position [3] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))),new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (135) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (136) TransformationProof (EQUIVALENT) 78.58/41.37 By narrowing [LPAR04] the rule new_showInt1ShowInt00(y0, Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.37 78.58/41.37 (new_showInt1ShowInt00(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 (new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.37 78.58/41.37 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (137) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Zero)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS2(Succ(x0), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.37 new_primModNatS3(Zero, Zero) 78.58/41.37 new_primDivNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.37 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.37 new_primModNatS2(Zero, Zero, x0) 78.58/41.37 new_primModNatS4(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.37 new_primModNatS04(x0) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primDivNatS02(x0, x1) 78.58/41.37 new_primPlusNat5(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.37 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.37 new_primModNatS3(Succ(x0), Zero) 78.58/41.37 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.37 78.58/41.37 We have to consider all minimal (P,Q,R)-chains. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (138) DependencyGraphProof (EQUIVALENT) 78.58/41.37 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.37 ---------------------------------------- 78.58/41.37 78.58/41.37 (139) 78.58/41.37 Obligation: 78.58/41.37 Q DP problem: 78.58/41.37 The TRS P consists of the following rules: 78.58/41.37 78.58/41.37 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.37 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.37 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.37 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.37 new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.37 78.58/41.37 The TRS R consists of the following rules: 78.58/41.37 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000)))) -> new_primPlusNat5(xv100, Succ(Succ(Zero))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.37 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.37 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.37 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.37 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.37 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.37 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.37 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.37 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.37 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.37 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.37 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.37 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.37 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.37 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.37 new_primModNatS4(xv145) -> Zero 78.58/41.37 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.37 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.37 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.37 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.37 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.37 new_primDivNatS4(xv170) -> Zero 78.58/41.37 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.37 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.37 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.37 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.37 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.37 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.37 78.58/41.37 The set Q consists of the following terms: 78.58/41.37 78.58/41.37 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.37 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.37 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.37 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.37 new_primDivNatS2(Zero, Zero) 78.58/41.37 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.37 new_primDivNatS3(Zero, Zero, x0) 78.58/41.37 new_primModNatS3(Zero, Succ(x0)) 78.58/41.37 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.37 new_primPlusNat2(Succ(x0), Zero) 78.58/41.37 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.37 new_primPlusNat2(Zero, Zero) 78.58/41.37 new_primModNatS03(x0, x1) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.37 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.37 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.37 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.37 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.37 new_primPlusNat3(x0, Zero, Zero) 78.58/41.37 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (140) UsableRulesProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (141) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (142) TransformationProof (EQUIVALENT) 78.58/41.38 By narrowing [LPAR04] the rule new_showInt1ShowInt00(y0, Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.38 78.58/41.38 (new_showInt1ShowInt00(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.38 (new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (143) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (144) DependencyGraphProof (EQUIVALENT) 78.58/41.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (145) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (146) UsableRulesProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (147) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (148) TransformationProof (EQUIVALENT) 78.58/41.38 By narrowing [LPAR04] the rule new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) at position [0,0] we obtained the following new rules [LPAR04]: 78.58/41.38 78.58/41.38 (new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.38 (new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)),new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0))) 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (149) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(Zero), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (150) DependencyGraphProof (EQUIVALENT) 78.58/41.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (151) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000)))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (152) UsableRulesProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (153) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (154) InductionCalculusProof (EQUIVALENT) 78.58/41.38 Note that final constraints are written in bold face. 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) the following chains were created: 78.58/41.38 *We consider the chain new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x3, Succ(Succ(Succ(Succ(Zero)))))), H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (H(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x3, Succ(Succ(Succ(Succ(Zero))))))=H(x4, Succ(x5), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x3, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (Succ(Succ(Succ(Succ(Zero))))=x130 & new_new_showInt1ShowInt00(x3, x130)=cons_new_showInt1ShowInt00(Zero, Zero) ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(x3))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x3, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x3, x130)=cons_new_showInt1ShowInt00(Zero, Zero) which results in the following new constraints: 78.58/41.38 78.58/41.38 (3) (new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Zero))))=Succ(x131) & (\/x133:new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Zero))))=x131 ==> new_showInt1ShowInt0(x133, Succ(Succ(Succ(Succ(Succ(x132))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x133, Succ(Succ(Succ(Succ(Succ(x132))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x132, Succ(Succ(Succ(Succ(Zero))))))) ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(x132)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(x132)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x132), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (4) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Zero, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (5) (cons_new_showInt1ShowInt00(Succ(x134), Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(x134)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(x134)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x134), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (6) (new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Zero)))=x131 ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(x132)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(x132)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x132), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x132, x131)=cons_new_showInt1ShowInt00(Zero, Zero) which results in the following new constraints: 78.58/41.38 78.58/41.38 (7) (new_new_showInt1ShowInt00(x136, x135)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Zero)))=Succ(x135) & (\/x137:new_new_showInt1ShowInt00(x136, x135)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Zero)))=x135 ==> new_showInt1ShowInt0(x137, Succ(Succ(Succ(Succ(Succ(Succ(x136)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x137, Succ(Succ(Succ(Succ(Succ(Succ(x136)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x136), Succ(Succ(Succ(Succ(Zero))))))) ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x136)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (8) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Zero), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (9) (cons_new_showInt1ShowInt00(Succ(x138), Zero)=cons_new_showInt1ShowInt00(Zero, Zero) & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x138))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x138))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x138)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (10) (new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x136)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II). 78.58/41.38 *We consider the chain new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Zero)))))), H(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x16), Zero)) -> new_showInt1ShowInt00(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x16), Zero) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (H(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Zero))))))=H(x14, Succ(x15), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x16), Zero)) ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (Succ(Succ(Succ(Succ(Zero))))=x139 & new_new_showInt1ShowInt00(x13, x139)=cons_new_showInt1ShowInt00(Succ(x16), Zero) ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(x13))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x13, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x13, x139)=cons_new_showInt1ShowInt00(Succ(x16), Zero) which results in the following new constraints: 78.58/41.38 78.58/41.38 (3) (new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Zero))))=Succ(x140) & (\/x142,x143:new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x142), Zero) & Succ(Succ(Succ(Succ(Zero))))=x140 ==> new_showInt1ShowInt0(x143, Succ(Succ(Succ(Succ(Succ(x141))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x143, Succ(Succ(Succ(Succ(Succ(x141))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x141, Succ(Succ(Succ(Succ(Zero))))))) ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(x141)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(x141)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x141), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (4) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Zero, Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (5) (cons_new_showInt1ShowInt00(Succ(x144), Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(x144)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(x144)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x144), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (6) (new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Zero)))=x140 ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(x141)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(x141)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x141), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_new_showInt1ShowInt00(x141, x140)=cons_new_showInt1ShowInt00(Succ(x16), Zero) which results in the following new constraints: 78.58/41.38 78.58/41.38 (7) (new_new_showInt1ShowInt00(x146, x145)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Zero)))=Succ(x145) & (\/x147,x148:new_new_showInt1ShowInt00(x146, x145)=cons_new_showInt1ShowInt00(Succ(x147), Zero) & Succ(Succ(Succ(Zero)))=x145 ==> new_showInt1ShowInt0(x148, Succ(Succ(Succ(Succ(Succ(Succ(x146)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x148, Succ(Succ(Succ(Succ(Succ(Succ(x146)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(x146), Succ(Succ(Succ(Succ(Zero))))))) ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x146)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (8) (cons_new_showInt1ShowInt00(Zero, Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Zero), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 (9) (cons_new_showInt1ShowInt00(Succ(x149), Zero)=cons_new_showInt1ShowInt00(Succ(x16), Zero) & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x149))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x149))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x149)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (10) (new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x146)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II). 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) the following chains were created: 78.58/41.38 *We consider the chain H(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero), new_showInt1ShowInt00(x25, Succ(x26), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(x25, Succ(x26), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt1ShowInt00(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)=new_showInt1ShowInt00(x25, Succ(x26), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) ==> H(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (H(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) the following chains were created: 78.58/41.38 *We consider the chain new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), new_showInt1ShowInt01(x43, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x44)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x44, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x43)) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt1ShowInt01(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=new_showInt1ShowInt01(x43, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) ==> new_showInt1ShowInt00(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x41, Succ(x42), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (new_showInt1ShowInt00(x41, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x41, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) the following chains were created: 78.58/41.38 *We consider the chain new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x60)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x60, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59)), new_showInt(Pos(Succ(x61)), x62) -> new_showInt1ShowInt0(x62, x61, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x60)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x60, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))=new_showInt(Pos(Succ(x61)), x62) ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x60)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x60, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (Succ(Succ(Succ(Succ(x60))))=x150 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x151 & Succ(Succ(Succ(Succ(Zero))))=x152 & new_primDivNatS01(x150, x151, x60, x152)=Succ(x61) ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x60)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x60, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x60))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x150, x151, x60, x152)=Succ(x61) which results in the following new constraints: 78.58/41.38 78.58/41.38 (3) (new_primDivNatS01(x159, x158, x157, x156)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(x157)))))=x159 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & Succ(Succ(Succ(Succ(Zero))))=Succ(x156) & (\/x160,x161:new_primDivNatS01(x159, x158, x157, x156)=Succ(x160) & Succ(Succ(Succ(Succ(x157))))=x159 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & Succ(Succ(Succ(Succ(Zero))))=x156 ==> new_showInt1ShowInt01(x161, Succ(Succ(Succ(Succ(Succ(x157))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x157)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x157, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x157))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x161))) ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(x157)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x157))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x157), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x157)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 (4) (new_primDivNatS02(x164, x163)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(x162)))))=x164 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x163 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(x162)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x162))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x162), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x162)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 (5) (new_primDivNatS02(x166, x165)=Succ(x61) & Succ(Succ(Succ(Succ(Zero))))=x166 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x165 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (6) (new_primDivNatS01(x159, x158, x157, x156)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(x157)))))=x159 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x158 & Succ(Succ(Succ(Zero)))=x156 ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(x157)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x157))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x157), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x157)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x159, x158, x157, x156)=Succ(x61) which results in the following new constraints: 78.58/41.38 78.58/41.38 (7) (new_primDivNatS01(x173, x172, x171, x170)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(Succ(x171))))))=x173 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x172 & Succ(Succ(Succ(Zero)))=Succ(x170) & (\/x174,x175:new_primDivNatS01(x173, x172, x171, x170)=Succ(x174) & Succ(Succ(Succ(Succ(Succ(x171)))))=x173 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x172 & Succ(Succ(Succ(Zero)))=x170 ==> new_showInt1ShowInt01(x175, Succ(Succ(Succ(Succ(Succ(Succ(x171)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x171))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x171), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x171)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x175))) ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x171)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x171)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 (8) (new_primDivNatS02(x178, x177)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(Succ(x176))))))=x178 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x177 & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x176))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x176)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x176)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x176))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 (9) (new_primDivNatS02(x180, x179)=Succ(x61) & Succ(Succ(Succ(Succ(Succ(Zero)))))=x180 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x179 & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (10) (new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x171)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x171)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II). 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) the following chains were created: 78.58/41.38 *We consider the chain new_showInt(Pos(Succ(x67)), x68) -> new_showInt1ShowInt0(x68, x67, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))), new_showInt1ShowInt0(x69, Succ(Succ(Succ(Succ(Succ(x70))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(x69, Succ(Succ(Succ(Succ(Succ(x70))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x70, Succ(Succ(Succ(Succ(Zero)))))) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt1ShowInt0(x68, x67, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))=new_showInt1ShowInt0(x69, Succ(Succ(Succ(Succ(Succ(x70))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) ==> new_showInt(Pos(Succ(x67)), x68)_>=_new_showInt1ShowInt0(x68, x67, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (new_showInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x70))))))), x68)_>=_new_showInt1ShowInt0(x68, Succ(Succ(Succ(Succ(Succ(x70))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) the following chains were created: 78.58/41.38 *We consider the chain H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero)) -> new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero), new_showInt1ShowInt00(x104, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x106), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x105)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x105, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x104)) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)=new_showInt1ShowInt00(x104, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x106), Zero) ==> H(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(x102), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (H(x101, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 For Pair new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) the following chains were created: 78.58/41.38 *We consider the chain new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x120)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x120, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119)), new_showInt(Pos(Succ(x122)), x123) -> new_showInt1ShowInt0(x123, x122, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) which results in the following constraint: 78.58/41.38 78.58/41.38 (1) (new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x120)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x120, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))=new_showInt(Pos(Succ(x122)), x123) ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x120)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x120, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: 78.58/41.38 78.58/41.38 (2) (Succ(Succ(Succ(Succ(x120))))=x181 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x182 & Succ(Succ(Succ(Succ(Zero))))=x183 & new_primDivNatS01(x181, x182, x120, x183)=Succ(x122) ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x120)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x120, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x120))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x181, x182, x120, x183)=Succ(x122) which results in the following new constraints: 78.58/41.38 78.58/41.38 (3) (new_primDivNatS01(x190, x189, x188, x187)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(x188)))))=x190 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x189 & Succ(Succ(Succ(Succ(Zero))))=Succ(x187) & (\/x191,x192,x193:new_primDivNatS01(x190, x189, x188, x187)=Succ(x191) & Succ(Succ(Succ(Succ(x188))))=x190 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x189 & Succ(Succ(Succ(Succ(Zero))))=x187 ==> new_showInt1ShowInt00(x192, Succ(Succ(Succ(Succ(Succ(x188))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x193), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x188)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x188, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x188))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x192))) ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(x188)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x188))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x188), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x188)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 (4) (new_primDivNatS02(x196, x195)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(x194)))))=x196 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x195 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(x194)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x194))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x194), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x194)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 (5) (new_primDivNatS02(x198, x197)=Succ(x122) & Succ(Succ(Succ(Succ(Zero))))=x198 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x197 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (6) (new_primDivNatS01(x190, x189, x188, x187)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(x188)))))=x190 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x189 & Succ(Succ(Succ(Zero)))=x187 ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(x188)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x188))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x188), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x188)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x190, x189, x188, x187)=Succ(x122) which results in the following new constraints: 78.58/41.38 78.58/41.38 (7) (new_primDivNatS01(x205, x204, x203, x202)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(Succ(x203))))))=x205 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x204 & Succ(Succ(Succ(Zero)))=Succ(x202) & (\/x206,x207,x208:new_primDivNatS01(x205, x204, x203, x202)=Succ(x206) & Succ(Succ(Succ(Succ(Succ(x203)))))=x205 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x204 & Succ(Succ(Succ(Zero)))=x202 ==> new_showInt1ShowInt00(x207, Succ(Succ(Succ(Succ(Succ(Succ(x203)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x208), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x203))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x203), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(x203)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x207))) ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x203)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x203)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 (8) (new_primDivNatS02(x211, x210)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(Succ(x209))))))=x211 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x210 & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x209))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x209)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x209)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x209))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 (9) (new_primDivNatS02(x213, x212)=Succ(x122) & Succ(Succ(Succ(Succ(Succ(Zero)))))=x213 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x212 & Succ(Succ(Succ(Zero)))=Zero ==> new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint: 78.58/41.38 78.58/41.38 (10) (new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x203)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x203)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II). 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 To summarize, we get the following constraints P__>=_ for the following pairs. 78.58/41.38 78.58/41.38 *new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 78.58/41.38 *(new_showInt1ShowInt0(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x136))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x136)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 *(new_showInt1ShowInt0(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))_>=_H(x12, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x146))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(Succ(Succ(x146)), Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 78.58/41.38 *(H(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero))_>=_new_showInt1ShowInt00(x23, Succ(x24), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 78.58/41.38 *(new_showInt1ShowInt00(x41, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero)_>=_new_showInt1ShowInt01(x41, Succ(Succ(Succ(Succ(Succ(x44))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 *(new_showInt1ShowInt01(x59, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x171)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x171)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x171))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x59))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 78.58/41.38 *(new_showInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x70))))))), x68)_>=_new_showInt1ShowInt0(x68, Succ(Succ(Succ(Succ(Succ(x70))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 78.58/41.38 *(H(x101, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x103), Zero))_>=_new_showInt1ShowInt00(x101, Succ(Succ(Succ(Succ(Succ(x105))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x103), Zero)) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 *new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 *(new_showInt1ShowInt00(x119, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x121), Zero)_>=_new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(Succ(x203)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x203)), Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x203))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), x119))) 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 78.58/41.38 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. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (155) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt0(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) -> H(y0, Succ(Succ(Succ(Succ(Succ(x0))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_new_showInt1ShowInt00(x0, Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Zero, Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) 78.58/41.38 new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Zero) -> new_showInt1ShowInt01(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 78.58/41.38 new_showInt1ShowInt01(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 new_showInt(Pos(Succ(xv300)), xv4) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 78.58/41.38 H(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), cons_new_showInt1ShowInt00(Succ(x3), Zero)) -> new_showInt1ShowInt00(z0, Succ(z1), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x3), Zero) 78.58/41.38 new_showInt1ShowInt00(y0, Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(y2), Zero) -> new_showInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x2)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Zero)))))), :(Char(new_primPlusNat3(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), Succ(Succ(Succ(Succ(Succ(x2))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))), y0)) 78.58/41.38 78.58/41.38 The TRS R consists of the following rules: 78.58/41.38 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280)) -> Zero 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280)) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000)))))))) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))))) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000)) 78.58/41.38 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000))))))) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat5(xv133, xv134) -> Succ(Succ(new_primPlusNat2(xv133, xv134))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Succ(xv1340)) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340))) 78.58/41.38 new_primPlusNat2(Succ(xv1330), Zero) -> Succ(xv1330) 78.58/41.38 new_primPlusNat2(Zero, Succ(xv1340)) -> Succ(xv1340) 78.58/41.38 new_primPlusNat2(Zero, Zero) -> Zero 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660)) -> Succ(Succ(xv163)) 78.58/41.38 new_primModNatS02(xv163, xv164, Zero, Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS03(xv163, xv164) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS2(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS2(Succ(xv1510), Zero, xv153) -> new_primModNatS3(xv1510, xv153) 78.58/41.38 new_primModNatS3(Succ(xv1390), Succ(xv1400)) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS02(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS03(xv163, xv164) 78.58/41.38 new_primModNatS3(Succ(xv1390), Zero) -> new_primModNatS2(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS2(Zero, Succ(xv1520), xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS2(Zero, Zero, xv153) -> new_primModNatS4(xv153) 78.58/41.38 new_primModNatS3(Zero, Succ(xv1400)) -> Succ(Zero) 78.58/41.38 new_primModNatS3(Zero, Zero) -> new_primModNatS2(Zero, Zero, Zero) 78.58/41.38 new_primModNatS4(xv145) -> Zero 78.58/41.38 new_primModNatS04(xv149) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS01(xv125, xv126, Zero, Zero) -> new_primDivNatS02(xv125, xv126) 78.58/41.38 new_primDivNatS02(xv125, xv126) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126))) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170) -> new_primDivNatS3(xv1680, xv1690, xv170) 78.58/41.38 new_primDivNatS3(Succ(xv1680), Zero, xv170) -> new_primDivNatS2(xv1680, xv170) 78.58/41.38 new_primDivNatS3(Zero, Zero, xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS3(Zero, Succ(xv1690), xv170) -> new_primDivNatS4(xv170) 78.58/41.38 new_primDivNatS4(xv170) -> Zero 78.58/41.38 new_primDivNatS2(Succ(xv830), Succ(xv840)) -> new_primDivNatS01(xv830, xv840, xv830, xv840) 78.58/41.38 new_primDivNatS2(Zero, Succ(xv840)) -> Zero 78.58/41.38 new_primDivNatS2(Zero, Zero) -> Succ(new_primDivNatS3(Zero, Zero, Zero)) 78.58/41.38 new_primDivNatS2(Succ(xv830), Zero) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero)) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Succ(xv980)) -> new_new_showInt1ShowInt00(xv970, xv980) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) -> cons_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(xv970), Zero) -> cons_new_showInt1ShowInt00(Succ(xv970), Zero) 78.58/41.38 78.58/41.38 The set Q consists of the following terms: 78.58/41.38 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primModNatS2(Succ(x0), Zero, x1) 78.58/41.38 new_primModNatS2(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primDivNatS2(Zero, Zero) 78.58/41.38 new_primPlusNat2(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS3(Zero, Zero, x0) 78.58/41.38 new_primModNatS3(Zero, Succ(x0)) 78.58/41.38 new_primDivNatS3(Succ(x0), Zero, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat2(Succ(x0), Zero) 78.58/41.38 new_primDivNatS2(Zero, Succ(x0)) 78.58/41.38 new_primPlusNat2(Zero, Zero) 78.58/41.38 new_primModNatS03(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(x1)))))) 78.58/41.38 new_primPlusNat3(x0, Zero, Succ(x1)) 78.58/41.38 new_primPlusNat3(x0, Succ(x1), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Succ(x1))) 78.58/41.38 new_primPlusNat3(x0, Zero, Zero) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Succ(x3)) 78.58/41.38 new_primDivNatS2(Succ(x0), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(x1))))) 78.58/41.38 new_primModNatS3(Zero, Zero) 78.58/41.38 new_primDivNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))) 78.58/41.38 new_primDivNatS3(Succ(x0), Succ(x1), x2) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Zero) 78.58/41.38 new_primDivNatS3(Zero, Succ(x0), x1) 78.58/41.38 new_primModNatS2(Zero, Zero, x0) 78.58/41.38 new_primModNatS4(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(x1)))))), Succ(Succ(Succ(Succ(Succ(Zero)))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(x1))))))) 78.58/41.38 new_primDivNatS01(x0, x1, Succ(x2), Zero) 78.58/41.38 new_primPlusNat3(x0, Succ(Zero), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(x1)), Succ(Zero)) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS3(Succ(x0), Succ(x1)) 78.58/41.38 new_primDivNatS01(x0, x1, Zero, Zero) 78.58/41.38 new_primModNatS04(x0) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(x1)))), Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(x1))), Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Zero)), Succ(Succ(Succ(x1)))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x2))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(x1))))), Succ(Succ(Succ(Succ(Zero))))) 78.58/41.38 new_primModNatS02(x0, x1, Zero, Succ(x2)) 78.58/41.38 new_primDivNatS02(x0, x1) 78.58/41.38 new_primPlusNat5(x0, x1) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x1))))))))) 78.58/41.38 new_primPlusNat3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.38 new_primModNatS2(Zero, Succ(x0), x1) 78.58/41.38 new_primDivNatS2(Succ(x0), Succ(x1)) 78.58/41.38 new_primModNatS3(Succ(x0), Zero) 78.58/41.38 new_primPlusNat2(Zero, Succ(x0)) 78.58/41.38 new_primModNatS02(x0, x1, Succ(x2), Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Succ(x1)) 78.58/41.38 new_new_showInt1ShowInt00(Zero, Zero) 78.58/41.38 new_new_showInt1ShowInt00(Succ(x0), Zero) 78.58/41.38 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (156) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (157) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980)) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 > 5 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (158) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (159) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primModNatS(Succ(xv1510), Zero, xv153) -> new_primModNatS1(xv1510, xv153) 78.58/41.38 new_primModNatS1(Zero, Zero) -> new_primModNatS(Zero, Zero, Zero) 78.58/41.38 new_primModNatS00(xv163, xv164) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS0(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS1(Succ(xv1390), Succ(xv1400)) -> new_primModNatS0(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS0(xv163, xv164, Zero, Zero) -> new_primModNatS00(xv163, xv164) 78.58/41.38 new_primModNatS1(Succ(xv1390), Zero) -> new_primModNatS(Succ(xv1390), Zero, Zero) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (160) DependencyGraphProof (EQUIVALENT) 78.58/41.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (161) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primModNatS1(Succ(xv1390), Succ(xv1400)) -> new_primModNatS0(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS(Succ(xv1510), Zero, xv153) -> new_primModNatS1(xv1510, xv153) 78.58/41.38 new_primModNatS1(Succ(xv1390), Zero) -> new_primModNatS(Succ(xv1390), Zero, Zero) 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS0(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS0(xv163, xv164, Zero, Zero) -> new_primModNatS00(xv163, xv164) 78.58/41.38 new_primModNatS00(xv163, xv164) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (162) QDPOrderProof (EQUIVALENT) 78.58/41.38 We use the reduction pair processor [LPAR04,JAR06]. 78.58/41.38 78.58/41.38 78.58/41.38 The following pairs can be oriented strictly and are deleted. 78.58/41.38 78.58/41.38 new_primModNatS1(Succ(xv1390), Succ(xv1400)) -> new_primModNatS0(xv1390, xv1400, xv1390, xv1400) 78.58/41.38 new_primModNatS(Succ(xv1510), Succ(xv1520), xv153) -> new_primModNatS(xv1510, xv1520, xv153) 78.58/41.38 new_primModNatS1(Succ(xv1390), Zero) -> new_primModNatS(Succ(xv1390), Zero, Zero) 78.58/41.38 The remaining pairs can at least be oriented weakly. 78.58/41.38 Used ordering: Polynomial interpretation [POLO]: 78.58/41.38 78.58/41.38 POL(Succ(x_1)) = 1 + x_1 78.58/41.38 POL(Zero) = 0 78.58/41.38 POL(new_primModNatS(x_1, x_2, x_3)) = x_1 78.58/41.38 POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1 78.58/41.38 POL(new_primModNatS00(x_1, x_2)) = 1 + x_1 78.58/41.38 POL(new_primModNatS1(x_1, x_2)) = 1 + x_1 78.58/41.38 78.58/41.38 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 78.58/41.38 none 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (163) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Zero) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 new_primModNatS(Succ(xv1510), Zero, xv153) -> new_primModNatS1(xv1510, xv153) 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS0(xv163, xv164, xv1650, xv1660) 78.58/41.38 new_primModNatS0(xv163, xv164, Zero, Zero) -> new_primModNatS00(xv163, xv164) 78.58/41.38 new_primModNatS00(xv163, xv164) -> new_primModNatS(Succ(xv163), Succ(xv164), Succ(xv164)) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (164) DependencyGraphProof (EQUIVALENT) 78.58/41.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (165) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primModNatS0(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS0(xv163, xv164, xv1650, xv1660) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (166) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primModNatS0(xv163, xv164, Succ(xv1650), Succ(xv1660)) -> new_primModNatS0(xv163, xv164, xv1650, xv1660) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (167) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (168) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat0(xv100, xv10200, Succ(Zero)) 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat0(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat0(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.38 new_primPlusNat0(xv100, Succ(xv1020), Zero) -> new_primPlusNat0(xv100, xv1020, Zero) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (169) DependencyGraphProof (EQUIVALENT) 78.58/41.38 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (170) 78.58/41.38 Complex Obligation (AND) 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (171) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat0(xv100, Succ(xv1020), Zero) -> new_primPlusNat0(xv100, xv1020, Zero) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (172) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primPlusNat0(xv100, Succ(xv1020), Zero) -> new_primPlusNat0(xv100, xv1020, Zero) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (173) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (174) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat0(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (175) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primPlusNat0(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero)))) -> new_primPlusNat0(xv100, xv1020000, Succ(Succ(Succ(Zero)))) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (176) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (177) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat0(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (178) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primPlusNat0(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero))) -> new_primPlusNat0(xv100, xv102000, Succ(Succ(Zero))) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (179) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (180) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat0(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat0(xv100, xv10200, Succ(Zero)) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (181) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primPlusNat0(xv100, Succ(Succ(xv10200)), Succ(Zero)) -> new_primPlusNat0(xv100, xv10200, Succ(Zero)) 78.58/41.38 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (182) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (183) 78.58/41.38 Obligation: 78.58/41.38 Q DP problem: 78.58/41.38 The TRS P consists of the following rules: 78.58/41.38 78.58/41.38 new_primPlusNat(Succ(xv1330), Succ(xv1340)) -> new_primPlusNat(xv1330, xv1340) 78.58/41.38 78.58/41.38 R is empty. 78.58/41.38 Q is empty. 78.58/41.38 We have to consider all minimal (P,Q,R)-chains. 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (184) QDPSizeChangeProof (EQUIVALENT) 78.58/41.38 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. 78.58/41.38 78.58/41.38 From the DPs we obtained the following set of size-change graphs: 78.58/41.38 *new_primPlusNat(Succ(xv1330), Succ(xv1340)) -> new_primPlusNat(xv1330, xv1340) 78.58/41.38 The graph contains the following edges 1 > 1, 2 > 2 78.58/41.38 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (185) 78.58/41.38 YES 78.58/41.38 78.58/41.38 ---------------------------------------- 78.58/41.38 78.58/41.38 (186) Narrow (COMPLETE) 78.58/41.38 Haskell To QDPs 78.58/41.38 78.58/41.38 digraph dp_graph { 78.58/41.38 node [outthreshold=100, inthreshold=100];1[label="showInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 78.58/41.38 3[label="showInt xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 78.58/41.38 4[label="showInt xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 78.58/41.38 5[label="showInt3 xv3 xv4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 78.58/41.38 6[label="showInt2 xv3 xv4 (xv3 < fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 78.58/41.38 7[label="showInt2 xv3 xv4 (compare xv3 (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 78.58/41.38 8[label="showInt2 xv3 xv4 (primCmpInt xv3 (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1445[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1445[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1445 -> 9[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1446[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 1446[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1446 -> 10[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 9[label="showInt2 (Pos xv30) xv4 (primCmpInt (Pos xv30) (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1447[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];9 -> 1447[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1447 -> 11[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1448[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 1448[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1448 -> 12[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 10[label="showInt2 (Neg xv30) xv4 (primCmpInt (Neg xv30) (fromInt (Pos Zero)) == LT)",fontsize=16,color="burlywood",shape="box"];1449[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];10 -> 1449[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1449 -> 13[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1450[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 1450[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1450 -> 14[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 11[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpInt (Pos (Succ xv300)) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 78.58/41.38 12[label="showInt2 (Pos Zero) xv4 (primCmpInt (Pos Zero) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 78.58/41.38 13[label="showInt2 (Neg (Succ xv300)) xv4 (primCmpInt (Neg (Succ xv300)) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 78.58/41.38 14[label="showInt2 (Neg Zero) xv4 (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 78.58/41.38 15[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpInt (Pos (Succ xv300)) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 78.58/41.38 16[label="showInt2 (Pos Zero) xv4 (primCmpInt (Pos Zero) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 78.58/41.38 17[label="showInt2 (Neg (Succ xv300)) xv4 (primCmpInt (Neg (Succ xv300)) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 78.58/41.38 18[label="showInt2 (Neg Zero) xv4 (primCmpInt (Neg Zero) (Pos Zero) == LT)",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 78.58/41.38 19[label="showInt2 (Pos (Succ xv300)) xv4 (primCmpNat (Succ xv300) Zero == LT)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 78.58/41.38 20[label="showInt2 (Pos Zero) xv4 (EQ == LT)",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 78.58/41.38 21[label="showInt2 (Neg (Succ xv300)) xv4 (LT == LT)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 78.58/41.38 22[label="showInt2 (Neg Zero) xv4 (EQ == LT)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 78.58/41.38 23[label="showInt2 (Pos (Succ xv300)) xv4 (GT == LT)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 78.58/41.38 24[label="showInt2 (Pos Zero) xv4 False",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 78.58/41.38 25[label="showInt2 (Neg (Succ xv300)) xv4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 78.58/41.38 26[label="showInt2 (Neg Zero) xv4 False",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 78.58/41.38 27[label="showInt2 (Pos (Succ xv300)) xv4 False",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3]; 78.58/41.38 28[label="showInt1 (Pos Zero) xv4 otherwise",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 78.58/41.38 29[label="error []",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 78.58/41.38 30[label="showInt1 (Neg Zero) xv4 otherwise",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3]; 78.58/41.38 31[label="showInt1 (Pos (Succ xv300)) xv4 otherwise",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3]; 78.58/41.38 32[label="showInt1 (Pos Zero) xv4 True",fontsize=16,color="black",shape="box"];32 -> 36[label="",style="solid", color="black", weight=3]; 78.58/41.38 33[label="error []",fontsize=16,color="red",shape="box"];34[label="showInt1 (Neg Zero) xv4 True",fontsize=16,color="black",shape="box"];34 -> 37[label="",style="solid", color="black", weight=3]; 78.58/41.38 35[label="showInt1 (Pos (Succ xv300)) xv4 True",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3]; 78.58/41.38 36[label="showInt1ShowInt0 xv4 (Pos Zero) (showInt1N' xv4 (Pos Zero) == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];36 -> 39[label="",style="solid", color="black", weight=3]; 78.58/41.38 37[label="showInt1ShowInt0 xv4 (Neg Zero) (showInt1N' xv4 (Neg Zero) == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3]; 78.58/41.38 38[label="showInt1ShowInt0 xv4 (Pos (Succ xv300)) (showInt1N' xv4 (Pos (Succ xv300)) == fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3]; 78.58/41.38 39[label="showInt1ShowInt0 xv4 (Pos Zero) (primEqInt (showInt1N' xv4 (Pos Zero)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3]; 78.58/41.38 40[label="showInt1ShowInt0 xv4 (Neg Zero) (primEqInt (showInt1N' xv4 (Neg Zero)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3]; 78.58/41.38 41[label="showInt1ShowInt0 xv4 (Pos (Succ xv300)) (primEqInt (showInt1N' xv4 (Pos (Succ xv300))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];41 -> 44[label="",style="solid", color="black", weight=3]; 78.58/41.38 42[label="showInt1ShowInt0 xv4 (Pos Zero) (primEqInt (showInt1N'0 xv4 (Pos Zero) (showInt1Vu76 xv4 (Pos Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3]; 78.58/41.38 43[label="showInt1ShowInt0 xv4 (Neg Zero) (primEqInt (showInt1N'0 xv4 (Neg Zero) (showInt1Vu76 xv4 (Neg Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];43 -> 46[label="",style="solid", color="black", weight=3]; 78.58/41.38 44[label="showInt1ShowInt0 xv4 (Pos (Succ xv300)) (primEqInt (showInt1N'0 xv4 (Pos (Succ xv300)) (showInt1Vu76 xv4 (Pos (Succ xv300)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];44 -> 47[label="",style="solid", color="black", weight=3]; 78.58/41.38 45 -> 48[label="",style="dashed", color="red", weight=0]; 78.58/41.38 45[label="showInt1ShowInt0 xv4 (Pos Zero) (primEqInt (showInt1N'0 xv4 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];45 -> 49[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 45 -> 50[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 46 -> 51[label="",style="dashed", color="red", weight=0]; 78.58/41.38 46[label="showInt1ShowInt0 xv4 (Neg Zero) (primEqInt (showInt1N'0 xv4 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];46 -> 52[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 46 -> 53[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 47 -> 54[label="",style="dashed", color="red", weight=0]; 78.58/41.38 47[label="showInt1ShowInt0 xv4 (Pos (Succ xv300)) (primEqInt (showInt1N'0 xv4 (Pos (Succ xv300)) (quotRem (Pos (Succ xv300)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ 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xv12 (Pos (Succ xv13)) (primEqInt (primQuotInt (Pos (Succ xv13)) (fromInt (Pos (Succ xv14)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];66 -> 69[label="",style="solid", color="black", weight=3]; 78.58/41.38 67[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (primQuotInt (Pos Zero) (Pos (Succ xv7))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];67 -> 70[label="",style="solid", color="black", weight=3]; 78.58/41.38 68[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (primQuotInt (Neg Zero) (Pos (Succ xv10))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];68 -> 71[label="",style="solid", color="black", weight=3]; 78.58/41.38 69[label="showInt1ShowInt0 xv12 (Pos (Succ xv13)) (primEqInt (primQuotInt (Pos (Succ xv13)) (Pos (Succ xv14))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];69 -> 72[label="",style="solid", color="black", weight=3]; 78.58/41.38 70[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (Pos (primDivNatS Zero (Succ xv7))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];70 -> 73[label="",style="solid", color="black", weight=3]; 78.58/41.38 71[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (Neg (primDivNatS Zero (Succ xv10))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];71 -> 74[label="",style="solid", color="black", weight=3]; 78.58/41.38 72[label="showInt1ShowInt0 xv12 (Pos (Succ xv13)) (primEqInt (Pos (primDivNatS (Succ xv13) (Succ xv14))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];72 -> 75[label="",style="solid", color="black", weight=3]; 78.58/41.38 73[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (Pos Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];73 -> 76[label="",style="solid", color="black", weight=3]; 78.58/41.38 74[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (Neg Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3]; 78.58/41.38 75[label="showInt1ShowInt0 xv12 (Pos (Succ xv13)) (primEqInt (Pos (primDivNatS0 xv13 xv14 (primGEqNatS xv13 xv14))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1451[label="xv13/Succ xv130",fontsize=10,color="white",style="solid",shape="box"];75 -> 1451[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1451 -> 78[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1452[label="xv13/Zero",fontsize=10,color="white",style="solid",shape="box"];75 -> 1452[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1452 -> 79[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 76[label="showInt1ShowInt0 xv6 (Pos Zero) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];76 -> 80[label="",style="solid", color="black", weight=3]; 78.58/41.38 77[label="showInt1ShowInt0 xv9 (Neg Zero) (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];77 -> 81[label="",style="solid", color="black", weight=3]; 78.58/41.38 78[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) xv14 (primGEqNatS (Succ xv130) xv14))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1453[label="xv14/Succ xv140",fontsize=10,color="white",style="solid",shape="box"];78 -> 1453[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1453 -> 82[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1454[label="xv14/Zero",fontsize=10,color="white",style="solid",shape="box"];78 -> 1454[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1454 -> 83[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 79[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero xv14 (primGEqNatS Zero xv14))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1455[label="xv14/Succ xv140",fontsize=10,color="white",style="solid",shape="box"];79 -> 1455[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1455 -> 84[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1456[label="xv14/Zero",fontsize=10,color="white",style="solid",shape="box"];79 -> 1456[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1456 -> 85[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 80[label="showInt1ShowInt0 xv6 (Pos Zero) True",fontsize=16,color="black",shape="box"];80 -> 86[label="",style="solid", color="black", weight=3]; 78.58/41.38 81[label="showInt1ShowInt0 xv9 (Neg Zero) True",fontsize=16,color="black",shape="box"];81 -> 87[label="",style="solid", color="black", weight=3]; 78.58/41.38 82[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) (Succ xv140) (primGEqNatS (Succ xv130) (Succ xv140)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];82 -> 88[label="",style="solid", color="black", weight=3]; 78.58/41.38 83[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) Zero (primGEqNatS (Succ xv130) Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];83 -> 89[label="",style="solid", color="black", weight=3]; 78.58/41.38 84[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero (Succ xv140) (primGEqNatS Zero (Succ xv140)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];84 -> 90[label="",style="solid", color="black", weight=3]; 78.58/41.38 85[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero Zero (primGEqNatS Zero Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];85 -> 91[label="",style="solid", color="black", weight=3]; 78.58/41.38 86[label="showInt1R' xv6 (Pos Zero)",fontsize=16,color="black",shape="box"];86 -> 92[label="",style="solid", color="black", weight=3]; 78.58/41.38 87[label="showInt1R' xv9 (Neg Zero)",fontsize=16,color="black",shape="box"];87 -> 93[label="",style="solid", color="black", weight=3]; 78.58/41.38 88 -> 569[label="",style="dashed", color="red", weight=0]; 78.58/41.38 88[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) (Succ xv140) (primGEqNatS xv130 xv140))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];88 -> 570[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 88 -> 571[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 88 -> 572[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 88 -> 573[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 88 -> 574[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 89[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (primDivNatS0 (Succ xv130) Zero True)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];89 -> 96[label="",style="solid", color="black", weight=3]; 78.58/41.38 90[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero (Succ xv140) False)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];90 -> 97[label="",style="solid", color="black", weight=3]; 78.58/41.38 91[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (primDivNatS0 Zero Zero True)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3]; 78.58/41.38 92[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv6 (Pos Zero))) : xv6",fontsize=16,color="green",shape="box"];92 -> 99[label="",style="dashed", color="green", weight=3]; 78.58/41.38 93[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv9 (Neg Zero))) : xv9",fontsize=16,color="green",shape="box"];93 -> 100[label="",style="dashed", color="green", weight=3]; 78.58/41.38 570[label="Succ xv130",fontsize=16,color="green",shape="box"];571[label="xv140",fontsize=16,color="green",shape="box"];572[label="xv130",fontsize=16,color="green",shape="box"];573[label="xv140",fontsize=16,color="green",shape="box"];574[label="xv12",fontsize=16,color="green",shape="box"];569[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS xv97 xv98))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="triangle"];1457[label="xv97/Succ xv970",fontsize=10,color="white",style="solid",shape="box"];569 -> 1457[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1457 -> 620[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1458[label="xv97/Zero",fontsize=10,color="white",style="solid",shape="box"];569 -> 1458[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1458 -> 621[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 96[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS (Succ xv130) Zero) (Succ Zero)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];96 -> 105[label="",style="solid", color="black", weight=3]; 78.58/41.38 97[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];97 -> 106[label="",style="solid", color="black", weight=3]; 78.58/41.38 98[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];98 -> 107[label="",style="solid", color="black", weight=3]; 78.58/41.38 99[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv6 (Pos Zero)))",fontsize=16,color="black",shape="box"];99 -> 147[label="",style="solid", color="black", weight=3]; 78.58/41.38 100[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv9 (Neg Zero)))",fontsize=16,color="black",shape="box"];100 -> 157[label="",style="solid", color="black", weight=3]; 78.58/41.38 620[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS (Succ xv970) xv98))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1459[label="xv98/Succ xv980",fontsize=10,color="white",style="solid",shape="box"];620 -> 1459[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1459 -> 624[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1460[label="xv98/Zero",fontsize=10,color="white",style="solid",shape="box"];620 -> 1460[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1460 -> 625[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 621[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero xv98))) (fromInt (Pos Zero)))",fontsize=16,color="burlywood",shape="box"];1461[label="xv98/Succ xv980",fontsize=10,color="white",style="solid",shape="box"];621 -> 1461[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1461 -> 626[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 1462[label="xv98/Zero",fontsize=10,color="white",style="solid",shape="box"];621 -> 1462[label="",style="solid", color="burlywood", weight=9]; 78.58/41.38 1462 -> 627[label="",style="solid", color="burlywood", weight=3]; 78.58/41.38 105[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS (Succ xv130) Zero) (Succ Zero)))) (Pos Zero))",fontsize=16,color="black",shape="box"];105 -> 118[label="",style="solid", color="black", weight=3]; 78.58/41.38 106[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];106 -> 119[label="",style="solid", color="black", weight=3]; 78.58/41.38 107[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero)))) (Pos Zero))",fontsize=16,color="black",shape="box"];107 -> 120[label="",style="solid", color="black", weight=3]; 78.58/41.38 147 -> 179[label="",style="dashed", color="red", weight=0]; 78.58/41.38 147[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv6 (Pos Zero)))",fontsize=16,color="magenta"];147 -> 180[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 147 -> 181[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 157 -> 182[label="",style="dashed", color="red", weight=0]; 78.58/41.38 157[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv9 (Neg Zero)))",fontsize=16,color="magenta"];157 -> 183[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 157 -> 184[label="",style="dashed", color="magenta", weight=3]; 78.58/41.38 624[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS (Succ xv970) (Succ xv980)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];624 -> 631[label="",style="solid", color="black", weight=3]; 78.58/41.38 625[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS (Succ xv970) Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];625 -> 632[label="",style="solid", color="black", weight=3]; 78.58/41.38 626[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero (Succ xv980)))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];626 -> 633[label="",style="solid", color="black", weight=3]; 78.58/41.38 627[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS Zero Zero))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];627 -> 634[label="",style="solid", color="black", weight=3]; 78.58/41.38 118[label="showInt1ShowInt0 xv12 (Pos (Succ (Succ xv130))) False",fontsize=16,color="black",shape="triangle"];118 -> 144[label="",style="solid", color="black", weight=3]; 78.58/41.38 119[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) True",fontsize=16,color="black",shape="box"];119 -> 145[label="",style="solid", color="black", weight=3]; 78.58/41.38 120[label="showInt1ShowInt0 xv12 (Pos (Succ Zero)) False",fontsize=16,color="black",shape="box"];120 -> 146[label="",style="solid", color="black", weight=3]; 78.58/41.39 180[label="xv6",fontsize=16,color="green",shape="box"];181[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];179[label="primIntToChar (fromEnum (Char (Succ xv22)) + fromIntegral (showInt1D xv23 (Pos Zero)))",fontsize=16,color="black",shape="triangle"];179 -> 185[label="",style="solid", color="black", weight=3]; 78.58/41.39 183[label="xv9",fontsize=16,color="green",shape="box"];184[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];182[label="primIntToChar (fromEnum (Char (Succ xv25)) + fromIntegral (showInt1D xv26 (Neg Zero)))",fontsize=16,color="black",shape="triangle"];182 -> 186[label="",style="solid", color="black", weight=3]; 78.58/41.39 631 -> 569[label="",style="dashed", color="red", weight=0]; 78.58/41.39 631[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) (primGEqNatS xv970 xv980))) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];631 -> 640[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 631 -> 641[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 632[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) True)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="triangle"];632 -> 642[label="",style="solid", color="black", weight=3]; 78.58/41.39 633[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) False)) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];633 -> 643[label="",style="solid", color="black", weight=3]; 78.58/41.39 634 -> 632[label="",style="dashed", color="red", weight=0]; 78.58/41.39 634[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (primDivNatS0 xv95 (Succ xv96) True)) (fromInt (Pos Zero)))",fontsize=16,color="magenta"];144 -> 4[label="",style="dashed", color="red", weight=0]; 78.58/41.39 144[label="showInt (showInt1N' xv12 (Pos (Succ (Succ xv130)))) (showInt1R' xv12 (Pos (Succ (Succ xv130))))",fontsize=16,color="magenta"];144 -> 174[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 144 -> 175[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 145[label="showInt1R' xv12 (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];145 -> 176[label="",style="solid", color="black", weight=3]; 78.58/41.39 146 -> 4[label="",style="dashed", color="red", weight=0]; 78.58/41.39 146[label="showInt (showInt1N' xv12 (Pos (Succ Zero))) (showInt1R' xv12 (Pos (Succ Zero)))",fontsize=16,color="magenta"];146 -> 177[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 146 -> 178[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 185[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv22))) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];185 -> 198[label="",style="solid", color="black", weight=3]; 78.58/41.39 186[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv25))) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];186 -> 199[label="",style="solid", color="black", weight=3]; 78.58/41.39 640[label="xv970",fontsize=16,color="green",shape="box"];641[label="xv980",fontsize=16,color="green",shape="box"];642[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS xv95 (Succ xv96)) (Succ (Succ xv96))))) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];642 -> 649[label="",style="solid", color="black", weight=3]; 78.58/41.39 643[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos Zero) (fromInt (Pos Zero)))",fontsize=16,color="black",shape="box"];643 -> 650[label="",style="solid", color="black", weight=3]; 78.58/41.39 174[label="showInt1R' xv12 (Pos (Succ (Succ xv130)))",fontsize=16,color="black",shape="triangle"];174 -> 194[label="",style="solid", color="black", weight=3]; 78.58/41.39 175[label="showInt1N' xv12 (Pos (Succ (Succ xv130)))",fontsize=16,color="black",shape="box"];175 -> 195[label="",style="solid", color="black", weight=3]; 78.58/41.39 176[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero)))) : xv12",fontsize=16,color="green",shape="box"];176 -> 196[label="",style="dashed", color="green", weight=3]; 78.58/41.39 177 -> 145[label="",style="dashed", color="red", weight=0]; 78.58/41.39 177[label="showInt1R' xv12 (Pos (Succ Zero))",fontsize=16,color="magenta"];178[label="showInt1N' xv12 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];178 -> 197[label="",style="solid", color="black", weight=3]; 78.58/41.39 198[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv22))) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];198 -> 213[label="",style="solid", color="black", weight=3]; 78.58/41.39 199[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv25))) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];199 -> 214[label="",style="solid", color="black", weight=3]; 78.58/41.39 649[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos (Succ (primDivNatS (primMinusNatS xv95 (Succ xv96)) (Succ (Succ xv96))))) (Pos Zero))",fontsize=16,color="black",shape="box"];649 -> 657[label="",style="solid", color="black", weight=3]; 78.58/41.39 650[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];650 -> 658[label="",style="solid", color="black", weight=3]; 78.58/41.39 194[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130))))) : xv12",fontsize=16,color="green",shape="box"];194 -> 208[label="",style="dashed", color="green", weight=3]; 78.58/41.39 195[label="showInt1N'0 xv12 (Pos (Succ (Succ xv130))) (showInt1Vu76 xv12 (Pos (Succ (Succ xv130))))",fontsize=16,color="black",shape="box"];195 -> 209[label="",style="solid", color="black", weight=3]; 78.58/41.39 196[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero))))",fontsize=16,color="black",shape="box"];196 -> 246[label="",style="solid", color="black", weight=3]; 78.58/41.39 197[label="showInt1N'0 xv12 (Pos (Succ Zero)) (showInt1Vu76 xv12 (Pos (Succ Zero)))",fontsize=16,color="black",shape="box"];197 -> 215[label="",style="solid", color="black", weight=3]; 78.58/41.39 213[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromIntegral (showInt1D xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];213 -> 228[label="",style="solid", color="black", weight=3]; 78.58/41.39 214[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromIntegral (showInt1D xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];214 -> 229[label="",style="solid", color="black", weight=3]; 78.58/41.39 657[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) False",fontsize=16,color="black",shape="box"];657 -> 666[label="",style="solid", color="black", weight=3]; 78.58/41.39 658[label="showInt1ShowInt0 xv94 (Pos (Succ xv95)) True",fontsize=16,color="black",shape="box"];658 -> 667[label="",style="solid", color="black", weight=3]; 78.58/41.39 208[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130)))))",fontsize=16,color="black",shape="box"];208 -> 270[label="",style="solid", color="black", weight=3]; 78.58/41.39 209 -> 474[label="",style="dashed", color="red", weight=0]; 78.58/41.39 209[label="showInt1N'0 xv12 (Pos (Succ (Succ xv130))) (quotRem (Pos (Succ (Succ xv130))) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];209 -> 475[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 209 -> 476[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 209 -> 477[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 246 -> 533[label="",style="dashed", color="red", weight=0]; 78.58/41.39 246[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ Zero))))",fontsize=16,color="magenta"];246 -> 534[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 246 -> 535[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 246 -> 536[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 215 -> 474[label="",style="dashed", color="red", weight=0]; 78.58/41.39 215[label="showInt1N'0 xv12 (Pos (Succ Zero)) (quotRem (Pos (Succ Zero)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];215 -> 478[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 215 -> 479[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 215 -> 480[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 228[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];228 -> 247[label="",style="solid", color="black", weight=3]; 78.58/41.39 229[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];229 -> 248[label="",style="solid", color="black", weight=3]; 78.58/41.39 666 -> 4[label="",style="dashed", color="red", weight=0]; 78.58/41.39 666[label="showInt (showInt1N' xv94 (Pos (Succ xv95))) (showInt1R' xv94 (Pos (Succ xv95)))",fontsize=16,color="magenta"];666 -> 673[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 666 -> 674[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 667[label="showInt1R' xv94 (Pos (Succ xv95))",fontsize=16,color="black",shape="triangle"];667 -> 675[label="",style="solid", color="black", weight=3]; 78.58/41.39 270 -> 533[label="",style="dashed", color="red", weight=0]; 78.58/41.39 270[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv12 (Pos (Succ (Succ xv130)))))",fontsize=16,color="magenta"];270 -> 537[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 270 -> 538[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 270 -> 539[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 475[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];476[label="xv12",fontsize=16,color="green",shape="box"];477[label="Succ xv130",fontsize=16,color="green",shape="box"];474[label="showInt1N'0 xv82 (Pos (Succ xv83)) (quotRem (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="triangle"];474 -> 490[label="",style="solid", color="black", weight=3]; 78.58/41.39 534[label="Zero",fontsize=16,color="green",shape="box"];535[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];536[label="xv12",fontsize=16,color="green",shape="box"];533[label="primIntToChar (fromEnum (Char (Succ xv86)) + fromIntegral (showInt1D xv87 (Pos (Succ xv88))))",fontsize=16,color="black",shape="triangle"];533 -> 549[label="",style="solid", color="black", weight=3]; 78.58/41.39 478[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];479[label="xv12",fontsize=16,color="green",shape="box"];480[label="Zero",fontsize=16,color="green",shape="box"];247[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger (toInteger (showInt1D xv23 (Pos Zero)))))",fontsize=16,color="black",shape="box"];247 -> 272[label="",style="solid", color="black", weight=3]; 78.58/41.39 248[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger (toInteger (showInt1D xv26 (Neg Zero)))))",fontsize=16,color="black",shape="box"];248 -> 273[label="",style="solid", color="black", weight=3]; 78.58/41.39 673 -> 667[label="",style="dashed", color="red", weight=0]; 78.58/41.39 673[label="showInt1R' xv94 (Pos (Succ xv95))",fontsize=16,color="magenta"];674[label="showInt1N' xv94 (Pos (Succ xv95))",fontsize=16,color="black",shape="box"];674 -> 683[label="",style="solid", color="black", weight=3]; 78.58/41.39 675[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95)))) : xv94",fontsize=16,color="green",shape="box"];675 -> 684[label="",style="dashed", color="green", weight=3]; 78.58/41.39 537[label="Succ xv130",fontsize=16,color="green",shape="box"];538[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];539[label="xv12",fontsize=16,color="green",shape="box"];490[label="showInt1N'0 xv82 (Pos (Succ xv83)) (primQrmInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="box"];490 -> 509[label="",style="solid", color="black", weight=3]; 78.58/41.39 549[label="primIntToChar (primPlusInt (fromEnum (Char (Succ xv86))) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];549 -> 560[label="",style="solid", color="black", weight=3]; 78.58/41.39 272[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (fromInteger (Integer (showInt1D xv23 (Pos Zero)))))",fontsize=16,color="black",shape="box"];272 -> 290[label="",style="solid", color="black", weight=3]; 78.58/41.39 273[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (fromInteger (Integer (showInt1D xv26 (Neg Zero)))))",fontsize=16,color="black",shape="box"];273 -> 291[label="",style="solid", color="black", weight=3]; 78.58/41.39 683[label="showInt1N'0 xv94 (Pos (Succ xv95)) (showInt1Vu76 xv94 (Pos (Succ xv95)))",fontsize=16,color="black",shape="box"];683 -> 694[label="",style="solid", color="black", weight=3]; 78.58/41.39 684[label="toEnum (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95))))",fontsize=16,color="black",shape="box"];684 -> 713[label="",style="solid", color="black", weight=3]; 78.58/41.39 509[label="showInt1N'0 xv82 (Pos (Succ xv83)) (primQuotInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))),primRemInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84))))",fontsize=16,color="black",shape="box"];509 -> 532[label="",style="solid", color="black", weight=3]; 78.58/41.39 560[label="primIntToChar (primPlusInt (primCharToInt (Char (Succ xv86))) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];560 -> 567[label="",style="solid", color="black", weight=3]; 78.58/41.39 290[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D xv23 (Pos Zero)))",fontsize=16,color="black",shape="box"];290 -> 304[label="",style="solid", color="black", weight=3]; 78.58/41.39 291[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D xv26 (Neg Zero)))",fontsize=16,color="black",shape="box"];291 -> 305[label="",style="solid", color="black", weight=3]; 78.58/41.39 694 -> 474[label="",style="dashed", color="red", weight=0]; 78.58/41.39 694[label="showInt1N'0 xv94 (Pos (Succ xv95)) (quotRem (Pos (Succ xv95)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))",fontsize=16,color="magenta"];694 -> 699[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 694 -> 700[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 694 -> 701[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 713 -> 533[label="",style="dashed", color="red", weight=0]; 78.58/41.39 713[label="primIntToChar (fromEnum (Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))))) + fromIntegral (showInt1D xv94 (Pos (Succ xv95))))",fontsize=16,color="magenta"];713 -> 723[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 713 -> 724[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 713 -> 725[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 532[label="primQuotInt (Pos (Succ xv83)) (fromInt (Pos (Succ xv84)))",fontsize=16,color="black",shape="triangle"];532 -> 550[label="",style="solid", color="black", weight=3]; 78.58/41.39 567[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromIntegral (showInt1D xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];567 -> 622[label="",style="solid", color="black", weight=3]; 78.58/41.39 304[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D0 xv23 (Pos Zero) (showInt1Vu76 xv23 (Pos Zero))))",fontsize=16,color="black",shape="box"];304 -> 317[label="",style="solid", color="black", weight=3]; 78.58/41.39 305[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D0 xv26 (Neg Zero) (showInt1Vu76 xv26 (Neg Zero))))",fontsize=16,color="black",shape="box"];305 -> 318[label="",style="solid", color="black", weight=3]; 78.58/41.39 699[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];700[label="xv94",fontsize=16,color="green",shape="box"];701[label="xv95",fontsize=16,color="green",shape="box"];723[label="xv95",fontsize=16,color="green",shape="box"];724[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))))",fontsize=16,color="green",shape="box"];725[label="xv94",fontsize=16,color="green",shape="box"];550[label="primQuotInt (Pos (Succ xv83)) (Pos (Succ xv84))",fontsize=16,color="black",shape="box"];550 -> 561[label="",style="solid", color="black", weight=3]; 78.58/41.39 622[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger . toInteger))",fontsize=16,color="black",shape="box"];622 -> 628[label="",style="solid", color="black", weight=3]; 78.58/41.39 317 -> 337[label="",style="dashed", color="red", weight=0]; 78.58/41.39 317[label="primIntToChar (primPlusInt (Pos (Succ xv22)) (showInt1D0 xv23 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];317 -> 338[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 317 -> 339[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 317 -> 340[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 318 -> 341[label="",style="dashed", color="red", weight=0]; 78.58/41.39 318[label="primIntToChar (primPlusInt (Pos (Succ xv25)) (showInt1D0 xv26 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];318 -> 342[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 318 -> 343[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 318 -> 344[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 561[label="Pos (primDivNatS (Succ xv83) (Succ xv84))",fontsize=16,color="green",shape="box"];561 -> 568[label="",style="dashed", color="green", weight=3]; 78.58/41.39 628[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger (toInteger (showInt1D xv87 (Pos (Succ xv88))))))",fontsize=16,color="black",shape="box"];628 -> 635[label="",style="solid", color="black", weight=3]; 78.58/41.39 338[label="xv22",fontsize=16,color="green",shape="box"];339[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];340[label="xv23",fontsize=16,color="green",shape="box"];337[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (quotRem (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="triangle"];337 -> 359[label="",style="solid", color="black", weight=3]; 78.58/41.39 342[label="xv25",fontsize=16,color="green",shape="box"];343[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];344[label="xv26",fontsize=16,color="green",shape="box"];341[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (quotRem (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="triangle"];341 -> 360[label="",style="solid", color="black", weight=3]; 78.58/41.39 568[label="primDivNatS (Succ xv83) (Succ xv84)",fontsize=16,color="black",shape="triangle"];568 -> 623[label="",style="solid", color="black", weight=3]; 78.58/41.39 635[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (fromInteger (Integer (showInt1D xv87 (Pos (Succ xv88))))))",fontsize=16,color="black",shape="box"];635 -> 644[label="",style="solid", color="black", weight=3]; 78.58/41.39 359[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (primQrmInt (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="box"];359 -> 371[label="",style="solid", color="black", weight=3]; 78.58/41.39 360[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (primQrmInt (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="box"];360 -> 372[label="",style="solid", color="black", weight=3]; 78.58/41.39 623[label="primDivNatS0 xv83 xv84 (primGEqNatS xv83 xv84)",fontsize=16,color="burlywood",shape="box"];1463[label="xv83/Succ xv830",fontsize=10,color="white",style="solid",shape="box"];623 -> 1463[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1463 -> 629[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1464[label="xv83/Zero",fontsize=10,color="white",style="solid",shape="box"];623 -> 1464[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1464 -> 630[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 644[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D xv87 (Pos (Succ xv88))))",fontsize=16,color="black",shape="box"];644 -> 651[label="",style="solid", color="black", weight=3]; 78.58/41.39 371[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (showInt1D0 xv59 (Pos Zero) (primQuotInt (Pos Zero) (fromInt (Pos (Succ xv60))),primRemInt (Pos Zero) (fromInt (Pos (Succ xv60))))))",fontsize=16,color="black",shape="box"];371 -> 385[label="",style="solid", color="black", weight=3]; 78.58/41.39 372[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (showInt1D0 xv63 (Neg Zero) (primQuotInt (Neg Zero) (fromInt (Pos (Succ xv64))),primRemInt (Neg Zero) (fromInt (Pos (Succ xv64))))))",fontsize=16,color="black",shape="box"];372 -> 386[label="",style="solid", color="black", weight=3]; 78.58/41.39 629[label="primDivNatS0 (Succ xv830) xv84 (primGEqNatS (Succ xv830) xv84)",fontsize=16,color="burlywood",shape="box"];1465[label="xv84/Succ xv840",fontsize=10,color="white",style="solid",shape="box"];629 -> 1465[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1465 -> 636[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1466[label="xv84/Zero",fontsize=10,color="white",style="solid",shape="box"];629 -> 1466[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1466 -> 637[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 630[label="primDivNatS0 Zero xv84 (primGEqNatS Zero xv84)",fontsize=16,color="burlywood",shape="box"];1467[label="xv84/Succ xv840",fontsize=10,color="white",style="solid",shape="box"];630 -> 1467[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1467 -> 638[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1468[label="xv84/Zero",fontsize=10,color="white",style="solid",shape="box"];630 -> 1468[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1468 -> 639[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 651[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D0 xv87 (Pos (Succ xv88)) (showInt1Vu76 xv87 (Pos (Succ xv88)))))",fontsize=16,color="black",shape="box"];651 -> 659[label="",style="solid", color="black", weight=3]; 78.58/41.39 385[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (primRemInt (Pos Zero) (fromInt (Pos (Succ xv60)))))",fontsize=16,color="black",shape="box"];385 -> 401[label="",style="solid", color="black", weight=3]; 78.58/41.39 386[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (primRemInt (Neg Zero) (fromInt (Pos (Succ xv64)))))",fontsize=16,color="black",shape="box"];386 -> 402[label="",style="solid", color="black", weight=3]; 78.58/41.39 636[label="primDivNatS0 (Succ xv830) (Succ xv840) (primGEqNatS (Succ xv830) (Succ xv840))",fontsize=16,color="black",shape="box"];636 -> 645[label="",style="solid", color="black", weight=3]; 78.58/41.39 637[label="primDivNatS0 (Succ xv830) Zero (primGEqNatS (Succ xv830) Zero)",fontsize=16,color="black",shape="box"];637 -> 646[label="",style="solid", color="black", weight=3]; 78.58/41.39 638[label="primDivNatS0 Zero (Succ xv840) (primGEqNatS Zero (Succ xv840))",fontsize=16,color="black",shape="box"];638 -> 647[label="",style="solid", color="black", weight=3]; 78.58/41.39 639[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];639 -> 648[label="",style="solid", color="black", weight=3]; 78.58/41.39 659 -> 668[label="",style="dashed", color="red", weight=0]; 78.58/41.39 659[label="primIntToChar (primPlusInt (Pos (Succ xv86)) (showInt1D0 xv87 (Pos (Succ xv88)) (quotRem (Pos (Succ xv88)) (fromInt (Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))",fontsize=16,color="magenta"];659 -> 669[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 659 -> 670[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 659 -> 671[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 659 -> 672[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 401[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (primRemInt (Pos Zero) (Pos (Succ xv60))))",fontsize=16,color="black",shape="box"];401 -> 421[label="",style="solid", color="black", weight=3]; 78.58/41.39 402[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (primRemInt (Neg Zero) (Pos (Succ xv64))))",fontsize=16,color="black",shape="box"];402 -> 422[label="",style="solid", color="black", weight=3]; 78.58/41.39 645 -> 917[label="",style="dashed", color="red", weight=0]; 78.58/41.39 645[label="primDivNatS0 (Succ xv830) (Succ xv840) (primGEqNatS xv830 xv840)",fontsize=16,color="magenta"];645 -> 918[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 645 -> 919[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 645 -> 920[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 645 -> 921[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 646[label="primDivNatS0 (Succ xv830) Zero True",fontsize=16,color="black",shape="box"];646 -> 654[label="",style="solid", color="black", weight=3]; 78.58/41.39 647[label="primDivNatS0 Zero (Succ xv840) False",fontsize=16,color="black",shape="box"];647 -> 655[label="",style="solid", color="black", weight=3]; 78.58/41.39 648[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];648 -> 656[label="",style="solid", color="black", weight=3]; 78.58/41.39 669[label="xv87",fontsize=16,color="green",shape="box"];670[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];671[label="xv88",fontsize=16,color="green",shape="box"];672[label="xv86",fontsize=16,color="green",shape="box"];668[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (quotRem (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="triangle"];668 -> 676[label="",style="solid", color="black", weight=3]; 78.58/41.39 421[label="primIntToChar (primPlusInt (Pos (Succ xv58)) (Pos (primModNatS Zero (Succ xv60))))",fontsize=16,color="black",shape="box"];421 -> 444[label="",style="solid", color="black", weight=3]; 78.58/41.39 422[label="primIntToChar (primPlusInt (Pos (Succ xv62)) (Neg (primModNatS Zero (Succ xv64))))",fontsize=16,color="black",shape="box"];422 -> 445[label="",style="solid", color="black", weight=3]; 78.58/41.39 918[label="xv840",fontsize=16,color="green",shape="box"];919[label="xv840",fontsize=16,color="green",shape="box"];920[label="xv830",fontsize=16,color="green",shape="box"];921[label="xv830",fontsize=16,color="green",shape="box"];917[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS xv127 xv128)",fontsize=16,color="burlywood",shape="triangle"];1469[label="xv127/Succ xv1270",fontsize=10,color="white",style="solid",shape="box"];917 -> 1469[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1469 -> 950[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1470[label="xv127/Zero",fontsize=10,color="white",style="solid",shape="box"];917 -> 1470[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1470 -> 951[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 654[label="Succ (primDivNatS (primMinusNatS (Succ xv830) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];654 -> 664[label="",style="dashed", color="green", weight=3]; 78.58/41.39 655[label="Zero",fontsize=16,color="green",shape="box"];656[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];656 -> 665[label="",style="dashed", color="green", weight=3]; 78.58/41.39 676[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (primQrmInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="box"];676 -> 685[label="",style="solid", color="black", weight=3]; 78.58/41.39 444[label="primIntToChar (Pos (primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60))))",fontsize=16,color="black",shape="box"];444 -> 467[label="",style="solid", color="black", weight=3]; 78.58/41.39 445[label="primIntToChar (primMinusNat (Succ xv62) (primModNatS Zero (Succ xv64)))",fontsize=16,color="black",shape="box"];445 -> 468[label="",style="solid", color="black", weight=3]; 78.58/41.39 950[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) xv128)",fontsize=16,color="burlywood",shape="box"];1471[label="xv128/Succ xv1280",fontsize=10,color="white",style="solid",shape="box"];950 -> 1471[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1471 -> 963[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1472[label="xv128/Zero",fontsize=10,color="white",style="solid",shape="box"];950 -> 1472[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1472 -> 964[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 951[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero xv128)",fontsize=16,color="burlywood",shape="box"];1473[label="xv128/Succ xv1280",fontsize=10,color="white",style="solid",shape="box"];951 -> 1473[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1473 -> 965[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1474[label="xv128/Zero",fontsize=10,color="white",style="solid",shape="box"];951 -> 1474[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1474 -> 966[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 664 -> 1379[label="",style="dashed", color="red", weight=0]; 78.58/41.39 664[label="primDivNatS (primMinusNatS (Succ xv830) Zero) (Succ Zero)",fontsize=16,color="magenta"];664 -> 1380[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 664 -> 1381[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 664 -> 1382[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 665 -> 1379[label="",style="dashed", color="red", weight=0]; 78.58/41.39 665[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];665 -> 1383[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 665 -> 1384[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 665 -> 1385[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 685 -> 711[label="",style="dashed", color="red", weight=0]; 78.58/41.39 685[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (primQuotInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))),primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="magenta"];685 -> 712[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 467[label="Char (primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60)))",fontsize=16,color="green",shape="box"];467 -> 502[label="",style="dashed", color="green", weight=3]; 78.58/41.39 468[label="primIntToChar (primMinusNat (Succ xv62) Zero)",fontsize=16,color="black",shape="box"];468 -> 503[label="",style="solid", color="black", weight=3]; 78.58/41.39 963[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) (Succ xv1280))",fontsize=16,color="black",shape="box"];963 -> 976[label="",style="solid", color="black", weight=3]; 78.58/41.39 964[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS (Succ xv1270) Zero)",fontsize=16,color="black",shape="box"];964 -> 977[label="",style="solid", color="black", weight=3]; 78.58/41.39 965[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero (Succ xv1280))",fontsize=16,color="black",shape="box"];965 -> 978[label="",style="solid", color="black", weight=3]; 78.58/41.39 966[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];966 -> 979[label="",style="solid", color="black", weight=3]; 78.58/41.39 1380[label="Zero",fontsize=16,color="green",shape="box"];1381[label="Succ xv830",fontsize=16,color="green",shape="box"];1382[label="Zero",fontsize=16,color="green",shape="box"];1379[label="primDivNatS (primMinusNatS xv168 xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="triangle"];1475[label="xv168/Succ xv1680",fontsize=10,color="white",style="solid",shape="box"];1379 -> 1475[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1475 -> 1413[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1476[label="xv168/Zero",fontsize=10,color="white",style="solid",shape="box"];1379 -> 1476[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1476 -> 1414[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1383[label="Zero",fontsize=16,color="green",shape="box"];1384[label="Zero",fontsize=16,color="green",shape="box"];1385[label="Zero",fontsize=16,color="green",shape="box"];712 -> 532[label="",style="dashed", color="red", weight=0]; 78.58/41.39 712[label="primQuotInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103)))",fontsize=16,color="magenta"];712 -> 714[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 712 -> 715[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 711[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (showInt1D0 xv101 (Pos (Succ xv102)) (xv108,primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103))))))",fontsize=16,color="black",shape="triangle"];711 -> 716[label="",style="solid", color="black", weight=3]; 78.58/41.39 502[label="primPlusNat (Succ xv58) (primModNatS Zero (Succ xv60))",fontsize=16,color="black",shape="triangle"];502 -> 522[label="",style="solid", color="black", weight=3]; 78.58/41.39 503[label="primIntToChar (Pos (Succ xv62))",fontsize=16,color="black",shape="box"];503 -> 523[label="",style="solid", color="black", weight=3]; 78.58/41.39 976 -> 917[label="",style="dashed", color="red", weight=0]; 78.58/41.39 976[label="primDivNatS0 (Succ xv125) (Succ xv126) (primGEqNatS xv1270 xv1280)",fontsize=16,color="magenta"];976 -> 988[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 976 -> 989[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 977[label="primDivNatS0 (Succ xv125) (Succ xv126) True",fontsize=16,color="black",shape="triangle"];977 -> 990[label="",style="solid", color="black", weight=3]; 78.58/41.39 978[label="primDivNatS0 (Succ xv125) (Succ xv126) False",fontsize=16,color="black",shape="box"];978 -> 991[label="",style="solid", color="black", weight=3]; 78.58/41.39 979 -> 977[label="",style="dashed", color="red", weight=0]; 78.58/41.39 979[label="primDivNatS0 (Succ xv125) (Succ xv126) True",fontsize=16,color="magenta"];1413[label="primDivNatS (primMinusNatS (Succ xv1680) xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="box"];1477[label="xv169/Succ xv1690",fontsize=10,color="white",style="solid",shape="box"];1413 -> 1477[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1477 -> 1417[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1478[label="xv169/Zero",fontsize=10,color="white",style="solid",shape="box"];1413 -> 1478[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1478 -> 1418[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1414[label="primDivNatS (primMinusNatS Zero xv169) (Succ xv170)",fontsize=16,color="burlywood",shape="box"];1479[label="xv169/Succ xv1690",fontsize=10,color="white",style="solid",shape="box"];1414 -> 1479[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1479 -> 1419[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1480[label="xv169/Zero",fontsize=10,color="white",style="solid",shape="box"];1414 -> 1480[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1480 -> 1420[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 714[label="xv103",fontsize=16,color="green",shape="box"];715[label="xv102",fontsize=16,color="green",shape="box"];716[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (primRemInt (Pos (Succ xv102)) (fromInt (Pos (Succ xv103)))))",fontsize=16,color="black",shape="box"];716 -> 726[label="",style="solid", color="black", weight=3]; 78.58/41.39 522[label="primPlusNat (Succ xv58) Zero",fontsize=16,color="black",shape="box"];522 -> 551[label="",style="solid", color="black", weight=3]; 78.58/41.39 523[label="Char (Succ xv62)",fontsize=16,color="green",shape="box"];988[label="xv1280",fontsize=16,color="green",shape="box"];989[label="xv1270",fontsize=16,color="green",shape="box"];990[label="Succ (primDivNatS (primMinusNatS (Succ xv125) (Succ xv126)) (Succ (Succ xv126)))",fontsize=16,color="green",shape="box"];990 -> 1002[label="",style="dashed", color="green", weight=3]; 78.58/41.39 991[label="Zero",fontsize=16,color="green",shape="box"];1417[label="primDivNatS (primMinusNatS (Succ xv1680) (Succ xv1690)) (Succ xv170)",fontsize=16,color="black",shape="box"];1417 -> 1425[label="",style="solid", color="black", weight=3]; 78.58/41.39 1418[label="primDivNatS (primMinusNatS (Succ xv1680) Zero) (Succ xv170)",fontsize=16,color="black",shape="box"];1418 -> 1426[label="",style="solid", color="black", weight=3]; 78.58/41.39 1419[label="primDivNatS (primMinusNatS Zero (Succ xv1690)) (Succ xv170)",fontsize=16,color="black",shape="box"];1419 -> 1427[label="",style="solid", color="black", weight=3]; 78.58/41.39 1420[label="primDivNatS (primMinusNatS Zero Zero) (Succ xv170)",fontsize=16,color="black",shape="box"];1420 -> 1428[label="",style="solid", color="black", weight=3]; 78.58/41.39 726[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (primRemInt (Pos (Succ xv102)) (Pos (Succ xv103))))",fontsize=16,color="black",shape="box"];726 -> 733[label="",style="solid", color="black", weight=3]; 78.58/41.39 551[label="Succ xv58",fontsize=16,color="green",shape="box"];1002 -> 1379[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1002[label="primDivNatS (primMinusNatS (Succ xv125) (Succ xv126)) (Succ (Succ xv126))",fontsize=16,color="magenta"];1002 -> 1386[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1002 -> 1387[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1002 -> 1388[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1425 -> 1379[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1425[label="primDivNatS (primMinusNatS xv1680 xv1690) (Succ xv170)",fontsize=16,color="magenta"];1425 -> 1433[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1425 -> 1434[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1426 -> 568[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1426[label="primDivNatS (Succ xv1680) (Succ xv170)",fontsize=16,color="magenta"];1426 -> 1435[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1426 -> 1436[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1427[label="primDivNatS Zero (Succ xv170)",fontsize=16,color="black",shape="triangle"];1427 -> 1437[label="",style="solid", color="black", weight=3]; 78.58/41.39 1428 -> 1427[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1428[label="primDivNatS Zero (Succ xv170)",fontsize=16,color="magenta"];733[label="primIntToChar (primPlusInt (Pos (Succ xv100)) (Pos (primModNatS (Succ xv102) (Succ xv103))))",fontsize=16,color="black",shape="box"];733 -> 741[label="",style="solid", color="black", weight=3]; 78.58/41.39 1386[label="Succ xv126",fontsize=16,color="green",shape="box"];1387[label="Succ xv125",fontsize=16,color="green",shape="box"];1388[label="Succ xv126",fontsize=16,color="green",shape="box"];1433[label="xv1680",fontsize=16,color="green",shape="box"];1434[label="xv1690",fontsize=16,color="green",shape="box"];1435[label="xv170",fontsize=16,color="green",shape="box"];1436[label="xv1680",fontsize=16,color="green",shape="box"];1437[label="Zero",fontsize=16,color="green",shape="box"];741[label="primIntToChar (Pos (primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103))))",fontsize=16,color="black",shape="box"];741 -> 751[label="",style="solid", color="black", weight=3]; 78.58/41.39 751[label="Char (primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103)))",fontsize=16,color="green",shape="box"];751 -> 758[label="",style="dashed", color="green", weight=3]; 78.58/41.39 758[label="primPlusNat (Succ xv100) (primModNatS (Succ xv102) (Succ xv103))",fontsize=16,color="black",shape="triangle"];758 -> 766[label="",style="solid", color="black", weight=3]; 78.58/41.39 766[label="primPlusNat (Succ xv100) (primModNatS0 xv102 xv103 (primGEqNatS xv102 xv103))",fontsize=16,color="burlywood",shape="box"];1481[label="xv102/Succ xv1020",fontsize=10,color="white",style="solid",shape="box"];766 -> 1481[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1481 -> 775[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1482[label="xv102/Zero",fontsize=10,color="white",style="solid",shape="box"];766 -> 1482[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1482 -> 776[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 775[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) xv103 (primGEqNatS (Succ xv1020) xv103))",fontsize=16,color="burlywood",shape="box"];1483[label="xv103/Succ xv1030",fontsize=10,color="white",style="solid",shape="box"];775 -> 1483[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1483 -> 786[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1484[label="xv103/Zero",fontsize=10,color="white",style="solid",shape="box"];775 -> 1484[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1484 -> 787[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 776[label="primPlusNat (Succ xv100) (primModNatS0 Zero xv103 (primGEqNatS Zero xv103))",fontsize=16,color="burlywood",shape="box"];1485[label="xv103/Succ xv1030",fontsize=10,color="white",style="solid",shape="box"];776 -> 1485[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1485 -> 788[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1486[label="xv103/Zero",fontsize=10,color="white",style="solid",shape="box"];776 -> 1486[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1486 -> 789[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 786[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) (Succ xv1030) (primGEqNatS (Succ xv1020) (Succ xv1030)))",fontsize=16,color="black",shape="box"];786 -> 797[label="",style="solid", color="black", weight=3]; 78.58/41.39 787[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) Zero (primGEqNatS (Succ xv1020) Zero))",fontsize=16,color="black",shape="box"];787 -> 798[label="",style="solid", color="black", weight=3]; 78.58/41.39 788[label="primPlusNat (Succ xv100) (primModNatS0 Zero (Succ xv1030) (primGEqNatS Zero (Succ xv1030)))",fontsize=16,color="black",shape="box"];788 -> 799[label="",style="solid", color="black", weight=3]; 78.58/41.39 789[label="primPlusNat (Succ xv100) (primModNatS0 Zero Zero (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];789 -> 800[label="",style="solid", color="black", weight=3]; 78.58/41.39 797[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) (Succ xv1030) (primGEqNatS xv1020 xv1030))",fontsize=16,color="burlywood",shape="box"];1487[label="xv1020/Succ xv10200",fontsize=10,color="white",style="solid",shape="box"];797 -> 1487[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1487 -> 809[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1488[label="xv1020/Zero",fontsize=10,color="white",style="solid",shape="box"];797 -> 1488[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1488 -> 810[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 798[label="primPlusNat (Succ xv100) (primModNatS0 (Succ xv1020) Zero True)",fontsize=16,color="black",shape="box"];798 -> 811[label="",style="solid", color="black", weight=3]; 78.58/41.39 799[label="primPlusNat (Succ xv100) (primModNatS0 Zero (Succ xv1030) False)",fontsize=16,color="black",shape="box"];799 -> 812[label="",style="solid", color="black", weight=3]; 78.58/41.39 800[label="primPlusNat (Succ xv100) (primModNatS0 Zero Zero True)",fontsize=16,color="black",shape="box"];800 -> 813[label="",style="solid", color="black", weight=3]; 78.58/41.39 809[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ xv1030) (primGEqNatS (Succ xv10200) xv1030))",fontsize=16,color="burlywood",shape="box"];1489[label="xv1030/Succ xv10300",fontsize=10,color="white",style="solid",shape="box"];809 -> 1489[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1489 -> 822[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1490[label="xv1030/Zero",fontsize=10,color="white",style="solid",shape="box"];809 -> 1490[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1490 -> 823[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 810[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ xv1030) (primGEqNatS Zero xv1030))",fontsize=16,color="burlywood",shape="box"];1491[label="xv1030/Succ xv10300",fontsize=10,color="white",style="solid",shape="box"];810 -> 1491[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1491 -> 824[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1492[label="xv1030/Zero",fontsize=10,color="white",style="solid",shape="box"];810 -> 1492[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1492 -> 825[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 811[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv1020) Zero) (Succ Zero))",fontsize=16,color="black",shape="box"];811 -> 826[label="",style="solid", color="black", weight=3]; 78.58/41.39 812 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 812[label="primPlusNat (Succ xv100) (Succ Zero)",fontsize=16,color="magenta"];812 -> 1098[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 812 -> 1099[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 813[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="black",shape="box"];813 -> 828[label="",style="solid", color="black", weight=3]; 78.58/41.39 822[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ (Succ xv10300)) (primGEqNatS (Succ xv10200) (Succ xv10300)))",fontsize=16,color="black",shape="box"];822 -> 846[label="",style="solid", color="black", weight=3]; 78.58/41.39 823[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ Zero) (primGEqNatS (Succ xv10200) Zero))",fontsize=16,color="black",shape="box"];823 -> 847[label="",style="solid", color="black", weight=3]; 78.58/41.39 824[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ (Succ xv10300)) (primGEqNatS Zero (Succ xv10300)))",fontsize=16,color="black",shape="box"];824 -> 848[label="",style="solid", color="black", weight=3]; 78.58/41.39 825[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ Zero) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];825 -> 849[label="",style="solid", color="black", weight=3]; 78.58/41.39 826 -> 758[label="",style="dashed", color="red", weight=0]; 78.58/41.39 826[label="primPlusNat (Succ xv100) (primModNatS (Succ xv1020) (Succ Zero))",fontsize=16,color="magenta"];826 -> 850[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 826 -> 851[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1098[label="xv100",fontsize=16,color="green",shape="box"];1099[label="Zero",fontsize=16,color="green",shape="box"];1097[label="primPlusNat (Succ xv133) (Succ xv134)",fontsize=16,color="black",shape="triangle"];1097 -> 1114[label="",style="solid", color="black", weight=3]; 78.58/41.39 828 -> 502[label="",style="dashed", color="red", weight=0]; 78.58/41.39 828[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ Zero))",fontsize=16,color="magenta"];828 -> 853[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 828 -> 854[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 846[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ (Succ xv10300)) (primGEqNatS xv10200 xv10300))",fontsize=16,color="burlywood",shape="box"];1493[label="xv10200/Succ xv102000",fontsize=10,color="white",style="solid",shape="box"];846 -> 1493[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1493 -> 884[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1494[label="xv10200/Zero",fontsize=10,color="white",style="solid",shape="box"];846 -> 1494[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1494 -> 885[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 847[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ xv10200)) (Succ Zero) True)",fontsize=16,color="black",shape="box"];847 -> 886[label="",style="solid", color="black", weight=3]; 78.58/41.39 848[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ (Succ xv10300)) False)",fontsize=16,color="black",shape="box"];848 -> 887[label="",style="solid", color="black", weight=3]; 78.58/41.39 849[label="primPlusNat (Succ xv100) (primModNatS0 (Succ Zero) (Succ Zero) True)",fontsize=16,color="black",shape="box"];849 -> 888[label="",style="solid", color="black", weight=3]; 78.58/41.39 850[label="Zero",fontsize=16,color="green",shape="box"];851[label="xv1020",fontsize=16,color="green",shape="box"];1114[label="Succ (Succ (primPlusNat xv133 xv134))",fontsize=16,color="green",shape="box"];1114 -> 1128[label="",style="dashed", color="green", weight=3]; 78.58/41.39 853[label="xv100",fontsize=16,color="green",shape="box"];854[label="Zero",fontsize=16,color="green",shape="box"];884[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ xv10300)) (primGEqNatS (Succ xv102000) xv10300))",fontsize=16,color="burlywood",shape="box"];1495[label="xv10300/Succ xv103000",fontsize=10,color="white",style="solid",shape="box"];884 -> 1495[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1495 -> 896[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1496[label="xv10300/Zero",fontsize=10,color="white",style="solid",shape="box"];884 -> 1496[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1496 -> 897[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 885[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ xv10300)) (primGEqNatS Zero xv10300))",fontsize=16,color="burlywood",shape="box"];1497[label="xv10300/Succ xv103000",fontsize=10,color="white",style="solid",shape="box"];885 -> 1497[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1497 -> 898[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1498[label="xv10300/Zero",fontsize=10,color="white",style="solid",shape="box"];885 -> 1498[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1498 -> 899[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 886[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv10200)) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];886 -> 900[label="",style="solid", color="black", weight=3]; 78.58/41.39 887 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 887[label="primPlusNat (Succ xv100) (Succ (Succ Zero))",fontsize=16,color="magenta"];887 -> 1100[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 887 -> 1101[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 888[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];888 -> 902[label="",style="solid", color="black", weight=3]; 78.58/41.39 1128[label="primPlusNat xv133 xv134",fontsize=16,color="burlywood",shape="triangle"];1499[label="xv133/Succ xv1330",fontsize=10,color="white",style="solid",shape="box"];1128 -> 1499[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1499 -> 1144[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1500[label="xv133/Zero",fontsize=10,color="white",style="solid",shape="box"];1128 -> 1500[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1500 -> 1145[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 896[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ (Succ xv103000))) (primGEqNatS (Succ xv102000) (Succ xv103000)))",fontsize=16,color="black",shape="box"];896 -> 910[label="",style="solid", color="black", weight=3]; 78.58/41.39 897[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ Zero)) (primGEqNatS (Succ xv102000) Zero))",fontsize=16,color="black",shape="box"];897 -> 911[label="",style="solid", color="black", weight=3]; 78.58/41.39 898[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ (Succ xv103000))) (primGEqNatS Zero (Succ xv103000)))",fontsize=16,color="black",shape="box"];898 -> 912[label="",style="solid", color="black", weight=3]; 78.58/41.39 899[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ Zero)) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];899 -> 913[label="",style="solid", color="black", weight=3]; 78.58/41.39 900[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv10200) Zero) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];900 -> 914[label="",style="solid", color="black", weight=3]; 78.58/41.39 1100[label="xv100",fontsize=16,color="green",shape="box"];1101[label="Succ Zero",fontsize=16,color="green",shape="box"];902[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ Zero)))",fontsize=16,color="black",shape="box"];902 -> 916[label="",style="solid", color="black", weight=3]; 78.58/41.39 1144[label="primPlusNat (Succ xv1330) xv134",fontsize=16,color="burlywood",shape="box"];1501[label="xv134/Succ xv1340",fontsize=10,color="white",style="solid",shape="box"];1144 -> 1501[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1501 -> 1164[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1502[label="xv134/Zero",fontsize=10,color="white",style="solid",shape="box"];1144 -> 1502[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1502 -> 1165[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1145[label="primPlusNat Zero xv134",fontsize=16,color="burlywood",shape="box"];1503[label="xv134/Succ xv1340",fontsize=10,color="white",style="solid",shape="box"];1145 -> 1503[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1503 -> 1166[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1504[label="xv134/Zero",fontsize=10,color="white",style="solid",shape="box"];1145 -> 1504[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1504 -> 1167[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 910[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ (Succ xv103000))) (primGEqNatS xv102000 xv103000))",fontsize=16,color="burlywood",shape="box"];1505[label="xv102000/Succ xv1020000",fontsize=10,color="white",style="solid",shape="box"];910 -> 1505[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1505 -> 952[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1506[label="xv102000/Zero",fontsize=10,color="white",style="solid",shape="box"];910 -> 1506[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1506 -> 953[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 911[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ xv102000))) (Succ (Succ Zero)) True)",fontsize=16,color="black",shape="box"];911 -> 954[label="",style="solid", color="black", weight=3]; 78.58/41.39 912[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ (Succ xv103000))) False)",fontsize=16,color="black",shape="box"];912 -> 955[label="",style="solid", color="black", weight=3]; 78.58/41.39 913[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ Zero)) (Succ (Succ Zero)) True)",fontsize=16,color="black",shape="box"];913 -> 956[label="",style="solid", color="black", weight=3]; 78.58/41.39 914 -> 758[label="",style="dashed", color="red", weight=0]; 78.58/41.39 914[label="primPlusNat (Succ xv100) (primModNatS (Succ xv10200) (Succ (Succ Zero)))",fontsize=16,color="magenta"];914 -> 957[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 914 -> 958[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 916 -> 502[label="",style="dashed", color="red", weight=0]; 78.58/41.39 916[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ Zero)))",fontsize=16,color="magenta"];916 -> 961[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 916 -> 962[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1164[label="primPlusNat (Succ xv1330) (Succ xv1340)",fontsize=16,color="black",shape="box"];1164 -> 1179[label="",style="solid", color="black", weight=3]; 78.58/41.39 1165[label="primPlusNat (Succ xv1330) Zero",fontsize=16,color="black",shape="box"];1165 -> 1180[label="",style="solid", color="black", weight=3]; 78.58/41.39 1166[label="primPlusNat Zero (Succ xv1340)",fontsize=16,color="black",shape="box"];1166 -> 1181[label="",style="solid", color="black", weight=3]; 78.58/41.39 1167[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];1167 -> 1182[label="",style="solid", color="black", weight=3]; 78.58/41.39 952[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ xv103000))) (primGEqNatS (Succ xv1020000) xv103000))",fontsize=16,color="burlywood",shape="box"];1507[label="xv103000/Succ xv1030000",fontsize=10,color="white",style="solid",shape="box"];952 -> 1507[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1507 -> 967[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1508[label="xv103000/Zero",fontsize=10,color="white",style="solid",shape="box"];952 -> 1508[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1508 -> 968[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 953[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ xv103000))) (primGEqNatS Zero xv103000))",fontsize=16,color="burlywood",shape="box"];1509[label="xv103000/Succ xv1030000",fontsize=10,color="white",style="solid",shape="box"];953 -> 1509[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1509 -> 969[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1510[label="xv103000/Zero",fontsize=10,color="white",style="solid",shape="box"];953 -> 1510[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1510 -> 970[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 954[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv102000))) (Succ (Succ Zero))) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];954 -> 971[label="",style="solid", color="black", weight=3]; 78.58/41.39 955 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 955[label="primPlusNat (Succ xv100) (Succ (Succ (Succ Zero)))",fontsize=16,color="magenta"];955 -> 1102[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 955 -> 1103[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 956[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];956 -> 973[label="",style="solid", color="black", weight=3]; 78.58/41.39 957[label="Succ Zero",fontsize=16,color="green",shape="box"];958[label="xv10200",fontsize=16,color="green",shape="box"];961[label="xv100",fontsize=16,color="green",shape="box"];962[label="Succ Zero",fontsize=16,color="green",shape="box"];1179[label="Succ (Succ (primPlusNat xv1330 xv1340))",fontsize=16,color="green",shape="box"];1179 -> 1196[label="",style="dashed", color="green", weight=3]; 78.58/41.39 1180[label="Succ xv1330",fontsize=16,color="green",shape="box"];1181[label="Succ xv1340",fontsize=16,color="green",shape="box"];1182[label="Zero",fontsize=16,color="green",shape="box"];967[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS (Succ xv1020000) (Succ xv1030000)))",fontsize=16,color="black",shape="box"];967 -> 980[label="",style="solid", color="black", weight=3]; 78.58/41.39 968[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero))) (primGEqNatS (Succ xv1020000) Zero))",fontsize=16,color="black",shape="box"];968 -> 981[label="",style="solid", color="black", weight=3]; 78.58/41.39 969[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS Zero (Succ xv1030000)))",fontsize=16,color="black",shape="box"];969 -> 982[label="",style="solid", color="black", weight=3]; 78.58/41.39 970[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];970 -> 983[label="",style="solid", color="black", weight=3]; 78.58/41.39 971[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv102000)) (Succ Zero)) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];971 -> 984[label="",style="solid", color="black", weight=3]; 78.58/41.39 1102[label="xv100",fontsize=16,color="green",shape="box"];1103[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];973[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];973 -> 986[label="",style="solid", color="black", weight=3]; 78.58/41.39 1196 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1196[label="primPlusNat xv1330 xv1340",fontsize=16,color="magenta"];1196 -> 1212[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1196 -> 1213[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 980[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS xv1020000 xv1030000))",fontsize=16,color="burlywood",shape="box"];1511[label="xv1020000/Succ xv10200000",fontsize=10,color="white",style="solid",shape="box"];980 -> 1511[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1511 -> 992[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1512[label="xv1020000/Zero",fontsize=10,color="white",style="solid",shape="box"];980 -> 1512[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1512 -> 993[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 981[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero))) True)",fontsize=16,color="black",shape="box"];981 -> 994[label="",style="solid", color="black", weight=3]; 78.58/41.39 982[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ (Succ xv1030000)))) False)",fontsize=16,color="black",shape="box"];982 -> 995[label="",style="solid", color="black", weight=3]; 78.58/41.39 983[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero))) True)",fontsize=16,color="black",shape="box"];983 -> 996[label="",style="solid", color="black", weight=3]; 78.58/41.39 984[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv102000) Zero) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];984 -> 997[label="",style="solid", color="black", weight=3]; 78.58/41.39 986[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];986 -> 1000[label="",style="solid", color="black", weight=3]; 78.58/41.39 1212[label="xv1330",fontsize=16,color="green",shape="box"];1213[label="xv1340",fontsize=16,color="green",shape="box"];992[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS (Succ xv10200000) xv1030000))",fontsize=16,color="burlywood",shape="box"];1513[label="xv1030000/Succ xv10300000",fontsize=10,color="white",style="solid",shape="box"];992 -> 1513[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1513 -> 1003[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1514[label="xv1030000/Zero",fontsize=10,color="white",style="solid",shape="box"];992 -> 1514[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1514 -> 1004[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 993[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ xv1030000)))) (primGEqNatS Zero xv1030000))",fontsize=16,color="burlywood",shape="box"];1515[label="xv1030000/Succ xv10300000",fontsize=10,color="white",style="solid",shape="box"];993 -> 1515[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1515 -> 1005[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1516[label="xv1030000/Zero",fontsize=10,color="white",style="solid",shape="box"];993 -> 1516[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1516 -> 1006[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 994[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ xv1020000)))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];994 -> 1007[label="",style="solid", color="black", weight=3]; 78.58/41.39 995 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 995[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];995 -> 1104[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 995 -> 1105[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 996[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];996 -> 1009[label="",style="solid", color="black", weight=3]; 78.58/41.39 997 -> 758[label="",style="dashed", color="red", weight=0]; 78.58/41.39 997[label="primPlusNat (Succ xv100) (primModNatS (Succ xv102000) (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];997 -> 1010[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 997 -> 1011[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1000 -> 502[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1000[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ Zero))))",fontsize=16,color="magenta"];1000 -> 1014[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1000 -> 1015[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1003[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS (Succ xv10200000) (Succ xv10300000)))",fontsize=16,color="black",shape="box"];1003 -> 1017[label="",style="solid", color="black", weight=3]; 78.58/41.39 1004[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero)))) (primGEqNatS (Succ xv10200000) Zero))",fontsize=16,color="black",shape="box"];1004 -> 1018[label="",style="solid", color="black", weight=3]; 78.58/41.39 1005[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS Zero (Succ xv10300000)))",fontsize=16,color="black",shape="box"];1005 -> 1019[label="",style="solid", color="black", weight=3]; 78.58/41.39 1006[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1006 -> 1020[label="",style="solid", color="black", weight=3]; 78.58/41.39 1007[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv1020000))) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1007 -> 1021[label="",style="solid", color="black", weight=3]; 78.58/41.39 1104[label="xv100",fontsize=16,color="green",shape="box"];1105[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1009[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1009 -> 1023[label="",style="solid", color="black", weight=3]; 78.58/41.39 1010[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];1011[label="xv102000",fontsize=16,color="green",shape="box"];1014[label="xv100",fontsize=16,color="green",shape="box"];1015[label="Succ (Succ Zero)",fontsize=16,color="green",shape="box"];1017[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS xv10200000 xv10300000))",fontsize=16,color="burlywood",shape="box"];1517[label="xv10200000/Succ xv102000000",fontsize=10,color="white",style="solid",shape="box"];1017 -> 1517[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1517 -> 1027[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1518[label="xv10200000/Zero",fontsize=10,color="white",style="solid",shape="box"];1017 -> 1518[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1518 -> 1028[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1018[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero)))) True)",fontsize=16,color="black",shape="box"];1018 -> 1029[label="",style="solid", color="black", weight=3]; 78.58/41.39 1019[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) False)",fontsize=16,color="black",shape="box"];1019 -> 1030[label="",style="solid", color="black", weight=3]; 78.58/41.39 1020[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero)))) True)",fontsize=16,color="black",shape="box"];1020 -> 1031[label="",style="solid", color="black", weight=3]; 78.58/41.39 1021[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv1020000)) (Succ Zero)) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1021 -> 1032[label="",style="solid", color="black", weight=3]; 78.58/41.39 1023[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1023 -> 1035[label="",style="solid", color="black", weight=3]; 78.58/41.39 1027[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS (Succ xv102000000) xv10300000))",fontsize=16,color="burlywood",shape="box"];1519[label="xv10300000/Succ xv103000000",fontsize=10,color="white",style="solid",shape="box"];1027 -> 1519[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1519 -> 1041[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1520[label="xv10300000/Zero",fontsize=10,color="white",style="solid",shape="box"];1027 -> 1520[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1520 -> 1042[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1028[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ xv10300000))))) (primGEqNatS Zero xv10300000))",fontsize=16,color="burlywood",shape="box"];1521[label="xv10300000/Succ xv103000000",fontsize=10,color="white",style="solid",shape="box"];1028 -> 1521[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1521 -> 1043[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1522[label="xv10300000/Zero",fontsize=10,color="white",style="solid",shape="box"];1028 -> 1522[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1522 -> 1044[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1029[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ (Succ xv10200000))))) (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1029 -> 1045[label="",style="solid", color="black", weight=3]; 78.58/41.39 1030 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1030[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1030 -> 1106[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1030 -> 1107[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1031[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1031 -> 1047[label="",style="solid", color="black", weight=3]; 78.58/41.39 1032[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv1020000) Zero) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1032 -> 1048[label="",style="solid", color="black", weight=3]; 78.58/41.39 1035[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1035 -> 1051[label="",style="solid", color="black", weight=3]; 78.58/41.39 1041[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS (Succ xv102000000) (Succ xv103000000)))",fontsize=16,color="black",shape="box"];1041 -> 1056[label="",style="solid", color="black", weight=3]; 78.58/41.39 1042[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ Zero))))) (primGEqNatS (Succ xv102000000) Zero))",fontsize=16,color="black",shape="box"];1042 -> 1057[label="",style="solid", color="black", weight=3]; 78.58/41.39 1043[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS Zero (Succ xv103000000)))",fontsize=16,color="black",shape="box"];1043 -> 1058[label="",style="solid", color="black", weight=3]; 78.58/41.39 1044[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1044 -> 1059[label="",style="solid", color="black", weight=3]; 78.58/41.39 1045[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ (Succ xv10200000)))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1045 -> 1060[label="",style="solid", color="black", weight=3]; 78.58/41.39 1106[label="xv100",fontsize=16,color="green",shape="box"];1107[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1047[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ Zero))) (Succ (Succ (Succ Zero)))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1047 -> 1062[label="",style="solid", color="black", weight=3]; 78.58/41.39 1048 -> 758[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1048[label="primPlusNat (Succ xv100) (primModNatS (Succ xv1020000) (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1048 -> 1063[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1048 -> 1064[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1051 -> 502[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1051[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1051 -> 1066[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1051 -> 1067[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1056[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS xv102000000 xv103000000))",fontsize=16,color="burlywood",shape="box"];1523[label="xv102000000/Succ xv1020000000",fontsize=10,color="white",style="solid",shape="box"];1056 -> 1523[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1523 -> 1074[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1524[label="xv102000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1056 -> 1524[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1524 -> 1075[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1057 -> 1078[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1057[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ xv102000000)))))) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="magenta"];1057 -> 1079[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1057 -> 1080[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1058[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) False)",fontsize=16,color="black",shape="box"];1058 -> 1077[label="",style="solid", color="black", weight=3]; 78.58/41.39 1059 -> 1078[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1059[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="magenta"];1059 -> 1081[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1059 -> 1082[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1060[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ (Succ xv10200000))) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1060 -> 1087[label="",style="solid", color="black", weight=3]; 78.58/41.39 1062[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ Zero)) (Succ (Succ Zero))) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1062 -> 1090[label="",style="solid", color="black", weight=3]; 78.58/41.39 1063[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1064[label="xv1020000",fontsize=16,color="green",shape="box"];1066[label="xv100",fontsize=16,color="green",shape="box"];1067[label="Succ (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];1074[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS (Succ xv1020000000) xv103000000))",fontsize=16,color="burlywood",shape="box"];1525[label="xv103000000/Succ xv1030000000",fontsize=10,color="white",style="solid",shape="box"];1074 -> 1525[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1525 -> 1092[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1526[label="xv103000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1074 -> 1526[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1526 -> 1093[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1075[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ xv103000000)))))) (primGEqNatS Zero xv103000000))",fontsize=16,color="burlywood",shape="box"];1527[label="xv103000000/Succ xv1030000000",fontsize=10,color="white",style="solid",shape="box"];1075 -> 1527[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1527 -> 1094[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1528[label="xv103000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1075 -> 1528[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1528 -> 1095[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1079[label="Succ (Succ (Succ (Succ (Succ xv102000000))))",fontsize=16,color="green",shape="box"];1080[label="xv100",fontsize=16,color="green",shape="box"];1078[label="primPlusNat (Succ xv130) (primModNatS0 (Succ xv131) (Succ (Succ (Succ (Succ (Succ Zero))))) True)",fontsize=16,color="black",shape="triangle"];1078 -> 1096[label="",style="solid", color="black", weight=3]; 78.58/41.39 1077 -> 1097[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1077[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1077 -> 1108[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1077 -> 1109[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1081[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1082[label="xv100",fontsize=16,color="green",shape="box"];1087[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ (Succ xv10200000)) (Succ Zero)) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1087 -> 1115[label="",style="solid", color="black", weight=3]; 78.58/41.39 1090[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1090 -> 1116[label="",style="solid", color="black", weight=3]; 78.58/41.39 1092[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS (Succ xv1020000000) (Succ xv1030000000)))",fontsize=16,color="black",shape="box"];1092 -> 1117[label="",style="solid", color="black", weight=3]; 78.58/41.39 1093[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (primGEqNatS (Succ xv1020000000) Zero))",fontsize=16,color="black",shape="box"];1093 -> 1118[label="",style="solid", color="black", weight=3]; 78.58/41.39 1094[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS Zero (Succ xv1030000000)))",fontsize=16,color="black",shape="box"];1094 -> 1119[label="",style="solid", color="black", weight=3]; 78.58/41.39 1095[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1095 -> 1120[label="",style="solid", color="black", weight=3]; 78.58/41.39 1096[label="primPlusNat (Succ xv130) (primModNatS (primMinusNatS (Succ xv131) (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="black",shape="box"];1096 -> 1121[label="",style="solid", color="black", weight=3]; 78.58/41.39 1108[label="xv100",fontsize=16,color="green",shape="box"];1109[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1115[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS (Succ xv10200000) Zero) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1115 -> 1129[label="",style="solid", color="black", weight=3]; 78.58/41.39 1116[label="primPlusNat (Succ xv100) (primModNatS (primMinusNatS Zero Zero) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1116 -> 1130[label="",style="solid", color="black", weight=3]; 78.58/41.39 1117[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS xv1020000000 xv1030000000))",fontsize=16,color="burlywood",shape="box"];1529[label="xv1020000000/Succ xv10200000000",fontsize=10,color="white",style="solid",shape="box"];1117 -> 1529[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1529 -> 1131[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1530[label="xv1020000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1117 -> 1530[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1530 -> 1132[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1118 -> 1135[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1118[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1020000000))))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1118 -> 1136[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1118 -> 1137[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1119[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) False)",fontsize=16,color="black",shape="box"];1119 -> 1134[label="",style="solid", color="black", weight=3]; 78.58/41.39 1120 -> 1135[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1120[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1120 -> 1138[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1120 -> 1139[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1121 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1121[label="primPlusNat (Succ xv130) (primModNatS (primMinusNatS xv131 (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1121 -> 1146[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1121 -> 1147[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1129 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1129[label="primPlusNat (Succ xv100) (primModNatS (Succ xv10200000) (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1129 -> 1148[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1129 -> 1149[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1130 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1130[label="primPlusNat (Succ xv100) (primModNatS Zero (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1130 -> 1150[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1130 -> 1151[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1131[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS (Succ xv10200000000) xv1030000000))",fontsize=16,color="burlywood",shape="box"];1531[label="xv1030000000/Succ xv10300000000",fontsize=10,color="white",style="solid",shape="box"];1131 -> 1531[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1531 -> 1152[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1532[label="xv1030000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1131 -> 1532[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1532 -> 1153[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1132[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv1030000000))))))) (primGEqNatS Zero xv1030000000))",fontsize=16,color="burlywood",shape="box"];1533[label="xv1030000000/Succ xv10300000000",fontsize=10,color="white",style="solid",shape="box"];1132 -> 1533[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1533 -> 1154[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1534[label="xv1030000000/Zero",fontsize=10,color="white",style="solid",shape="box"];1132 -> 1534[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1534 -> 1155[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1136[label="Succ (Succ (Succ (Succ (Succ (Succ xv1020000000)))))",fontsize=16,color="green",shape="box"];1137[label="xv100",fontsize=16,color="green",shape="box"];1135 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1135[label="primPlusNat (Succ xv136) (primModNatS0 (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True)",fontsize=16,color="magenta"];1135 -> 1156[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1135 -> 1157[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1134 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1134[label="primPlusNat (Succ xv100) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1134 -> 1158[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1134 -> 1159[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1138[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1139[label="xv100",fontsize=16,color="green",shape="box"];1146[label="Succ xv130",fontsize=16,color="green",shape="box"];1147 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1147[label="primModNatS (primMinusNatS xv131 (Succ (Succ (Succ (Succ Zero))))) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="magenta"];1147 -> 1283[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1147 -> 1284[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1147 -> 1285[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1148[label="Succ xv100",fontsize=16,color="green",shape="box"];1149 -> 1170[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1149[label="primModNatS (Succ xv10200000) (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1149 -> 1171[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1149 -> 1172[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1150[label="Succ xv100",fontsize=16,color="green",shape="box"];1151 -> 1216[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1151[label="primModNatS Zero (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1151 -> 1217[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1152[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS (Succ xv10200000000) (Succ xv10300000000)))",fontsize=16,color="black",shape="box"];1152 -> 1184[label="",style="solid", color="black", weight=3]; 78.58/41.39 1153[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (primGEqNatS (Succ xv10200000000) Zero))",fontsize=16,color="black",shape="box"];1153 -> 1185[label="",style="solid", color="black", weight=3]; 78.58/41.39 1154[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS Zero (Succ xv10300000000)))",fontsize=16,color="black",shape="box"];1154 -> 1186[label="",style="solid", color="black", weight=3]; 78.58/41.39 1155[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (primGEqNatS Zero Zero))",fontsize=16,color="black",shape="box"];1155 -> 1187[label="",style="solid", color="black", weight=3]; 78.58/41.39 1156[label="Succ xv136",fontsize=16,color="green",shape="box"];1157[label="primModNatS0 (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))) True",fontsize=16,color="black",shape="box"];1157 -> 1188[label="",style="solid", color="black", weight=3]; 78.58/41.39 1158[label="Succ xv100",fontsize=16,color="green",shape="box"];1159[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1283[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1284[label="xv131",fontsize=16,color="green",shape="box"];1285[label="Succ (Succ (Succ (Succ (Succ Zero))))",fontsize=16,color="green",shape="box"];1282[label="primModNatS (primMinusNatS xv151 xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="triangle"];1535[label="xv151/Succ xv1510",fontsize=10,color="white",style="solid",shape="box"];1282 -> 1535[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1535 -> 1308[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1536[label="xv151/Zero",fontsize=10,color="white",style="solid",shape="box"];1282 -> 1536[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1536 -> 1309[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1171[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1172[label="xv10200000",fontsize=16,color="green",shape="box"];1170[label="primModNatS (Succ xv139) (Succ xv140)",fontsize=16,color="black",shape="triangle"];1170 -> 1191[label="",style="solid", color="black", weight=3]; 78.58/41.39 1217[label="Succ (Succ (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];1216[label="primModNatS Zero (Succ xv145)",fontsize=16,color="black",shape="triangle"];1216 -> 1225[label="",style="solid", color="black", weight=3]; 78.58/41.39 1184 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1184[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS xv10200000000 xv10300000000))",fontsize=16,color="magenta"];1184 -> 1197[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1184 -> 1198[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1185 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1185[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True)",fontsize=16,color="magenta"];1185 -> 1199[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1185 -> 1200[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1186 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1186[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) False)",fontsize=16,color="magenta"];1186 -> 1201[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1186 -> 1202[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1187 -> 1128[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1187[label="primPlusNat (Succ xv100) (primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True)",fontsize=16,color="magenta"];1187 -> 1203[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1187 -> 1204[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1188 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1188[label="primModNatS (primMinusNatS (Succ xv137) (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="magenta"];1188 -> 1286[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1188 -> 1287[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1188 -> 1288[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1308[label="primModNatS (primMinusNatS (Succ xv1510) xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="box"];1537[label="xv152/Succ xv1520",fontsize=10,color="white",style="solid",shape="box"];1308 -> 1537[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1537 -> 1325[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1538[label="xv152/Zero",fontsize=10,color="white",style="solid",shape="box"];1308 -> 1538[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1538 -> 1326[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1309[label="primModNatS (primMinusNatS Zero xv152) (Succ xv153)",fontsize=16,color="burlywood",shape="box"];1539[label="xv152/Succ xv1520",fontsize=10,color="white",style="solid",shape="box"];1309 -> 1539[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1539 -> 1327[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1540[label="xv152/Zero",fontsize=10,color="white",style="solid",shape="box"];1309 -> 1540[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1540 -> 1328[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1191[label="primModNatS0 xv139 xv140 (primGEqNatS xv139 xv140)",fontsize=16,color="burlywood",shape="box"];1541[label="xv139/Succ xv1390",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1541[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1541 -> 1226[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1542[label="xv139/Zero",fontsize=10,color="white",style="solid",shape="box"];1191 -> 1542[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1542 -> 1227[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1225[label="Zero",fontsize=16,color="green",shape="box"];1197[label="Succ xv100",fontsize=16,color="green",shape="box"];1198 -> 1362[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1198[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) (primGEqNatS xv10200000000 xv10300000000)",fontsize=16,color="magenta"];1198 -> 1363[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1198 -> 1364[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1198 -> 1365[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1198 -> 1366[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1199[label="Succ xv100",fontsize=16,color="green",shape="box"];1200 -> 1230[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1200[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="magenta"];1200 -> 1231[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1201[label="Succ xv100",fontsize=16,color="green",shape="box"];1202 -> 1237[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1202[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000)))))))) False",fontsize=16,color="magenta"];1202 -> 1238[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1203[label="Succ xv100",fontsize=16,color="green",shape="box"];1204 -> 1230[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1204[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="magenta"];1204 -> 1232[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1286[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1287[label="Succ xv137",fontsize=16,color="green",shape="box"];1288[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1325[label="primModNatS (primMinusNatS (Succ xv1510) (Succ xv1520)) (Succ xv153)",fontsize=16,color="black",shape="box"];1325 -> 1336[label="",style="solid", color="black", weight=3]; 78.58/41.39 1326[label="primModNatS (primMinusNatS (Succ xv1510) Zero) (Succ xv153)",fontsize=16,color="black",shape="box"];1326 -> 1337[label="",style="solid", color="black", weight=3]; 78.58/41.39 1327[label="primModNatS (primMinusNatS Zero (Succ xv1520)) (Succ xv153)",fontsize=16,color="black",shape="box"];1327 -> 1338[label="",style="solid", color="black", weight=3]; 78.58/41.39 1328[label="primModNatS (primMinusNatS Zero Zero) (Succ xv153)",fontsize=16,color="black",shape="box"];1328 -> 1339[label="",style="solid", color="black", weight=3]; 78.58/41.39 1226[label="primModNatS0 (Succ xv1390) xv140 (primGEqNatS (Succ xv1390) xv140)",fontsize=16,color="burlywood",shape="box"];1543[label="xv140/Succ xv1400",fontsize=10,color="white",style="solid",shape="box"];1226 -> 1543[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1543 -> 1247[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1544[label="xv140/Zero",fontsize=10,color="white",style="solid",shape="box"];1226 -> 1544[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1544 -> 1248[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1227[label="primModNatS0 Zero xv140 (primGEqNatS Zero xv140)",fontsize=16,color="burlywood",shape="box"];1545[label="xv140/Succ xv1400",fontsize=10,color="white",style="solid",shape="box"];1227 -> 1545[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1545 -> 1249[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1546[label="xv140/Zero",fontsize=10,color="white",style="solid",shape="box"];1227 -> 1546[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1546 -> 1250[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1363[label="xv10300000000",fontsize=16,color="green",shape="box"];1364[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000))))))",fontsize=16,color="green",shape="box"];1365[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000))))))",fontsize=16,color="green",shape="box"];1366[label="xv10200000000",fontsize=16,color="green",shape="box"];1362[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS xv165 xv166)",fontsize=16,color="burlywood",shape="triangle"];1547[label="xv165/Succ xv1650",fontsize=10,color="white",style="solid",shape="box"];1362 -> 1547[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1547 -> 1415[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1548[label="xv165/Zero",fontsize=10,color="white",style="solid",shape="box"];1362 -> 1548[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1548 -> 1416[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1231[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10200000000))))))",fontsize=16,color="green",shape="box"];1230[label="primModNatS0 (Succ xv147) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) True",fontsize=16,color="black",shape="triangle"];1230 -> 1255[label="",style="solid", color="black", weight=3]; 78.58/41.39 1238[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ xv10300000000))))))",fontsize=16,color="green",shape="box"];1237[label="primModNatS0 (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))) (Succ xv149) False",fontsize=16,color="black",shape="triangle"];1237 -> 1256[label="",style="solid", color="black", weight=3]; 78.58/41.39 1232[label="Succ (Succ (Succ (Succ (Succ (Succ Zero)))))",fontsize=16,color="green",shape="box"];1336 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1336[label="primModNatS (primMinusNatS xv1510 xv1520) (Succ xv153)",fontsize=16,color="magenta"];1336 -> 1346[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1336 -> 1347[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1337 -> 1170[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1337[label="primModNatS (Succ xv1510) (Succ xv153)",fontsize=16,color="magenta"];1337 -> 1348[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1337 -> 1349[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1338 -> 1216[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1338[label="primModNatS Zero (Succ xv153)",fontsize=16,color="magenta"];1338 -> 1350[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1339 -> 1216[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1339[label="primModNatS Zero (Succ xv153)",fontsize=16,color="magenta"];1339 -> 1351[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1247[label="primModNatS0 (Succ xv1390) (Succ xv1400) (primGEqNatS (Succ xv1390) (Succ xv1400))",fontsize=16,color="black",shape="box"];1247 -> 1268[label="",style="solid", color="black", weight=3]; 78.58/41.39 1248[label="primModNatS0 (Succ xv1390) Zero (primGEqNatS (Succ xv1390) Zero)",fontsize=16,color="black",shape="box"];1248 -> 1269[label="",style="solid", color="black", weight=3]; 78.58/41.39 1249[label="primModNatS0 Zero (Succ xv1400) (primGEqNatS Zero (Succ xv1400))",fontsize=16,color="black",shape="box"];1249 -> 1270[label="",style="solid", color="black", weight=3]; 78.58/41.39 1250[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1250 -> 1271[label="",style="solid", color="black", weight=3]; 78.58/41.39 1415[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) xv166)",fontsize=16,color="burlywood",shape="box"];1549[label="xv166/Succ xv1660",fontsize=10,color="white",style="solid",shape="box"];1415 -> 1549[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1549 -> 1421[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1550[label="xv166/Zero",fontsize=10,color="white",style="solid",shape="box"];1415 -> 1550[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1550 -> 1422[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1416[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero xv166)",fontsize=16,color="burlywood",shape="box"];1551[label="xv166/Succ xv1660",fontsize=10,color="white",style="solid",shape="box"];1416 -> 1551[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1551 -> 1423[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1552[label="xv166/Zero",fontsize=10,color="white",style="solid",shape="box"];1416 -> 1552[label="",style="solid", color="burlywood", weight=9]; 78.58/41.39 1552 -> 1424[label="",style="solid", color="burlywood", weight=3]; 78.58/41.39 1255 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1255[label="primModNatS (primMinusNatS (Succ xv147) (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))) (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="magenta"];1255 -> 1298[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1255 -> 1299[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1255 -> 1300[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1256[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))",fontsize=16,color="green",shape="box"];1346[label="xv1520",fontsize=16,color="green",shape="box"];1347[label="xv1510",fontsize=16,color="green",shape="box"];1348[label="xv153",fontsize=16,color="green",shape="box"];1349[label="xv1510",fontsize=16,color="green",shape="box"];1350[label="xv153",fontsize=16,color="green",shape="box"];1351[label="xv153",fontsize=16,color="green",shape="box"];1268 -> 1362[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1268[label="primModNatS0 (Succ xv1390) (Succ xv1400) (primGEqNatS xv1390 xv1400)",fontsize=16,color="magenta"];1268 -> 1367[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1268 -> 1368[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1268 -> 1369[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1268 -> 1370[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1269[label="primModNatS0 (Succ xv1390) Zero True",fontsize=16,color="black",shape="box"];1269 -> 1312[label="",style="solid", color="black", weight=3]; 78.58/41.39 1270[label="primModNatS0 Zero (Succ xv1400) False",fontsize=16,color="black",shape="box"];1270 -> 1313[label="",style="solid", color="black", weight=3]; 78.58/41.39 1271[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];1271 -> 1314[label="",style="solid", color="black", weight=3]; 78.58/41.39 1421[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) (Succ xv1660))",fontsize=16,color="black",shape="box"];1421 -> 1429[label="",style="solid", color="black", weight=3]; 78.58/41.39 1422[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS (Succ xv1650) Zero)",fontsize=16,color="black",shape="box"];1422 -> 1430[label="",style="solid", color="black", weight=3]; 78.58/41.39 1423[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero (Succ xv1660))",fontsize=16,color="black",shape="box"];1423 -> 1431[label="",style="solid", color="black", weight=3]; 78.58/41.39 1424[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];1424 -> 1432[label="",style="solid", color="black", weight=3]; 78.58/41.39 1298[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1299[label="Succ xv147",fontsize=16,color="green",shape="box"];1300[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))",fontsize=16,color="green",shape="box"];1367[label="xv1400",fontsize=16,color="green",shape="box"];1368[label="xv1390",fontsize=16,color="green",shape="box"];1369[label="xv1400",fontsize=16,color="green",shape="box"];1370[label="xv1390",fontsize=16,color="green",shape="box"];1312 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1312[label="primModNatS (primMinusNatS (Succ xv1390) Zero) (Succ Zero)",fontsize=16,color="magenta"];1312 -> 1356[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1312 -> 1357[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1312 -> 1358[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1313[label="Succ Zero",fontsize=16,color="green",shape="box"];1314 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1314[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];1314 -> 1359[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1314 -> 1360[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1314 -> 1361[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1429 -> 1362[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1429[label="primModNatS0 (Succ xv163) (Succ xv164) (primGEqNatS xv1650 xv1660)",fontsize=16,color="magenta"];1429 -> 1438[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1429 -> 1439[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1430[label="primModNatS0 (Succ xv163) (Succ xv164) True",fontsize=16,color="black",shape="triangle"];1430 -> 1440[label="",style="solid", color="black", weight=3]; 78.58/41.39 1431[label="primModNatS0 (Succ xv163) (Succ xv164) False",fontsize=16,color="black",shape="box"];1431 -> 1441[label="",style="solid", color="black", weight=3]; 78.58/41.39 1432 -> 1430[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1432[label="primModNatS0 (Succ xv163) (Succ xv164) True",fontsize=16,color="magenta"];1356[label="Zero",fontsize=16,color="green",shape="box"];1357[label="Succ xv1390",fontsize=16,color="green",shape="box"];1358[label="Zero",fontsize=16,color="green",shape="box"];1359[label="Zero",fontsize=16,color="green",shape="box"];1360[label="Zero",fontsize=16,color="green",shape="box"];1361[label="Zero",fontsize=16,color="green",shape="box"];1438[label="xv1660",fontsize=16,color="green",shape="box"];1439[label="xv1650",fontsize=16,color="green",shape="box"];1440 -> 1282[label="",style="dashed", color="red", weight=0]; 78.58/41.39 1440[label="primModNatS (primMinusNatS (Succ xv163) (Succ xv164)) (Succ (Succ xv164))",fontsize=16,color="magenta"];1440 -> 1442[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1440 -> 1443[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1440 -> 1444[label="",style="dashed", color="magenta", weight=3]; 78.58/41.39 1441[label="Succ (Succ xv163)",fontsize=16,color="green",shape="box"];1442[label="Succ xv164",fontsize=16,color="green",shape="box"];1443[label="Succ xv163",fontsize=16,color="green",shape="box"];1444[label="Succ xv164",fontsize=16,color="green",shape="box"];} 78.58/41.39 78.58/41.39 ---------------------------------------- 78.58/41.39 78.58/41.39 (187) 78.58/41.39 Obligation: 78.58/41.39 Q DP problem: 78.58/41.39 The TRS P consists of the following rules: 78.58/41.39 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero, []) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'1(xv94, xv95, []), []) 78.58/41.39 new_showInt1ShowInt0(xv12, Succ(xv130), Succ(xv140), []) -> new_showInt1ShowInt00(xv12, Succ(xv130), xv140, xv130, xv140, []) 78.58/41.39 new_showInt(Pos(Succ(xv300)), xv4, []) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []) 78.58/41.39 new_showInt1ShowInt0(xv12, Succ(xv130), Zero, []) -> new_showInt(new_showInt1N'0(xv12, Succ(xv130), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'(xv12, xv130, []), []) 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980), []) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980, []) 78.58/41.39 new_showInt1ShowInt02(xv12, xv130, []) -> new_showInt(new_showInt1N'0(xv12, Succ(xv130), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'(xv12, xv130, []), []) 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero, []) -> new_showInt1ShowInt01(xv94, xv95, xv96, []) 78.58/41.39 new_showInt1ShowInt0(xv12, Zero, Zero, []) -> new_showInt(new_showInt1N'0(xv12, Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'0(xv12, []), []) 78.58/41.39 new_showInt1ShowInt01(xv94, xv95, xv96, []) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'1(xv94, xv95, []), []) 78.58/41.39 78.58/41.39 The TRS R consists of the following rules: 78.58/41.39 78.58/41.39 new_primPlusNat1(xv136, xv137, []) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []), []) 78.58/41.39 new_showInt1N'0(xv82, xv83, xv84, []) -> new_primQuotInt(xv83, xv84, []) 78.58/41.39 new_primDivNatS3(Zero, Succ(xv1690), xv170, []) -> new_primDivNatS4(xv170, []) 78.58/41.39 new_primDivNatS4(xv170, []) -> Zero 78.58/41.39 new_primModNatS01(xv147, []) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) 78.58/41.39 new_primModNatS2(Succ(xv1510), Zero, xv153, []) -> new_primModNatS3(xv1510, xv153, []) 78.58/41.39 new_primPlusNat2(Succ(xv1330), Zero, []) -> Succ(xv1330) 78.58/41.39 new_primPlusNat2(Zero, Succ(xv1340), []) -> Succ(xv1340) 78.58/41.39 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero, []) -> new_primDivNatS02(xv125, xv126, []) 78.58/41.39 new_primIntToChar1(xv100, xv101, xv102, xv103, []) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103, []), xv103, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), []), []) 78.58/41.39 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280), []) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000)))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000)))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), []) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primModNatS2(Zero, Zero, xv153, []) -> new_primModNatS4(xv153, []) 78.58/41.39 new_primDivNatS2(Zero, Succ(xv840), []) -> Zero 78.58/41.39 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153, []) -> new_primModNatS2(xv1510, xv1520, xv153, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000))))))), []) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero))), []) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero, []) -> new_primModNatS03(xv163, xv164, []) 78.58/41.39 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170, []) -> new_primDivNatS3(xv1680, xv1690, xv170, []) 78.58/41.39 new_primModNatS2(Zero, Succ(xv1520), xv153, []) -> new_primModNatS4(xv153, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000)))))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), []) 78.58/41.39 new_primModNatS02(xv163, xv164, Zero, Zero, []) -> new_primModNatS03(xv163, xv164, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero), []) -> new_primPlusNat6(xv100, Succ(Zero), []) 78.58/41.39 new_primModNatS04(xv149, []) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.39 new_primPlusNat6(xv58, xv60, []) -> Succ(xv58) 78.58/41.39 new_primPlusNat2(Zero, Zero, []) -> Zero 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero), []) -> new_primPlusNat3(xv100, xv10200, Succ(Zero), []) 78.58/41.39 new_primDivNatS01(xv125, xv126, Zero, Zero, []) -> new_primDivNatS02(xv125, xv126, []) 78.58/41.39 new_showInt1R'1(xv94, xv95, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, []), xv94) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero))))), []) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000))))), []) 78.58/41.39 new_primDivNatS2(Succ(xv830), Succ(xv840), []) -> new_primDivNatS01(xv830, xv840, xv830, xv840, []) 78.58/41.39 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660), []) -> new_primModNatS02(xv163, xv164, xv1650, xv1660, []) 78.58/41.39 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660), []) -> Succ(Succ(xv163)) 78.58/41.39 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300)), []) -> new_primPlusNat5(xv100, Succ(Zero), []) 78.58/41.39 new_primPlusNat3(xv100, Zero, Succ(xv1030), []) -> new_primPlusNat5(xv100, Zero, []) 78.58/41.39 new_primDivNatS02(xv125, xv126, []) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126), [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(xv1020), Zero, []) -> new_primPlusNat3(xv100, xv1020, Zero, []) 78.58/41.39 new_primDivNatS3(Succ(xv1680), Zero, xv170, []) -> new_primDivNatS2(xv1680, xv170, []) 78.58/41.39 new_primModNatS3(Zero, Zero, []) -> new_primModNatS2(Zero, Zero, Zero, []) 78.58/41.39 new_showInt1R'(xv12, xv130, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Succ(xv130), []), xv12) 78.58/41.39 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280), []) -> Zero 78.58/41.39 new_primPlusNat5(xv133, xv134, []) -> Succ(Succ(new_primPlusNat2(xv133, xv134, []))) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero)))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))), []), []) 78.58/41.39 new_primModNatS3(Succ(xv1390), Zero, []) -> new_primModNatS2(Succ(xv1390), Zero, Zero, []) 78.58/41.39 new_primPlusNat4(xv130, xv131, []) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))), []), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000))))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero)))), []) 78.58/41.39 new_primModNatS4(xv145, []) -> Zero 78.58/41.39 new_primModNatS03(xv163, xv164, []) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164), []) 78.58/41.39 new_primQuotInt(xv83, xv84, []) -> Pos(new_primDivNatS2(xv83, xv84, [])) 78.58/41.39 new_primModNatS3(Succ(xv1390), Succ(xv1400), []) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000)))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), []), []) 78.58/41.39 new_primDivNatS2(Zero, Zero, []) -> Succ(new_primDivNatS3(Zero, Zero, Zero, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000)))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000, []), []) 78.58/41.39 new_primPlusNat3(xv100, Zero, Zero, []) -> new_primPlusNat6(xv100, Zero, []) 78.58/41.39 new_primDivNatS3(Zero, Zero, xv170, []) -> new_primDivNatS4(xv170, []) 78.58/41.39 new_showInt1R'0(xv12, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Zero, []), xv12) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []), []) 78.58/41.39 new_primModNatS3(Zero, Succ(xv1400), []) -> Succ(Zero) 78.58/41.39 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103, []) -> Char(new_primPlusNat3(xv100, xv102, xv103, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero)), []) -> new_primPlusNat6(xv100, Succ(Succ(Zero)), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero))))), []) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero)))), []) 78.58/41.39 new_primDivNatS2(Succ(xv830), Zero, []) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000))), []) -> new_primPlusNat5(xv100, Succ(Succ(Zero)), []) 78.58/41.39 new_primPlusNat2(Succ(xv1330), Succ(xv1340), []) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340, []))) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero)), []) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero)), []) 78.58/41.39 new_primIntToChar(xv86, xv87, xv88, []) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero)))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))), []), []) 78.58/41.39 78.58/41.39 Q is empty. 78.58/41.39 We have to consider all (P,Q,R)-chains. 78.58/41.39 ---------------------------------------- 78.58/41.39 78.58/41.39 (188) DependencyGraphProof (EQUIVALENT) 78.58/41.39 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. 78.58/41.39 ---------------------------------------- 78.58/41.39 78.58/41.39 (189) 78.58/41.39 Obligation: 78.58/41.39 Q DP problem: 78.58/41.39 The TRS P consists of the following rules: 78.58/41.39 78.58/41.39 new_showInt(Pos(Succ(xv300)), xv4, []) -> new_showInt1ShowInt0(xv4, xv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []) 78.58/41.39 new_showInt1ShowInt0(xv12, Succ(xv130), Succ(xv140), []) -> new_showInt1ShowInt00(xv12, Succ(xv130), xv140, xv130, xv140, []) 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Zero, []) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'1(xv94, xv95, []), []) 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Succ(xv970), Succ(xv980), []) -> new_showInt1ShowInt00(xv94, xv95, xv96, xv970, xv980, []) 78.58/41.39 new_showInt1ShowInt00(xv94, xv95, xv96, Zero, Zero, []) -> new_showInt1ShowInt01(xv94, xv95, xv96, []) 78.58/41.39 new_showInt1ShowInt01(xv94, xv95, xv96, []) -> new_showInt(new_showInt1N'0(xv94, xv95, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []), new_showInt1R'1(xv94, xv95, []), []) 78.58/41.39 78.58/41.39 The TRS R consists of the following rules: 78.58/41.39 78.58/41.39 new_primPlusNat1(xv136, xv137, []) -> new_primPlusNat2(Succ(xv136), new_primModNatS2(Succ(xv137), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []), []) 78.58/41.39 new_showInt1N'0(xv82, xv83, xv84, []) -> new_primQuotInt(xv83, xv84, []) 78.58/41.39 new_primDivNatS3(Zero, Succ(xv1690), xv170, []) -> new_primDivNatS4(xv170, []) 78.58/41.39 new_primDivNatS4(xv170, []) -> Zero 78.58/41.39 new_primModNatS01(xv147, []) -> new_primModNatS2(Succ(xv147), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) 78.58/41.39 new_primModNatS2(Succ(xv1510), Zero, xv153, []) -> new_primModNatS3(xv1510, xv153, []) 78.58/41.39 new_primPlusNat2(Succ(xv1330), Zero, []) -> Succ(xv1330) 78.58/41.39 new_primPlusNat2(Zero, Succ(xv1340), []) -> Succ(xv1340) 78.58/41.39 new_primDivNatS01(xv125, xv126, Succ(xv1270), Zero, []) -> new_primDivNatS02(xv125, xv126, []) 78.58/41.39 new_primIntToChar1(xv100, xv101, xv102, xv103, []) -> new_primIntToChar0(xv100, xv101, xv102, new_primQuotInt(xv102, xv103, []), xv103, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), []), []) 78.58/41.39 new_primDivNatS01(xv125, xv126, Succ(xv1270), Succ(xv1280), []) -> new_primDivNatS01(xv125, xv126, xv1270, xv1280, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(xv1030000)))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv1020000000)))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Zero))), []) -> new_primPlusNat6(xv100, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primModNatS2(Zero, Zero, xv153, []) -> new_primModNatS4(xv153, []) 78.58/41.39 new_primDivNatS2(Zero, Succ(xv840), []) -> Zero 78.58/41.39 new_primModNatS2(Succ(xv1510), Succ(xv1520), xv153, []) -> new_primModNatS2(xv1510, xv1520, xv153, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv1030000000))))))), []) -> new_primPlusNat2(Succ(xv100), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(xv1020000)))), Succ(Succ(Succ(Zero))), []) -> new_primPlusNat3(xv100, xv1020000, Succ(Succ(Succ(Zero))), []) 78.58/41.39 new_primModNatS02(xv163, xv164, Succ(xv1650), Zero, []) -> new_primModNatS03(xv163, xv164, []) 78.58/41.39 new_primDivNatS3(Succ(xv1680), Succ(xv1690), xv170, []) -> new_primDivNatS3(xv1680, xv1690, xv170, []) 78.58/41.39 new_primModNatS2(Zero, Succ(xv1520), xv153, []) -> new_primModNatS4(xv153, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Succ(xv103000000)))))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), []) 78.58/41.39 new_primModNatS02(xv163, xv164, Zero, Zero, []) -> new_primModNatS03(xv163, xv164, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Zero), Succ(Zero), []) -> new_primPlusNat6(xv100, Succ(Zero), []) 78.58/41.39 new_primModNatS04(xv149, []) -> Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))) 78.58/41.39 new_primPlusNat6(xv58, xv60, []) -> Succ(xv58) 78.58/41.39 new_primPlusNat2(Zero, Zero, []) -> Zero 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(xv10200)), Succ(Zero), []) -> new_primPlusNat3(xv100, xv10200, Succ(Zero), []) 78.58/41.39 new_primDivNatS01(xv125, xv126, Zero, Zero, []) -> new_primDivNatS02(xv125, xv126, []) 78.58/41.39 new_showInt1R'1(xv94, xv95, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv94, xv95, []), xv94) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(xv102000000)))))), Succ(Succ(Succ(Succ(Succ(Zero))))), []) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Succ(xv102000000))))), []) 78.58/41.39 new_primDivNatS2(Succ(xv830), Succ(xv840), []) -> new_primDivNatS01(xv830, xv840, xv830, xv840, []) 78.58/41.39 new_primModNatS02(xv163, xv164, Succ(xv1650), Succ(xv1660), []) -> new_primModNatS02(xv163, xv164, xv1650, xv1660, []) 78.58/41.39 new_primModNatS02(xv163, xv164, Zero, Succ(xv1660), []) -> Succ(Succ(xv163)) 78.58/41.39 new_primPlusNat3(xv100, Succ(Zero), Succ(Succ(xv10300)), []) -> new_primPlusNat5(xv100, Succ(Zero), []) 78.58/41.39 new_primPlusNat3(xv100, Zero, Succ(xv1030), []) -> new_primPlusNat5(xv100, Zero, []) 78.58/41.39 new_primDivNatS02(xv125, xv126, []) -> Succ(new_primDivNatS3(Succ(xv125), Succ(xv126), Succ(xv126), [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(xv1020), Zero, []) -> new_primPlusNat3(xv100, xv1020, Zero, []) 78.58/41.39 new_primDivNatS3(Succ(xv1680), Zero, xv170, []) -> new_primDivNatS2(xv1680, xv170, []) 78.58/41.39 new_primModNatS3(Zero, Zero, []) -> new_primModNatS2(Zero, Zero, Zero, []) 78.58/41.39 new_showInt1R'(xv12, xv130, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Succ(xv130), []), xv12) 78.58/41.39 new_primDivNatS01(xv125, xv126, Zero, Succ(xv1280), []) -> Zero 78.58/41.39 new_primPlusNat5(xv133, xv134, []) -> Succ(Succ(new_primPlusNat2(xv133, xv134, []))) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Zero)))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS4(Succ(Succ(Succ(Succ(Zero)))), []), []) 78.58/41.39 new_primModNatS3(Succ(xv1390), Zero, []) -> new_primModNatS2(Succ(xv1390), Zero, Zero, []) 78.58/41.39 new_primPlusNat4(xv130, xv131, []) -> new_primPlusNat2(Succ(xv130), new_primModNatS2(xv131, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Zero))))), []), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(xv10300000))))), []) -> new_primPlusNat5(xv100, Succ(Succ(Succ(Succ(Zero)))), []) 78.58/41.39 new_primModNatS4(xv145, []) -> Zero 78.58/41.39 new_primModNatS03(xv163, xv164, []) -> new_primModNatS2(Succ(xv163), Succ(xv164), Succ(xv164), []) 78.58/41.39 new_primQuotInt(xv83, xv84, []) -> Pos(new_primDivNatS2(xv83, xv84, [])) 78.58/41.39 new_primModNatS3(Succ(xv1390), Succ(xv1400), []) -> new_primModNatS02(xv1390, xv1400, xv1390, xv1400, []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []) -> new_primPlusNat1(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000)))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS04(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), []), []) 78.58/41.39 new_primDivNatS2(Zero, Zero, []) -> Succ(new_primDivNatS3(Zero, Zero, Zero, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000)))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000)))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS02(Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10200000000))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(xv10300000000))))))), xv10200000000, xv10300000000, []), []) 78.58/41.39 new_primPlusNat3(xv100, Zero, Zero, []) -> new_primPlusNat6(xv100, Zero, []) 78.58/41.39 new_primDivNatS3(Zero, Zero, xv170, []) -> new_primDivNatS4(xv170, []) 78.58/41.39 new_showInt1R'0(xv12, []) -> :(new_primIntToChar(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))))))))))))))))))))))))))))))))))))), xv12, Zero, []), xv12) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS01(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))), []), []) 78.58/41.39 new_primModNatS3(Zero, Succ(xv1400), []) -> Succ(Zero) 78.58/41.39 new_primIntToChar0(xv100, xv101, xv102, xv108, xv103, []) -> Char(new_primPlusNat3(xv100, xv102, xv103, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Zero)), []) -> new_primPlusNat6(xv100, Succ(Succ(Zero)), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(Zero))))), Succ(Succ(Succ(Succ(Succ(Zero))))), []) -> new_primPlusNat4(xv100, Succ(Succ(Succ(Succ(Zero)))), []) 78.58/41.39 new_primDivNatS2(Succ(xv830), Zero, []) -> Succ(new_primDivNatS3(Succ(xv830), Zero, Zero, [])) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Zero)), Succ(Succ(Succ(xv103000))), []) -> new_primPlusNat5(xv100, Succ(Succ(Zero)), []) 78.58/41.39 new_primPlusNat2(Succ(xv1330), Succ(xv1340), []) -> Succ(Succ(new_primPlusNat2(xv1330, xv1340, []))) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(xv102000))), Succ(Succ(Zero)), []) -> new_primPlusNat3(xv100, xv102000, Succ(Succ(Zero)), []) 78.58/41.39 new_primIntToChar(xv86, xv87, xv88, []) -> new_primIntToChar1(xv86, xv87, xv88, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))), []) 78.58/41.39 new_primPlusNat3(xv100, Succ(Succ(Succ(Succ(Succ(xv10200000))))), Succ(Succ(Succ(Succ(Zero)))), []) -> new_primPlusNat2(Succ(xv100), new_primModNatS3(xv10200000, Succ(Succ(Succ(Succ(Zero)))), []), []) 78.58/41.39 78.58/41.39 Q is empty. 78.58/41.39 We have to consider all (P,Q,R)-chains. 78.70/41.43 EOF