/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- MAYBE proof of /export/starexec/sandbox/benchmark/theBenchmark.hs # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could not be shown: (0) HASKELL (1) IFR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) LetRed [EQUIVALENT, 0 ms] (8) HASKELL (9) NumRed [SOUND, 0 ms] (10) HASKELL (11) Narrow [SOUND, 0 ms] (12) AND (13) QDP (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] (15) YES (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES (19) QDP (20) DependencyGraphProof [EQUIVALENT, 0 ms] (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) YES (24) QDP (25) DependencyGraphProof [EQUIVALENT, 0 ms] (26) QDP (27) QDPOrderProof [EQUIVALENT, 195 ms] (28) QDP (29) DependencyGraphProof [EQUIVALENT, 0 ms] (30) AND (31) QDP (32) QDPOrderProof [EQUIVALENT, 0 ms] (33) QDP (34) MNOCProof [EQUIVALENT, 0 ms] (35) QDP (36) InductionCalculusProof [EQUIVALENT, 49 ms] (37) QDP (38) QDPPairToRuleProof [EQUIVALENT, 0 ms] (39) AND (40) QDP (41) MNOCProof [EQUIVALENT, 0 ms] (42) QDP (43) InductionCalculusProof [EQUIVALENT, 24 ms] (44) QDP (45) QDPPairToRuleProof [EQUIVALENT, 7 ms] (46) AND (47) QDP (48) MNOCProof [EQUIVALENT, 0 ms] (49) QDP (50) InductionCalculusProof [EQUIVALENT, 28 ms] (51) QDP (52) QDP (53) QDPSizeChangeProof [EQUIVALENT, 0 ms] (54) YES (55) QDP (56) QDPSizeChangeProof [EQUIVALENT, 0 ms] (57) YES (58) QDP (59) TransformationProof [EQUIVALENT, 0 ms] (60) QDP (61) UsableRulesProof [EQUIVALENT, 0 ms] (62) QDP (63) QReductionProof [EQUIVALENT, 0 ms] (64) QDP (65) TransformationProof [EQUIVALENT, 0 ms] (66) QDP (67) TransformationProof [EQUIVALENT, 0 ms] (68) QDP (69) TransformationProof [EQUIVALENT, 0 ms] (70) QDP (71) TransformationProof [EQUIVALENT, 0 ms] (72) QDP (73) TransformationProof [EQUIVALENT, 0 ms] (74) QDP (75) UsableRulesProof [EQUIVALENT, 0 ms] (76) QDP (77) QReductionProof [EQUIVALENT, 0 ms] (78) QDP (79) TransformationProof [EQUIVALENT, 0 ms] (80) QDP (81) TransformationProof [EQUIVALENT, 0 ms] (82) QDP (83) DependencyGraphProof [EQUIVALENT, 0 ms] (84) AND (85) QDP (86) TransformationProof [EQUIVALENT, 0 ms] (87) QDP (88) TransformationProof [EQUIVALENT, 0 ms] (89) QDP (90) DependencyGraphProof [EQUIVALENT, 0 ms] (91) AND (92) QDP (93) QDPOrderProof [EQUIVALENT, 0 ms] (94) QDP (95) DependencyGraphProof [EQUIVALENT, 0 ms] (96) AND (97) QDP (98) QDPSizeChangeProof [EQUIVALENT, 0 ms] (99) YES (100) QDP (101) MNOCProof [EQUIVALENT, 0 ms] (102) QDP (103) InductionCalculusProof [EQUIVALENT, 0 ms] (104) QDP (105) QDPPairToRuleProof [EQUIVALENT, 0 ms] (106) AND (107) QDP (108) MNOCProof [EQUIVALENT, 0 ms] (109) QDP (110) InductionCalculusProof [EQUIVALENT, 0 ms] (111) QDP (112) QDP (113) QDPSizeChangeProof [EQUIVALENT, 0 ms] (114) YES (115) QDP (116) UsableRulesProof [EQUIVALENT, 0 ms] (117) QDP (118) QReductionProof [EQUIVALENT, 0 ms] (119) QDP (120) QDPSizeChangeProof [EQUIVALENT, 0 ms] (121) YES (122) QDP (123) UsableRulesProof [EQUIVALENT, 0 ms] (124) QDP (125) QReductionProof [EQUIVALENT, 0 ms] (126) QDP (127) QDPSizeChangeProof [EQUIVALENT, 0 ms] (128) YES (129) QDP (130) QDPSizeChangeProof [EQUIVALENT, 0 ms] (131) YES (132) QDP (133) QDPSizeChangeProof [EQUIVALENT, 0 ms] (134) YES (135) QDP (136) QDPSizeChangeProof [EQUIVALENT, 0 ms] (137) YES (138) Narrow [COMPLETE, 0 ms] (139) TRUE ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) IFR (EQUIVALENT) If Reductions: The following If expression "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" is transformed to "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); primDivNatS0 x y False = Zero; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " The following Function with conditions "g x n|even ng (x * x) (n `quot` 2)|otherwisef x (n - 1) (x * y); " is transformed to "g x n = g2 x n; " "g0 x n True = f x (n - 1) (x * y); " "g1 x n True = g (x * x) (n `quot` 2); g1 x n False = g0 x n otherwise; " "g2 x n = g1 x n (even n); " The following Function with conditions "f wz 0 y = y; f x n y = g x n where { g x n|even ng (x * x) (n `quot` 2)|otherwisef x (n - 1) (x * y); } ; " is transformed to "f wz yu y = f4 wz yu y; f x n y = f0 x n y; " "f0 x n y = g x n where { g x n = g2 x n; ; g0 x n True = f x (n - 1) (x * y); ; g1 x n True = g (x * x) (n `quot` 2); g1 x n False = g0 x n otherwise; ; g2 x n = g1 x n (even n); } ; " "f3 True wz yu y = y; f3 yv yw yx yy = f0 yw yx yy; " "f4 wz yu y = f3 (yu == 0) wz yu y; f4 yz zu zv = f0 yz zu zv; " The following Function with conditions "^ x 0 = 1; ^ x n|n > 0f x (n - 1) x where { f wz 0 y = y; f x n y = g x n where { g x n|even ng (x * x) (n `quot` 2)|otherwisef x (n - 1) (x * y); } ; } ; ^ xu xv = error []; " is transformed to "^ x zy = pr4 x zy; ^ x n = pr2 x n; ^ xu xv = pr0 xu xv; " "pr0 xu xv = error []; " "pr2 x n = pr1 x n (n > 0) where { f wz yu y = f4 wz yu y; f x n y = f0 x n y; ; f0 x n y = g x n where { g x n = g2 x n; ; g0 x n True = f x (n - 1) (x * y); ; g1 x n True = g (x * x) (n `quot` 2); g1 x n False = g0 x n otherwise; ; g2 x n = g1 x n (even n); } ; ; f3 True wz yu y = y; f3 yv yw yx yy = f0 yw yx yy; ; f4 wz yu y = f3 (yu == 0) wz yu y; f4 yz zu zv = f0 yz zu zv; ; pr1 x n True = f x (n - 1) x; pr1 x n False = pr0 x n; } ; pr2 zw zx = pr0 zw zx; " "pr3 True x zy = 1; pr3 zz vuu vuv = pr2 vuu vuv; " "pr4 x zy = pr3 (zy == 0) x zy; pr4 vuw vux = pr2 vuw vux; " ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "pr1 x n (n > 0) where { f wz yu y = f4 wz yu y; f x n y = f0 x n y; ; f0 x n y = g x n where { g x n = g2 x n; ; g0 x n True = f x (n - 1) (x * y); ; g1 x n True = g (x * x) (n `quot` 2); g1 x n False = g0 x n otherwise; ; g2 x n = g1 x n (even n); } ; ; f3 True wz yu y = y; f3 yv yw yx yy = f0 yw yx yy; ; f4 wz yu y = f3 (yu == 0) wz yu y; f4 yz zu zv = f0 yz zu zv; ; pr1 x n True = f x (n - 1) x; pr1 x n False = pr0 x n; } " are unpacked to the following functions on top level "pr2Pr1 x n True = pr2F x (n - 1) x; pr2Pr1 x n False = pr0 x n; " "pr2F wz yu y = pr2F4 wz yu y; pr2F x n y = pr2F0 x n y; " "pr2F3 True wz yu y = y; pr2F3 yv yw yx yy = pr2F0 yw yx yy; " "pr2F0 x n y = pr2F0G y x n; " "pr2F4 wz yu y = pr2F3 (yu == 0) wz yu y; pr2F4 yz zu zv = pr2F0 yz zu zv; " The bindings of the following Let/Where expression "g x n where { g x n = g2 x n; ; g0 x n True = f x (n - 1) (x * y); ; g1 x n True = g (x * x) (n `quot` 2); g1 x n False = g0 x n otherwise; ; g2 x n = g1 x n (even n); } " are unpacked to the following functions on top level "pr2F0G1 vuy x n True = pr2F0G vuy (x * x) (n `quot` 2); pr2F0G1 vuy x n False = pr2F0G0 vuy x n otherwise; " "pr2F0G vuy x n = pr2F0G2 vuy x n; " "pr2F0G0 vuy x n True = pr2F x (n - 1) (x * vuy); " "pr2F0G2 vuy x n = pr2F0G1 vuy x n (even n); " ---------------------------------------- (8) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (9) NumRed (SOUND) Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. ---------------------------------------- (10) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (11) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="(^)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="(^) vuz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="(^) vuz3 vuz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="pr4 vuz3 vuz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="pr3 (vuz4 == fromInt (Pos Zero)) vuz3 vuz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="pr3 (primEqInt vuz4 (fromInt (Pos Zero))) vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];4738[label="vuz4/Pos vuz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 4738[label="",style="solid", color="burlywood", weight=9]; 4738 -> 8[label="",style="solid", color="burlywood", weight=3]; 4739[label="vuz4/Neg vuz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 4739[label="",style="solid", color="burlywood", weight=9]; 4739 -> 9[label="",style="solid", color="burlywood", weight=3]; 8[label="pr3 (primEqInt (Pos vuz40) (fromInt (Pos Zero))) vuz3 (Pos vuz40)",fontsize=16,color="burlywood",shape="box"];4740[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 4740[label="",style="solid", color="burlywood", weight=9]; 4740 -> 10[label="",style="solid", color="burlywood", weight=3]; 4741[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 4741[label="",style="solid", color="burlywood", weight=9]; 4741 -> 11[label="",style="solid", color="burlywood", weight=3]; 9[label="pr3 (primEqInt (Neg vuz40) (fromInt (Pos Zero))) vuz3 (Neg vuz40)",fontsize=16,color="burlywood",shape="box"];4742[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 4742[label="",style="solid", color="burlywood", weight=9]; 4742 -> 12[label="",style="solid", color="burlywood", weight=3]; 4743[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 4743[label="",style="solid", color="burlywood", weight=9]; 4743 -> 13[label="",style="solid", color="burlywood", weight=3]; 10[label="pr3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3]; 11[label="pr3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 12[label="pr3 (primEqInt (Neg (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 13[label="pr3 (primEqInt (Neg Zero) (fromInt (Pos Zero))) vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 14[label="pr3 (primEqInt (Pos (Succ vuz400)) (Pos Zero)) vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 15[label="pr3 (primEqInt (Pos Zero) (Pos Zero)) vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 16[label="pr3 (primEqInt (Neg (Succ vuz400)) (Pos Zero)) vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 17[label="pr3 (primEqInt (Neg Zero) (Pos Zero)) vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 18[label="pr3 False vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 19[label="pr3 True vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 20[label="pr3 False vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 21[label="pr3 True vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 22[label="pr2 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 23[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];23 -> 27[label="",style="solid", color="black", weight=3]; 24[label="pr2 vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 25 -> 23[label="",style="dashed", color="red", weight=0]; 25[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];26[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (Pos (Succ vuz400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3]; 27[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];28[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (Neg (Succ vuz400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3]; 29[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (compare (Pos (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3]; 30[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (compare (Neg (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3]; 31[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpInt (Pos (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3]; 32[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (primCmpInt (Neg (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3]; 33[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpInt (Pos (Succ vuz400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3]; 34[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (primCmpInt (Neg (Succ vuz400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3]; 35[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpNat (Succ vuz400) Zero == GT)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 36[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (LT == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3]; 37[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (GT == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3]; 38[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) False",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3]; 39[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) True",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3]; 40[label="pr0 vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3]; 41 -> 43[label="",style="dashed", color="red", weight=0]; 41[label="pr2F vuz3 (Pos (Succ vuz400) - fromInt (Pos (Succ Zero))) vuz3",fontsize=16,color="magenta"];41 -> 44[label="",style="dashed", color="magenta", weight=3]; 42[label="error []",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3]; 44 -> 23[label="",style="dashed", color="red", weight=0]; 44[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];43[label="pr2F vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="triangle"];43 -> 46[label="",style="solid", color="black", weight=3]; 45[label="error []",fontsize=16,color="red",shape="box"];46[label="pr2F4 vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];46 -> 47[label="",style="solid", color="black", weight=3]; 47[label="pr2F3 (Pos (Succ vuz400) - vuz5 == fromInt (Pos Zero)) vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];47 -> 48[label="",style="solid", color="black", weight=3]; 48[label="pr2F3 (primEqInt (Pos (Succ vuz400) - vuz5) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];48 -> 49[label="",style="solid", color="black", weight=3]; 49[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) vuz5) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) vuz5) vuz3",fontsize=16,color="burlywood",shape="box"];4744[label="vuz5/Pos vuz50",fontsize=10,color="white",style="solid",shape="box"];49 -> 4744[label="",style="solid", color="burlywood", weight=9]; 4744 -> 50[label="",style="solid", color="burlywood", weight=3]; 4745[label="vuz5/Neg vuz50",fontsize=10,color="white",style="solid",shape="box"];49 -> 4745[label="",style="solid", color="burlywood", weight=9]; 4745 -> 51[label="",style="solid", color="burlywood", weight=3]; 50[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) (Pos vuz50)) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) (Pos vuz50)) vuz3",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3]; 51[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) (Neg vuz50)) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) (Neg vuz50)) vuz3",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3]; 52[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) vuz50) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) vuz50) vuz3",fontsize=16,color="burlywood",shape="box"];4746[label="vuz50/Succ vuz500",fontsize=10,color="white",style="solid",shape="box"];52 -> 4746[label="",style="solid", color="burlywood", weight=9]; 4746 -> 54[label="",style="solid", color="burlywood", weight=3]; 4747[label="vuz50/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 4747[label="",style="solid", color="burlywood", weight=9]; 4747 -> 55[label="",style="solid", color="burlywood", weight=3]; 53[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) vuz50)) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) vuz50)) vuz3",fontsize=16,color="burlywood",shape="box"];4748[label="vuz50/Succ vuz500",fontsize=10,color="white",style="solid",shape="box"];53 -> 4748[label="",style="solid", color="burlywood", weight=9]; 4748 -> 56[label="",style="solid", color="burlywood", weight=3]; 4749[label="vuz50/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 4749[label="",style="solid", color="burlywood", weight=9]; 4749 -> 57[label="",style="solid", color="burlywood", weight=3]; 54[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) (Succ vuz500)) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) (Succ vuz500)) vuz3",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3]; 55[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) Zero) vuz3",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 56[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) (Succ vuz500))) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) (Succ vuz500))) vuz3",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 57[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) Zero)) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) Zero)) vuz3",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3]; 58[label="pr2F3 (primEqInt (primMinusNat vuz400 vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat vuz400 vuz500) vuz3",fontsize=16,color="burlywood",shape="triangle"];4750[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 4750[label="",style="solid", color="burlywood", weight=9]; 4750 -> 62[label="",style="solid", color="burlywood", weight=3]; 4751[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 4751[label="",style="solid", color="burlywood", weight=9]; 4751 -> 63[label="",style="solid", color="burlywood", weight=3]; 59[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="triangle"];59 -> 64[label="",style="solid", color="black", weight=3]; 60 -> 59[label="",style="dashed", color="red", weight=0]; 60[label="pr2F3 (primEqInt (Pos (Succ (Succ (primPlusNat vuz400 vuz500)))) (fromInt (Pos Zero))) vuz3 (Pos (Succ (Succ (primPlusNat vuz400 vuz500)))) vuz3",fontsize=16,color="magenta"];60 -> 65[label="",style="dashed", color="magenta", weight=3]; 61 -> 59[label="",style="dashed", color="red", weight=0]; 61[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="magenta"];62[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) vuz500) vuz3",fontsize=16,color="burlywood",shape="box"];4752[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];62 -> 4752[label="",style="solid", color="burlywood", weight=9]; 4752 -> 66[label="",style="solid", color="burlywood", weight=3]; 4753[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 4753[label="",style="solid", color="burlywood", weight=9]; 4753 -> 67[label="",style="solid", color="burlywood", weight=3]; 63[label="pr2F3 (primEqInt (primMinusNat Zero vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero vuz500) vuz3",fontsize=16,color="burlywood",shape="box"];4754[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];63 -> 4754[label="",style="solid", color="burlywood", weight=9]; 4754 -> 68[label="",style="solid", color="burlywood", weight=3]; 4755[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 4755[label="",style="solid", color="burlywood", weight=9]; 4755 -> 69[label="",style="solid", color="burlywood", weight=3]; 64[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (Pos Zero)) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];64 -> 70[label="",style="solid", color="black", weight=3]; 65[label="Succ (primPlusNat vuz400 vuz500)",fontsize=16,color="green",shape="box"];65 -> 71[label="",style="dashed", color="green", weight=3]; 66[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3]; 67[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) Zero) vuz3",fontsize=16,color="black",shape="box"];67 -> 73[label="",style="solid", color="black", weight=3]; 68[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];68 -> 74[label="",style="solid", color="black", weight=3]; 69[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero Zero) vuz3",fontsize=16,color="black",shape="box"];69 -> 75[label="",style="solid", color="black", weight=3]; 70[label="pr2F3 False vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];70 -> 76[label="",style="solid", color="black", weight=3]; 71[label="primPlusNat vuz400 vuz500",fontsize=16,color="burlywood",shape="triangle"];4756[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];71 -> 4756[label="",style="solid", color="burlywood", weight=9]; 4756 -> 77[label="",style="solid", color="burlywood", weight=3]; 4757[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];71 -> 4757[label="",style="solid", color="burlywood", weight=9]; 4757 -> 78[label="",style="solid", color="burlywood", weight=3]; 72 -> 58[label="",style="dashed", color="red", weight=0]; 72[label="pr2F3 (primEqInt (primMinusNat vuz4000 vuz5000) (fromInt (Pos Zero))) vuz3 (primMinusNat vuz4000 vuz5000) vuz3",fontsize=16,color="magenta"];72 -> 79[label="",style="dashed", color="magenta", weight=3]; 72 -> 80[label="",style="dashed", color="magenta", weight=3]; 73 -> 59[label="",style="dashed", color="red", weight=0]; 73[label="pr2F3 (primEqInt (Pos (Succ vuz4000)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz4000)) vuz3",fontsize=16,color="magenta"];73 -> 81[label="",style="dashed", color="magenta", weight=3]; 74[label="pr2F3 (primEqInt (Neg (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];74 -> 82[label="",style="solid", color="black", weight=3]; 75[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];75 -> 83[label="",style="solid", color="black", weight=3]; 76[label="pr2F0 vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];76 -> 84[label="",style="solid", color="black", weight=3]; 77[label="primPlusNat (Succ vuz4000) vuz500",fontsize=16,color="burlywood",shape="box"];4758[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];77 -> 4758[label="",style="solid", color="burlywood", weight=9]; 4758 -> 85[label="",style="solid", color="burlywood", weight=3]; 4759[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];77 -> 4759[label="",style="solid", color="burlywood", weight=9]; 4759 -> 86[label="",style="solid", color="burlywood", weight=3]; 78[label="primPlusNat Zero vuz500",fontsize=16,color="burlywood",shape="box"];4760[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];78 -> 4760[label="",style="solid", color="burlywood", weight=9]; 4760 -> 87[label="",style="solid", color="burlywood", weight=3]; 4761[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];78 -> 4761[label="",style="solid", color="burlywood", weight=9]; 4761 -> 88[label="",style="solid", color="burlywood", weight=3]; 79[label="vuz4000",fontsize=16,color="green",shape="box"];80[label="vuz5000",fontsize=16,color="green",shape="box"];81[label="vuz4000",fontsize=16,color="green",shape="box"];82[label="pr2F3 (primEqInt (Neg (Succ vuz5000)) (Pos Zero)) vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];82 -> 89[label="",style="solid", color="black", weight=3]; 83[label="pr2F3 (primEqInt (Pos Zero) (Pos Zero)) vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];83 -> 90[label="",style="solid", color="black", weight=3]; 84[label="pr2F0G vuz3 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];84 -> 91[label="",style="solid", color="black", weight=3]; 85[label="primPlusNat (Succ vuz4000) (Succ vuz5000)",fontsize=16,color="black",shape="box"];85 -> 92[label="",style="solid", color="black", weight=3]; 86[label="primPlusNat (Succ vuz4000) Zero",fontsize=16,color="black",shape="box"];86 -> 93[label="",style="solid", color="black", weight=3]; 87[label="primPlusNat Zero (Succ vuz5000)",fontsize=16,color="black",shape="box"];87 -> 94[label="",style="solid", color="black", weight=3]; 88[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];88 -> 95[label="",style="solid", color="black", weight=3]; 89[label="pr2F3 False vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];89 -> 96[label="",style="solid", color="black", weight=3]; 90[label="pr2F3 True vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];90 -> 97[label="",style="solid", color="black", weight=3]; 91[label="pr2F0G2 vuz3 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3]; 92[label="Succ (Succ (primPlusNat vuz4000 vuz5000))",fontsize=16,color="green",shape="box"];92 -> 99[label="",style="dashed", color="green", weight=3]; 93[label="Succ vuz4000",fontsize=16,color="green",shape="box"];94[label="Succ vuz5000",fontsize=16,color="green",shape="box"];95[label="Zero",fontsize=16,color="green",shape="box"];96[label="pr2F0 vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3]; 97[label="vuz3",fontsize=16,color="green",shape="box"];98[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (even (Pos (Succ vuz400)))",fontsize=16,color="black",shape="box"];98 -> 101[label="",style="solid", color="black", weight=3]; 99 -> 71[label="",style="dashed", color="red", weight=0]; 99[label="primPlusNat vuz4000 vuz5000",fontsize=16,color="magenta"];99 -> 102[label="",style="dashed", color="magenta", weight=3]; 99 -> 103[label="",style="dashed", color="magenta", weight=3]; 100[label="pr2F0G vuz3 vuz3 (Neg (Succ vuz5000))",fontsize=16,color="black",shape="box"];100 -> 104[label="",style="solid", color="black", weight=3]; 101[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (primEvenInt (Pos (Succ vuz400)))",fontsize=16,color="black",shape="box"];101 -> 105[label="",style="solid", color="black", weight=3]; 102[label="vuz4000",fontsize=16,color="green",shape="box"];103[label="vuz5000",fontsize=16,color="green",shape="box"];104[label="pr2F0G2 vuz3 vuz3 (Neg (Succ vuz5000))",fontsize=16,color="black",shape="box"];104 -> 106[label="",style="solid", color="black", weight=3]; 105 -> 256[label="",style="dashed", color="red", weight=0]; 105[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (primEvenNat (Succ vuz400))",fontsize=16,color="magenta"];105 -> 257[label="",style="dashed", color="magenta", weight=3]; 105 -> 258[label="",style="dashed", color="magenta", weight=3]; 105 -> 259[label="",style="dashed", color="magenta", weight=3]; 106[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (even (Neg (Succ vuz5000)))",fontsize=16,color="black",shape="box"];106 -> 109[label="",style="solid", color="black", weight=3]; 257[label="vuz3",fontsize=16,color="green",shape="box"];258[label="vuz400",fontsize=16,color="green",shape="box"];259[label="Succ vuz400",fontsize=16,color="green",shape="box"];256[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat vuz22)",fontsize=16,color="burlywood",shape="triangle"];4762[label="vuz22/Succ vuz220",fontsize=10,color="white",style="solid",shape="box"];256 -> 4762[label="",style="solid", color="burlywood", weight=9]; 4762 -> 275[label="",style="solid", color="burlywood", weight=3]; 4763[label="vuz22/Zero",fontsize=10,color="white",style="solid",shape="box"];256 -> 4763[label="",style="solid", color="burlywood", weight=9]; 4763 -> 276[label="",style="solid", color="burlywood", weight=3]; 109[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (primEvenInt (Neg (Succ vuz5000)))",fontsize=16,color="black",shape="box"];109 -> 112[label="",style="solid", color="black", weight=3]; 275[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ vuz220))",fontsize=16,color="burlywood",shape="box"];4764[label="vuz220/Succ vuz2200",fontsize=10,color="white",style="solid",shape="box"];275 -> 4764[label="",style="solid", color="burlywood", weight=9]; 4764 -> 279[label="",style="solid", color="burlywood", weight=3]; 4765[label="vuz220/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 4765[label="",style="solid", color="burlywood", weight=9]; 4765 -> 280[label="",style="solid", color="burlywood", weight=3]; 276[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];276 -> 281[label="",style="solid", color="black", weight=3]; 112 -> 199[label="",style="dashed", color="red", weight=0]; 112[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (primEvenNat (Succ vuz5000))",fontsize=16,color="magenta"];112 -> 200[label="",style="dashed", color="magenta", weight=3]; 112 -> 201[label="",style="dashed", color="magenta", weight=3]; 112 -> 202[label="",style="dashed", color="magenta", weight=3]; 279[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ (Succ vuz2200)))",fontsize=16,color="black",shape="box"];279 -> 284[label="",style="solid", color="black", weight=3]; 280[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];280 -> 285[label="",style="solid", color="black", weight=3]; 281[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) True",fontsize=16,color="black",shape="box"];281 -> 286[label="",style="solid", color="black", weight=3]; 200[label="Succ vuz5000",fontsize=16,color="green",shape="box"];201[label="vuz3",fontsize=16,color="green",shape="box"];202[label="vuz5000",fontsize=16,color="green",shape="box"];199[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat vuz14)",fontsize=16,color="burlywood",shape="triangle"];4766[label="vuz14/Succ vuz140",fontsize=10,color="white",style="solid",shape="box"];199 -> 4766[label="",style="solid", color="burlywood", weight=9]; 4766 -> 212[label="",style="solid", color="burlywood", weight=3]; 4767[label="vuz14/Zero",fontsize=10,color="white",style="solid",shape="box"];199 -> 4767[label="",style="solid", color="burlywood", weight=9]; 4767 -> 213[label="",style="solid", color="burlywood", weight=3]; 284 -> 256[label="",style="dashed", color="red", weight=0]; 284[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat vuz2200)",fontsize=16,color="magenta"];284 -> 289[label="",style="dashed", color="magenta", weight=3]; 285[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) False",fontsize=16,color="black",shape="box"];285 -> 290[label="",style="solid", color="black", weight=3]; 286[label="pr2F0G vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];286 -> 291[label="",style="solid", color="black", weight=3]; 212[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ vuz140))",fontsize=16,color="burlywood",shape="box"];4768[label="vuz140/Succ vuz1400",fontsize=10,color="white",style="solid",shape="box"];212 -> 4768[label="",style="solid", color="burlywood", weight=9]; 4768 -> 216[label="",style="solid", color="burlywood", weight=3]; 4769[label="vuz140/Zero",fontsize=10,color="white",style="solid",shape="box"];212 -> 4769[label="",style="solid", color="burlywood", weight=9]; 4769 -> 217[label="",style="solid", color="burlywood", weight=3]; 213[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];213 -> 218[label="",style="solid", color="black", weight=3]; 289[label="vuz2200",fontsize=16,color="green",shape="box"];290[label="pr2F0G0 vuz20 vuz20 (Pos (Succ vuz21)) otherwise",fontsize=16,color="black",shape="box"];290 -> 294[label="",style="solid", color="black", weight=3]; 291[label="pr2F0G2 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];291 -> 295[label="",style="solid", color="black", weight=3]; 216[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ (Succ vuz1400)))",fontsize=16,color="black",shape="box"];216 -> 227[label="",style="solid", color="black", weight=3]; 217[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];217 -> 228[label="",style="solid", color="black", weight=3]; 218[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) True",fontsize=16,color="black",shape="box"];218 -> 229[label="",style="solid", color="black", weight=3]; 294[label="pr2F0G0 vuz20 vuz20 (Pos (Succ vuz21)) True",fontsize=16,color="black",shape="box"];294 -> 298[label="",style="solid", color="black", weight=3]; 295[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];295 -> 299[label="",style="solid", color="black", weight=3]; 227 -> 199[label="",style="dashed", color="red", weight=0]; 227[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat vuz1400)",fontsize=16,color="magenta"];227 -> 237[label="",style="dashed", color="magenta", weight=3]; 228[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) False",fontsize=16,color="black",shape="box"];228 -> 238[label="",style="solid", color="black", weight=3]; 229[label="pr2F0G vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];229 -> 239[label="",style="solid", color="black", weight=3]; 298 -> 302[label="",style="dashed", color="red", weight=0]; 298[label="pr2F vuz20 (Pos (Succ vuz21) - fromInt (Pos (Succ Zero))) (vuz20 * vuz20)",fontsize=16,color="magenta"];298 -> 303[label="",style="dashed", color="magenta", weight=3]; 299[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];299 -> 304[label="",style="solid", color="black", weight=3]; 237[label="vuz1400",fontsize=16,color="green",shape="box"];238[label="pr2F0G0 vuz12 vuz12 (Neg (Succ vuz13)) otherwise",fontsize=16,color="black",shape="box"];238 -> 253[label="",style="solid", color="black", weight=3]; 239[label="pr2F0G2 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];239 -> 254[label="",style="solid", color="black", weight=3]; 303 -> 23[label="",style="dashed", color="red", weight=0]; 303[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];302[label="pr2F vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="triangle"];302 -> 305[label="",style="solid", color="black", weight=3]; 304[label="pr2F0G1 vuz20 (vuz20 * vuz20) (primQuotInt (Pos (Succ vuz21)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos (Succ vuz21)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];304 -> 309[label="",style="solid", color="black", weight=3]; 253[label="pr2F0G0 vuz12 vuz12 (Neg (Succ vuz13)) True",fontsize=16,color="black",shape="box"];253 -> 277[label="",style="solid", color="black", weight=3]; 254[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];254 -> 278[label="",style="solid", color="black", weight=3]; 305[label="pr2F4 vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="box"];305 -> 310[label="",style="solid", color="black", weight=3]; 309[label="pr2F0G1 vuz20 (vuz20 * vuz20) (primQuotInt (Pos (Succ vuz21)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos (Succ vuz21)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];309 -> 315[label="",style="solid", color="black", weight=3]; 277 -> 282[label="",style="dashed", color="red", weight=0]; 277[label="pr2F vuz12 (Neg (Succ vuz13) - fromInt (Pos (Succ Zero))) (vuz12 * vuz12)",fontsize=16,color="magenta"];277 -> 283[label="",style="dashed", color="magenta", weight=3]; 278[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];278 -> 287[label="",style="solid", color="black", weight=3]; 310[label="pr2F3 (Pos (Succ vuz21) - vuz24 == fromInt (Pos Zero)) vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="box"];310 -> 316[label="",style="solid", color="black", weight=3]; 315 -> 1605[label="",style="dashed", color="red", weight=0]; 315[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (primDivNatS (Succ vuz21) (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS (Succ vuz21) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];315 -> 1606[label="",style="dashed", color="magenta", weight=3]; 315 -> 1607[label="",style="dashed", color="magenta", weight=3]; 315 -> 1608[label="",style="dashed", color="magenta", weight=3]; 315 -> 1609[label="",style="dashed", color="magenta", weight=3]; 283 -> 23[label="",style="dashed", color="red", weight=0]; 283[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];282[label="pr2F vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="triangle"];282 -> 288[label="",style="solid", color="black", weight=3]; 287[label="pr2F0G1 vuz12 (vuz12 * vuz12) (primQuotInt (Neg (Succ vuz13)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg (Succ vuz13)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];287 -> 292[label="",style="solid", color="black", weight=3]; 316 -> 3789[label="",style="dashed", color="red", weight=0]; 316[label="pr2F3 (primEqInt (Pos (Succ vuz21) - vuz24) (fromInt (Pos Zero))) vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="magenta"];316 -> 3790[label="",style="dashed", color="magenta", weight=3]; 316 -> 3791[label="",style="dashed", color="magenta", weight=3]; 316 -> 3792[label="",style="dashed", color="magenta", weight=3]; 316 -> 3793[label="",style="dashed", color="magenta", weight=3]; 1606 -> 1226[label="",style="dashed", color="red", weight=0]; 1606[label="primDivNatS (Succ vuz21) (Succ (Succ Zero))",fontsize=16,color="magenta"];1606 -> 1624[label="",style="dashed", color="magenta", weight=3]; 1607[label="vuz20",fontsize=16,color="green",shape="box"];1608[label="vuz20",fontsize=16,color="green",shape="box"];1609 -> 1226[label="",style="dashed", color="red", weight=0]; 1609[label="primDivNatS (Succ vuz21) (Succ (Succ Zero))",fontsize=16,color="magenta"];1609 -> 1625[label="",style="dashed", color="magenta", weight=3]; 1605[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenInt (Pos vuz106))",fontsize=16,color="black",shape="triangle"];1605 -> 1626[label="",style="solid", color="black", weight=3]; 288[label="pr2F4 vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="box"];288 -> 293[label="",style="solid", color="black", weight=3]; 292[label="pr2F0G1 vuz12 (vuz12 * vuz12) (primQuotInt (Neg (Succ vuz13)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg (Succ vuz13)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];292 -> 296[label="",style="solid", color="black", weight=3]; 3790[label="vuz24",fontsize=16,color="green",shape="box"];3791[label="vuz21",fontsize=16,color="green",shape="box"];3792[label="vuz20",fontsize=16,color="green",shape="box"];3793[label="vuz20",fontsize=16,color="green",shape="box"];3789[label="pr2F3 (primEqInt (Pos (Succ vuz202) - vuz203) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz202) - vuz203) (vuz204 * vuz205)",fontsize=16,color="black",shape="triangle"];3789 -> 3814[label="",style="solid", color="black", weight=3]; 1624[label="Succ vuz21",fontsize=16,color="green",shape="box"];1226[label="primDivNatS vuz55 (Succ (Succ Zero))",fontsize=16,color="burlywood",shape="triangle"];4770[label="vuz55/Succ vuz550",fontsize=10,color="white",style="solid",shape="box"];1226 -> 4770[label="",style="solid", color="burlywood", weight=9]; 4770 -> 1241[label="",style="solid", color="burlywood", weight=3]; 4771[label="vuz55/Zero",fontsize=10,color="white",style="solid",shape="box"];1226 -> 4771[label="",style="solid", color="burlywood", weight=9]; 4771 -> 1242[label="",style="solid", color="burlywood", weight=3]; 1625[label="Succ vuz21",fontsize=16,color="green",shape="box"];1626[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat vuz106)",fontsize=16,color="burlywood",shape="triangle"];4772[label="vuz106/Succ vuz1060",fontsize=10,color="white",style="solid",shape="box"];1626 -> 4772[label="",style="solid", color="burlywood", weight=9]; 4772 -> 1659[label="",style="solid", color="burlywood", weight=3]; 4773[label="vuz106/Zero",fontsize=10,color="white",style="solid",shape="box"];1626 -> 4773[label="",style="solid", color="burlywood", weight=9]; 4773 -> 1660[label="",style="solid", color="burlywood", weight=3]; 293[label="pr2F3 (Neg (Succ vuz13) - vuz23 == fromInt (Pos Zero)) vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="box"];293 -> 297[label="",style="solid", color="black", weight=3]; 296 -> 1755[label="",style="dashed", color="red", weight=0]; 296[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (primDivNatS (Succ vuz13) (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS (Succ vuz13) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];296 -> 1756[label="",style="dashed", color="magenta", weight=3]; 296 -> 1757[label="",style="dashed", color="magenta", weight=3]; 296 -> 1758[label="",style="dashed", color="magenta", weight=3]; 296 -> 1759[label="",style="dashed", color="magenta", weight=3]; 3814[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) vuz203) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) vuz203) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4774[label="vuz203/Pos vuz2030",fontsize=10,color="white",style="solid",shape="box"];3814 -> 4774[label="",style="solid", color="burlywood", weight=9]; 4774 -> 3843[label="",style="solid", color="burlywood", weight=3]; 4775[label="vuz203/Neg vuz2030",fontsize=10,color="white",style="solid",shape="box"];3814 -> 4775[label="",style="solid", color="burlywood", weight=9]; 4775 -> 3844[label="",style="solid", color="burlywood", weight=3]; 1241[label="primDivNatS (Succ vuz550) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];1241 -> 1287[label="",style="solid", color="black", weight=3]; 1242[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];1242 -> 1288[label="",style="solid", color="black", weight=3]; 1659[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ vuz1060))",fontsize=16,color="burlywood",shape="box"];4776[label="vuz1060/Succ vuz10600",fontsize=10,color="white",style="solid",shape="box"];1659 -> 4776[label="",style="solid", color="burlywood", weight=9]; 4776 -> 1713[label="",style="solid", color="burlywood", weight=3]; 4777[label="vuz1060/Zero",fontsize=10,color="white",style="solid",shape="box"];1659 -> 4777[label="",style="solid", color="burlywood", weight=9]; 4777 -> 1714[label="",style="solid", color="burlywood", weight=3]; 1660[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];1660 -> 1715[label="",style="solid", color="black", weight=3]; 297 -> 4256[label="",style="dashed", color="red", weight=0]; 297[label="pr2F3 (primEqInt (Neg (Succ vuz13) - vuz23) (fromInt (Pos Zero))) vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="magenta"];297 -> 4257[label="",style="dashed", color="magenta", weight=3]; 297 -> 4258[label="",style="dashed", color="magenta", weight=3]; 297 -> 4259[label="",style="dashed", color="magenta", weight=3]; 297 -> 4260[label="",style="dashed", color="magenta", weight=3]; 1756[label="vuz12",fontsize=16,color="green",shape="box"];1757 -> 1226[label="",style="dashed", color="red", weight=0]; 1757[label="primDivNatS (Succ vuz13) (Succ (Succ Zero))",fontsize=16,color="magenta"];1757 -> 1770[label="",style="dashed", color="magenta", weight=3]; 1758[label="vuz12",fontsize=16,color="green",shape="box"];1759 -> 1226[label="",style="dashed", color="red", weight=0]; 1759[label="primDivNatS (Succ vuz13) (Succ (Succ Zero))",fontsize=16,color="magenta"];1759 -> 1771[label="",style="dashed", color="magenta", weight=3]; 1755[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenInt (Neg vuz114))",fontsize=16,color="black",shape="triangle"];1755 -> 1772[label="",style="solid", color="black", weight=3]; 3843[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) (Pos vuz2030)) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) (Pos vuz2030)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];3843 -> 3919[label="",style="solid", color="black", weight=3]; 3844[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) (Neg vuz2030)) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) (Neg vuz2030)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];3844 -> 3920[label="",style="solid", color="black", weight=3]; 1287 -> 562[label="",style="dashed", color="red", weight=0]; 1287[label="primDivNatS0 vuz550 (Succ Zero) (primGEqNatS vuz550 (Succ Zero))",fontsize=16,color="magenta"];1287 -> 1313[label="",style="dashed", color="magenta", weight=3]; 1288[label="Zero",fontsize=16,color="green",shape="box"];1713[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ (Succ vuz10600)))",fontsize=16,color="black",shape="box"];1713 -> 1773[label="",style="solid", color="black", weight=3]; 1714[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];1714 -> 1774[label="",style="solid", color="black", weight=3]; 1715[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) True",fontsize=16,color="black",shape="box"];1715 -> 1775[label="",style="solid", color="black", weight=3]; 4257[label="vuz13",fontsize=16,color="green",shape="box"];4258[label="vuz12",fontsize=16,color="green",shape="box"];4259[label="vuz23",fontsize=16,color="green",shape="box"];4260[label="vuz12",fontsize=16,color="green",shape="box"];4256[label="pr2F3 (primEqInt (Neg (Succ vuz214) - vuz215) (fromInt (Pos Zero))) vuz216 (Neg (Succ vuz214) - vuz215) (vuz216 * vuz217)",fontsize=16,color="black",shape="triangle"];4256 -> 4281[label="",style="solid", color="black", weight=3]; 1770[label="Succ vuz13",fontsize=16,color="green",shape="box"];1771[label="Succ vuz13",fontsize=16,color="green",shape="box"];1772[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat vuz114)",fontsize=16,color="burlywood",shape="triangle"];4778[label="vuz114/Succ vuz1140",fontsize=10,color="white",style="solid",shape="box"];1772 -> 4778[label="",style="solid", color="burlywood", weight=9]; 4778 -> 1793[label="",style="solid", color="burlywood", weight=3]; 4779[label="vuz114/Zero",fontsize=10,color="white",style="solid",shape="box"];1772 -> 4779[label="",style="solid", color="burlywood", weight=9]; 4779 -> 1794[label="",style="solid", color="burlywood", weight=3]; 3919[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) vuz2030) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) vuz2030) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4780[label="vuz2030/Succ vuz20300",fontsize=10,color="white",style="solid",shape="box"];3919 -> 4780[label="",style="solid", color="burlywood", weight=9]; 4780 -> 4005[label="",style="solid", color="burlywood", weight=3]; 4781[label="vuz2030/Zero",fontsize=10,color="white",style="solid",shape="box"];3919 -> 4781[label="",style="solid", color="burlywood", weight=9]; 4781 -> 4006[label="",style="solid", color="burlywood", weight=3]; 3920 -> 4007[label="",style="dashed", color="red", weight=0]; 3920[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz202) vuz2030)) (fromInt (Pos Zero))) vuz204 (Pos (primPlusNat (Succ vuz202) vuz2030)) (vuz204 * vuz205)",fontsize=16,color="magenta"];3920 -> 4008[label="",style="dashed", color="magenta", weight=3]; 3920 -> 4009[label="",style="dashed", color="magenta", weight=3]; 1313[label="vuz550",fontsize=16,color="green",shape="box"];562[label="primDivNatS0 vuz1300 (Succ Zero) (primGEqNatS vuz1300 (Succ Zero))",fontsize=16,color="burlywood",shape="triangle"];4782[label="vuz1300/Succ vuz13000",fontsize=10,color="white",style="solid",shape="box"];562 -> 4782[label="",style="solid", color="burlywood", weight=9]; 4782 -> 570[label="",style="solid", color="burlywood", weight=3]; 4783[label="vuz1300/Zero",fontsize=10,color="white",style="solid",shape="box"];562 -> 4783[label="",style="solid", color="burlywood", weight=9]; 4783 -> 571[label="",style="solid", color="burlywood", weight=3]; 1773 -> 1626[label="",style="dashed", color="red", weight=0]; 1773[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat vuz10600)",fontsize=16,color="magenta"];1773 -> 1795[label="",style="dashed", color="magenta", weight=3]; 1774[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) False",fontsize=16,color="black",shape="box"];1774 -> 1796[label="",style="solid", color="black", weight=3]; 1775[label="pr2F0G vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1775 -> 1797[label="",style="solid", color="black", weight=3]; 4281[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) vuz215) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) vuz215) (vuz216 * vuz217)",fontsize=16,color="burlywood",shape="box"];4784[label="vuz215/Pos vuz2150",fontsize=10,color="white",style="solid",shape="box"];4281 -> 4784[label="",style="solid", color="burlywood", weight=9]; 4784 -> 4288[label="",style="solid", color="burlywood", weight=3]; 4785[label="vuz215/Neg vuz2150",fontsize=10,color="white",style="solid",shape="box"];4281 -> 4785[label="",style="solid", color="burlywood", weight=9]; 4785 -> 4289[label="",style="solid", color="burlywood", weight=3]; 1793[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ vuz1140))",fontsize=16,color="burlywood",shape="box"];4786[label="vuz1140/Succ vuz11400",fontsize=10,color="white",style="solid",shape="box"];1793 -> 4786[label="",style="solid", color="burlywood", weight=9]; 4786 -> 1807[label="",style="solid", color="burlywood", weight=3]; 4787[label="vuz1140/Zero",fontsize=10,color="white",style="solid",shape="box"];1793 -> 4787[label="",style="solid", color="burlywood", weight=9]; 4787 -> 1808[label="",style="solid", color="burlywood", weight=3]; 1794[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];1794 -> 1809[label="",style="solid", color="black", weight=3]; 4005[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) (Succ vuz20300)) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) (Succ vuz20300)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4005 -> 4086[label="",style="solid", color="black", weight=3]; 4006[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4006 -> 4087[label="",style="solid", color="black", weight=3]; 4008 -> 71[label="",style="dashed", color="red", weight=0]; 4008[label="primPlusNat (Succ vuz202) vuz2030",fontsize=16,color="magenta"];4008 -> 4088[label="",style="dashed", color="magenta", weight=3]; 4008 -> 4089[label="",style="dashed", color="magenta", weight=3]; 4009 -> 71[label="",style="dashed", color="red", weight=0]; 4009[label="primPlusNat (Succ vuz202) vuz2030",fontsize=16,color="magenta"];4009 -> 4090[label="",style="dashed", color="magenta", weight=3]; 4009 -> 4091[label="",style="dashed", color="magenta", weight=3]; 4007[label="pr2F3 (primEqInt (Pos vuz212) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="triangle"];4788[label="vuz212/Succ vuz2120",fontsize=10,color="white",style="solid",shape="box"];4007 -> 4788[label="",style="solid", color="burlywood", weight=9]; 4788 -> 4092[label="",style="solid", color="burlywood", weight=3]; 4789[label="vuz212/Zero",fontsize=10,color="white",style="solid",shape="box"];4007 -> 4789[label="",style="solid", color="burlywood", weight=9]; 4789 -> 4093[label="",style="solid", color="burlywood", weight=3]; 570[label="primDivNatS0 (Succ vuz13000) (Succ Zero) (primGEqNatS (Succ vuz13000) (Succ Zero))",fontsize=16,color="black",shape="box"];570 -> 584[label="",style="solid", color="black", weight=3]; 571[label="primDivNatS0 Zero (Succ Zero) (primGEqNatS Zero (Succ Zero))",fontsize=16,color="black",shape="box"];571 -> 585[label="",style="solid", color="black", weight=3]; 1795[label="vuz10600",fontsize=16,color="green",shape="box"];1796[label="pr2F0G0 vuz102 (vuz103 * vuz103) (Pos vuz105) otherwise",fontsize=16,color="black",shape="box"];1796 -> 1810[label="",style="solid", color="black", weight=3]; 1797[label="pr2F0G2 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1797 -> 1811[label="",style="solid", color="black", weight=3]; 4288[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) (Pos vuz2150)) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) (Pos vuz2150)) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4288 -> 4296[label="",style="solid", color="black", weight=3]; 4289[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) (Neg vuz2150)) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) (Neg vuz2150)) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4289 -> 4297[label="",style="solid", color="black", weight=3]; 1807[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ (Succ vuz11400)))",fontsize=16,color="black",shape="box"];1807 -> 1817[label="",style="solid", color="black", weight=3]; 1808[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];1808 -> 1818[label="",style="solid", color="black", weight=3]; 1809[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) True",fontsize=16,color="black",shape="box"];1809 -> 1819[label="",style="solid", color="black", weight=3]; 4086[label="pr2F3 (primEqInt (primMinusNat vuz202 vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat vuz202 vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="triangle"];4790[label="vuz202/Succ vuz2020",fontsize=10,color="white",style="solid",shape="box"];4086 -> 4790[label="",style="solid", color="burlywood", weight=9]; 4790 -> 4165[label="",style="solid", color="burlywood", weight=3]; 4791[label="vuz202/Zero",fontsize=10,color="white",style="solid",shape="box"];4086 -> 4791[label="",style="solid", color="burlywood", weight=9]; 4791 -> 4166[label="",style="solid", color="burlywood", weight=3]; 4087 -> 4007[label="",style="dashed", color="red", weight=0]; 4087[label="pr2F3 (primEqInt (Pos (Succ vuz202)) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz202)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4087 -> 4167[label="",style="dashed", color="magenta", weight=3]; 4087 -> 4168[label="",style="dashed", color="magenta", weight=3]; 4088[label="Succ vuz202",fontsize=16,color="green",shape="box"];4089[label="vuz2030",fontsize=16,color="green",shape="box"];4090[label="Succ vuz202",fontsize=16,color="green",shape="box"];4091[label="vuz2030",fontsize=16,color="green",shape="box"];4092[label="pr2F3 (primEqInt (Pos (Succ vuz2120)) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4092 -> 4169[label="",style="solid", color="black", weight=3]; 4093[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4093 -> 4170[label="",style="solid", color="black", weight=3]; 584[label="primDivNatS0 (Succ vuz13000) (Succ Zero) (primGEqNatS vuz13000 Zero)",fontsize=16,color="burlywood",shape="box"];4792[label="vuz13000/Succ vuz130000",fontsize=10,color="white",style="solid",shape="box"];584 -> 4792[label="",style="solid", color="burlywood", weight=9]; 4792 -> 606[label="",style="solid", color="burlywood", weight=3]; 4793[label="vuz13000/Zero",fontsize=10,color="white",style="solid",shape="box"];584 -> 4793[label="",style="solid", color="burlywood", weight=9]; 4793 -> 607[label="",style="solid", color="burlywood", weight=3]; 585[label="primDivNatS0 Zero (Succ Zero) False",fontsize=16,color="black",shape="box"];585 -> 608[label="",style="solid", color="black", weight=3]; 1810[label="pr2F0G0 vuz102 (vuz103 * vuz103) (Pos vuz105) True",fontsize=16,color="black",shape="box"];1810 -> 1820[label="",style="solid", color="black", weight=3]; 1811[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1811 -> 1821[label="",style="solid", color="black", weight=3]; 4296 -> 4311[label="",style="dashed", color="red", weight=0]; 4296[label="pr2F3 (primEqInt (Neg (primPlusNat (Succ vuz214) vuz2150)) (fromInt (Pos Zero))) vuz216 (Neg (primPlusNat (Succ vuz214) vuz2150)) (vuz216 * vuz217)",fontsize=16,color="magenta"];4296 -> 4312[label="",style="dashed", color="magenta", weight=3]; 4296 -> 4313[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4086[label="",style="dashed", color="red", weight=0]; 4297[label="pr2F3 (primEqInt (primMinusNat vuz2150 (Succ vuz214)) (fromInt (Pos Zero))) vuz216 (primMinusNat vuz2150 (Succ vuz214)) (vuz216 * vuz217)",fontsize=16,color="magenta"];4297 -> 4330[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4331[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4332[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4333[label="",style="dashed", color="magenta", weight=3]; 1817 -> 1772[label="",style="dashed", color="red", weight=0]; 1817[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat vuz11400)",fontsize=16,color="magenta"];1817 -> 1829[label="",style="dashed", color="magenta", weight=3]; 1818[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) False",fontsize=16,color="black",shape="box"];1818 -> 1830[label="",style="solid", color="black", weight=3]; 1819[label="pr2F0G vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1819 -> 1831[label="",style="solid", color="black", weight=3]; 4165[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4794[label="vuz20300/Succ vuz203000",fontsize=10,color="white",style="solid",shape="box"];4165 -> 4794[label="",style="solid", color="burlywood", weight=9]; 4794 -> 4282[label="",style="solid", color="burlywood", weight=3]; 4795[label="vuz20300/Zero",fontsize=10,color="white",style="solid",shape="box"];4165 -> 4795[label="",style="solid", color="burlywood", weight=9]; 4795 -> 4283[label="",style="solid", color="burlywood", weight=3]; 4166[label="pr2F3 (primEqInt (primMinusNat Zero vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4796[label="vuz20300/Succ vuz203000",fontsize=10,color="white",style="solid",shape="box"];4166 -> 4796[label="",style="solid", color="burlywood", weight=9]; 4796 -> 4284[label="",style="solid", color="burlywood", weight=3]; 4797[label="vuz20300/Zero",fontsize=10,color="white",style="solid",shape="box"];4166 -> 4797[label="",style="solid", color="burlywood", weight=9]; 4797 -> 4285[label="",style="solid", color="burlywood", weight=3]; 4167[label="Succ vuz202",fontsize=16,color="green",shape="box"];4168[label="Succ vuz202",fontsize=16,color="green",shape="box"];4169[label="pr2F3 (primEqInt (Pos (Succ vuz2120)) (Pos Zero)) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4169 -> 4286[label="",style="solid", color="black", weight=3]; 4170[label="pr2F3 (primEqInt (Pos Zero) (Pos Zero)) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4170 -> 4287[label="",style="solid", color="black", weight=3]; 606[label="primDivNatS0 (Succ (Succ vuz130000)) (Succ Zero) (primGEqNatS (Succ vuz130000) Zero)",fontsize=16,color="black",shape="box"];606 -> 619[label="",style="solid", color="black", weight=3]; 607[label="primDivNatS0 (Succ Zero) (Succ Zero) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];607 -> 620[label="",style="solid", color="black", weight=3]; 608[label="Zero",fontsize=16,color="green",shape="box"];1820 -> 1832[label="",style="dashed", color="red", weight=0]; 1820[label="pr2F (vuz103 * vuz103) (Pos vuz105 - fromInt (Pos (Succ Zero))) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1820 -> 1833[label="",style="dashed", color="magenta", weight=3]; 1821[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1821 -> 1834[label="",style="solid", color="black", weight=3]; 4312 -> 71[label="",style="dashed", color="red", weight=0]; 4312[label="primPlusNat (Succ vuz214) vuz2150",fontsize=16,color="magenta"];4312 -> 4334[label="",style="dashed", color="magenta", weight=3]; 4312 -> 4335[label="",style="dashed", color="magenta", weight=3]; 4313 -> 71[label="",style="dashed", color="red", weight=0]; 4313[label="primPlusNat (Succ vuz214) vuz2150",fontsize=16,color="magenta"];4313 -> 4336[label="",style="dashed", color="magenta", weight=3]; 4313 -> 4337[label="",style="dashed", color="magenta", weight=3]; 4311[label="pr2F3 (primEqInt (Neg vuz219) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="burlywood",shape="triangle"];4798[label="vuz219/Succ vuz2190",fontsize=10,color="white",style="solid",shape="box"];4311 -> 4798[label="",style="solid", color="burlywood", weight=9]; 4798 -> 4338[label="",style="solid", color="burlywood", weight=3]; 4799[label="vuz219/Zero",fontsize=10,color="white",style="solid",shape="box"];4311 -> 4799[label="",style="solid", color="burlywood", weight=9]; 4799 -> 4339[label="",style="solid", color="burlywood", weight=3]; 4330[label="Succ vuz214",fontsize=16,color="green",shape="box"];4331[label="vuz2150",fontsize=16,color="green",shape="box"];4332[label="vuz217",fontsize=16,color="green",shape="box"];4333[label="vuz216",fontsize=16,color="green",shape="box"];1829[label="vuz11400",fontsize=16,color="green",shape="box"];1830[label="pr2F0G0 vuz110 (vuz111 * vuz111) (Neg vuz113) otherwise",fontsize=16,color="black",shape="box"];1830 -> 1835[label="",style="solid", color="black", weight=3]; 1831[label="pr2F0G2 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1831 -> 1836[label="",style="solid", color="black", weight=3]; 4282[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4282 -> 4290[label="",style="solid", color="black", weight=3]; 4283[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4283 -> 4291[label="",style="solid", color="black", weight=3]; 4284[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4284 -> 4292[label="",style="solid", color="black", weight=3]; 4285[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4285 -> 4293[label="",style="solid", color="black", weight=3]; 4286[label="pr2F3 False vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4286 -> 4294[label="",style="solid", color="black", weight=3]; 4287[label="pr2F3 True vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4287 -> 4295[label="",style="solid", color="black", weight=3]; 619[label="primDivNatS0 (Succ (Succ vuz130000)) (Succ Zero) True",fontsize=16,color="black",shape="box"];619 -> 645[label="",style="solid", color="black", weight=3]; 620[label="primDivNatS0 (Succ Zero) (Succ Zero) True",fontsize=16,color="black",shape="box"];620 -> 646[label="",style="solid", color="black", weight=3]; 1833 -> 23[label="",style="dashed", color="red", weight=0]; 1833[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];1832[label="pr2F (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="triangle"];1832 -> 1837[label="",style="solid", color="black", weight=3]; 1834[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (primQuotInt (Pos vuz105) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos vuz105) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1834 -> 1846[label="",style="solid", color="black", weight=3]; 4334[label="Succ vuz214",fontsize=16,color="green",shape="box"];4335[label="vuz2150",fontsize=16,color="green",shape="box"];4336[label="Succ vuz214",fontsize=16,color="green",shape="box"];4337[label="vuz2150",fontsize=16,color="green",shape="box"];4338[label="pr2F3 (primEqInt (Neg (Succ vuz2190)) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4338 -> 4365[label="",style="solid", color="black", weight=3]; 4339[label="pr2F3 (primEqInt (Neg Zero) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4339 -> 4366[label="",style="solid", color="black", weight=3]; 1835[label="pr2F0G0 vuz110 (vuz111 * vuz111) (Neg vuz113) True",fontsize=16,color="black",shape="box"];1835 -> 1847[label="",style="solid", color="black", weight=3]; 1836[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1836 -> 1848[label="",style="solid", color="black", weight=3]; 4290 -> 4086[label="",style="dashed", color="red", weight=0]; 4290[label="pr2F3 (primEqInt (primMinusNat vuz2020 vuz203000) (fromInt (Pos Zero))) vuz204 (primMinusNat vuz2020 vuz203000) (vuz204 * vuz205)",fontsize=16,color="magenta"];4290 -> 4298[label="",style="dashed", color="magenta", weight=3]; 4290 -> 4299[label="",style="dashed", color="magenta", weight=3]; 4291 -> 4007[label="",style="dashed", color="red", weight=0]; 4291[label="pr2F3 (primEqInt (Pos (Succ vuz2020)) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz2020)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4291 -> 4300[label="",style="dashed", color="magenta", weight=3]; 4291 -> 4301[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4311[label="",style="dashed", color="red", weight=0]; 4292[label="pr2F3 (primEqInt (Neg (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (Neg (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4292 -> 4314[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4315[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4316[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4317[label="",style="dashed", color="magenta", weight=3]; 4293 -> 4007[label="",style="dashed", color="red", weight=0]; 4293[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz204 (Pos Zero) (vuz204 * vuz205)",fontsize=16,color="magenta"];4293 -> 4303[label="",style="dashed", color="magenta", weight=3]; 4293 -> 4304[label="",style="dashed", color="magenta", weight=3]; 4294[label="pr2F0 vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4294 -> 4305[label="",style="solid", color="black", weight=3]; 4295[label="vuz204 * vuz205",fontsize=16,color="blue",shape="box"];4800[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4800[label="",style="solid", color="blue", weight=9]; 4800 -> 4306[label="",style="solid", color="blue", weight=3]; 4801[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4801[label="",style="solid", color="blue", weight=9]; 4801 -> 4307[label="",style="solid", color="blue", weight=3]; 4802[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4802[label="",style="solid", color="blue", weight=9]; 4802 -> 4308[label="",style="solid", color="blue", weight=3]; 4803[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4803[label="",style="solid", color="blue", weight=9]; 4803 -> 4309[label="",style="solid", color="blue", weight=3]; 4804[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4804[label="",style="solid", color="blue", weight=9]; 4804 -> 4310[label="",style="solid", color="blue", weight=3]; 645[label="Succ (primDivNatS (primMinusNatS (Succ (Succ vuz130000)) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];645 -> 673[label="",style="dashed", color="green", weight=3]; 646[label="Succ (primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];646 -> 674[label="",style="dashed", color="green", weight=3]; 1837[label="pr2F4 (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1837 -> 1849[label="",style="solid", color="black", weight=3]; 1846[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (primQuotInt (Pos vuz105) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos vuz105) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1846 -> 1859[label="",style="solid", color="black", weight=3]; 4365[label="pr2F3 (primEqInt (Neg (Succ vuz2190)) (Pos Zero)) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4365 -> 4378[label="",style="solid", color="black", weight=3]; 4366[label="pr2F3 (primEqInt (Neg Zero) (Pos Zero)) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4366 -> 4379[label="",style="solid", color="black", weight=3]; 1847 -> 1860[label="",style="dashed", color="red", weight=0]; 1847[label="pr2F (vuz111 * vuz111) (Neg vuz113 - fromInt (Pos (Succ Zero))) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1847 -> 1861[label="",style="dashed", color="magenta", weight=3]; 1848[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1848 -> 1862[label="",style="solid", color="black", weight=3]; 4298[label="vuz203000",fontsize=16,color="green",shape="box"];4299[label="vuz2020",fontsize=16,color="green",shape="box"];4300[label="Succ vuz2020",fontsize=16,color="green",shape="box"];4301[label="Succ vuz2020",fontsize=16,color="green",shape="box"];4314[label="Succ vuz203000",fontsize=16,color="green",shape="box"];4315[label="vuz204",fontsize=16,color="green",shape="box"];4316[label="Succ vuz203000",fontsize=16,color="green",shape="box"];4317[label="vuz205",fontsize=16,color="green",shape="box"];4303[label="Zero",fontsize=16,color="green",shape="box"];4304[label="Zero",fontsize=16,color="green",shape="box"];4305[label="pr2F0G (vuz204 * vuz205) vuz204 (Pos vuz211)",fontsize=16,color="black",shape="box"];4305 -> 4340[label="",style="solid", color="black", weight=3]; 4306 -> 1024[label="",style="dashed", color="red", weight=0]; 4306[label="vuz204 * vuz205",fontsize=16,color="magenta"];4306 -> 4341[label="",style="dashed", color="magenta", weight=3]; 4306 -> 4342[label="",style="dashed", color="magenta", weight=3]; 4307 -> 1041[label="",style="dashed", color="red", weight=0]; 4307[label="vuz204 * vuz205",fontsize=16,color="magenta"];4307 -> 4343[label="",style="dashed", color="magenta", weight=3]; 4307 -> 4344[label="",style="dashed", color="magenta", weight=3]; 4308 -> 1051[label="",style="dashed", color="red", weight=0]; 4308[label="vuz204 * vuz205",fontsize=16,color="magenta"];4308 -> 4345[label="",style="dashed", color="magenta", weight=3]; 4308 -> 4346[label="",style="dashed", color="magenta", weight=3]; 4309 -> 1061[label="",style="dashed", color="red", weight=0]; 4309[label="vuz204 * vuz205",fontsize=16,color="magenta"];4309 -> 4347[label="",style="dashed", color="magenta", weight=3]; 4309 -> 4348[label="",style="dashed", color="magenta", weight=3]; 4310 -> 1073[label="",style="dashed", color="red", weight=0]; 4310[label="vuz204 * vuz205",fontsize=16,color="magenta"];4310 -> 4349[label="",style="dashed", color="magenta", weight=3]; 4310 -> 4350[label="",style="dashed", color="magenta", weight=3]; 673 -> 508[label="",style="dashed", color="red", weight=0]; 673[label="primDivNatS (primMinusNatS (Succ (Succ vuz130000)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];673 -> 739[label="",style="dashed", color="magenta", weight=3]; 674 -> 510[label="",style="dashed", color="red", weight=0]; 674[label="primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];1849[label="pr2F3 (Pos vuz105 - vuz115 == fromInt (Pos Zero)) (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1849 -> 1863[label="",style="solid", color="black", weight=3]; 1859 -> 1605[label="",style="dashed", color="red", weight=0]; 1859[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos (primDivNatS vuz105 (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS vuz105 (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1859 -> 1864[label="",style="dashed", color="magenta", weight=3]; 1859 -> 1865[label="",style="dashed", color="magenta", weight=3]; 1859 -> 1866[label="",style="dashed", color="magenta", weight=3]; 4378[label="pr2F3 False vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4378 -> 4381[label="",style="solid", color="black", weight=3]; 4379[label="pr2F3 True vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4379 -> 4382[label="",style="solid", color="black", weight=3]; 1861 -> 23[label="",style="dashed", color="red", weight=0]; 1861[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];1860[label="pr2F (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="triangle"];1860 -> 1867[label="",style="solid", color="black", weight=3]; 1862[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (primQuotInt (Neg vuz113) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg vuz113) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1862 -> 1874[label="",style="solid", color="black", weight=3]; 4340[label="pr2F0G2 (vuz204 * vuz205) vuz204 (Pos vuz211)",fontsize=16,color="black",shape="box"];4340 -> 4367[label="",style="solid", color="black", weight=3]; 4341[label="vuz204",fontsize=16,color="green",shape="box"];4342[label="vuz205",fontsize=16,color="green",shape="box"];1024[label="vuz69 * vuz20",fontsize=16,color="black",shape="triangle"];1024 -> 1029[label="",style="solid", color="black", weight=3]; 4343[label="vuz204",fontsize=16,color="green",shape="box"];4344[label="vuz205",fontsize=16,color="green",shape="box"];1041[label="vuz70 * vuz20",fontsize=16,color="black",shape="triangle"];1041 -> 1046[label="",style="solid", color="black", weight=3]; 4345[label="vuz205",fontsize=16,color="green",shape="box"];4346[label="vuz204",fontsize=16,color="green",shape="box"];1051[label="vuz71 * vuz20",fontsize=16,color="black",shape="triangle"];1051 -> 1056[label="",style="solid", color="black", weight=3]; 4347[label="vuz204",fontsize=16,color="green",shape="box"];4348[label="vuz205",fontsize=16,color="green",shape="box"];1061[label="vuz72 * vuz20",fontsize=16,color="black",shape="triangle"];1061 -> 1066[label="",style="solid", color="black", weight=3]; 4349[label="vuz205",fontsize=16,color="green",shape="box"];4350[label="vuz204",fontsize=16,color="green",shape="box"];1073[label="vuz73 * vuz20",fontsize=16,color="black",shape="triangle"];1073 -> 1078[label="",style="solid", color="black", weight=3]; 739[label="vuz130000",fontsize=16,color="green",shape="box"];508[label="primDivNatS (primMinusNatS (Succ (Succ vuz1300)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];508 -> 538[label="",style="solid", color="black", weight=3]; 510[label="primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];510 -> 541[label="",style="solid", color="black", weight=3]; 1863[label="pr2F3 (primEqInt (Pos vuz105 - vuz115) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1863 -> 1875[label="",style="solid", color="black", weight=3]; 1864 -> 1226[label="",style="dashed", color="red", weight=0]; 1864[label="primDivNatS vuz105 (Succ (Succ Zero))",fontsize=16,color="magenta"];1864 -> 1876[label="",style="dashed", color="magenta", weight=3]; 1865[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4805[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4805[label="",style="solid", color="blue", weight=9]; 4805 -> 1877[label="",style="solid", color="blue", weight=3]; 4806[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4806[label="",style="solid", color="blue", weight=9]; 4806 -> 1878[label="",style="solid", color="blue", weight=3]; 4807[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4807[label="",style="solid", color="blue", weight=9]; 4807 -> 1879[label="",style="solid", color="blue", weight=3]; 4808[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4808[label="",style="solid", color="blue", weight=9]; 4808 -> 1880[label="",style="solid", color="blue", weight=3]; 4809[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4809[label="",style="solid", color="blue", weight=9]; 4809 -> 1881[label="",style="solid", color="blue", weight=3]; 1866 -> 1226[label="",style="dashed", color="red", weight=0]; 1866[label="primDivNatS vuz105 (Succ (Succ Zero))",fontsize=16,color="magenta"];1866 -> 1882[label="",style="dashed", color="magenta", weight=3]; 4381[label="pr2F0 vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4381 -> 4384[label="",style="solid", color="black", weight=3]; 4382[label="vuz216 * vuz217",fontsize=16,color="blue",shape="box"];4810[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4810[label="",style="solid", color="blue", weight=9]; 4810 -> 4385[label="",style="solid", color="blue", weight=3]; 4811[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4811[label="",style="solid", color="blue", weight=9]; 4811 -> 4386[label="",style="solid", color="blue", weight=3]; 4812[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4812[label="",style="solid", color="blue", weight=9]; 4812 -> 4387[label="",style="solid", color="blue", weight=3]; 4813[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4813[label="",style="solid", color="blue", weight=9]; 4813 -> 4388[label="",style="solid", color="blue", weight=3]; 4814[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4814[label="",style="solid", color="blue", weight=9]; 4814 -> 4389[label="",style="solid", color="blue", weight=3]; 1867[label="pr2F4 (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1867 -> 1883[label="",style="solid", color="black", weight=3]; 1874[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (primQuotInt (Neg vuz113) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg vuz113) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1874 -> 1896[label="",style="solid", color="black", weight=3]; 4367[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (even (Pos vuz211))",fontsize=16,color="black",shape="box"];4367 -> 4380[label="",style="solid", color="black", weight=3]; 1029[label="error []",fontsize=16,color="red",shape="box"];1046[label="primMulInt vuz70 vuz20",fontsize=16,color="burlywood",shape="box"];4815[label="vuz70/Pos vuz700",fontsize=10,color="white",style="solid",shape="box"];1046 -> 4815[label="",style="solid", color="burlywood", weight=9]; 4815 -> 1057[label="",style="solid", color="burlywood", weight=3]; 4816[label="vuz70/Neg vuz700",fontsize=10,color="white",style="solid",shape="box"];1046 -> 4816[label="",style="solid", color="burlywood", weight=9]; 4816 -> 1058[label="",style="solid", color="burlywood", weight=3]; 1056[label="error []",fontsize=16,color="red",shape="box"];1066[label="error []",fontsize=16,color="red",shape="box"];1078[label="error []",fontsize=16,color="red",shape="box"];538[label="primDivNatS (primMinusNatS (Succ vuz1300) Zero) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];538 -> 554[label="",style="solid", color="black", weight=3]; 541[label="primDivNatS (primMinusNatS Zero Zero) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];541 -> 557[label="",style="solid", color="black", weight=3]; 1875[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) vuz115) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4817[label="vuz115/Pos vuz1150",fontsize=10,color="white",style="solid",shape="box"];1875 -> 4817[label="",style="solid", color="burlywood", weight=9]; 4817 -> 1897[label="",style="solid", color="burlywood", weight=3]; 4818[label="vuz115/Neg vuz1150",fontsize=10,color="white",style="solid",shape="box"];1875 -> 4818[label="",style="solid", color="burlywood", weight=9]; 4818 -> 1898[label="",style="solid", color="burlywood", weight=3]; 1876[label="vuz105",fontsize=16,color="green",shape="box"];1877 -> 397[label="",style="dashed", color="red", weight=0]; 1877[label="vuz103 * vuz103",fontsize=16,color="magenta"];1877 -> 1899[label="",style="dashed", color="magenta", weight=3]; 1878 -> 398[label="",style="dashed", color="red", weight=0]; 1878[label="vuz103 * vuz103",fontsize=16,color="magenta"];1878 -> 1900[label="",style="dashed", color="magenta", weight=3]; 1879 -> 399[label="",style="dashed", color="red", weight=0]; 1879[label="vuz103 * vuz103",fontsize=16,color="magenta"];1879 -> 1901[label="",style="dashed", color="magenta", weight=3]; 1880 -> 400[label="",style="dashed", color="red", weight=0]; 1880[label="vuz103 * vuz103",fontsize=16,color="magenta"];1880 -> 1902[label="",style="dashed", color="magenta", weight=3]; 1881 -> 401[label="",style="dashed", color="red", weight=0]; 1881[label="vuz103 * vuz103",fontsize=16,color="magenta"];1881 -> 1903[label="",style="dashed", color="magenta", weight=3]; 1882[label="vuz105",fontsize=16,color="green",shape="box"];4384[label="pr2F0G (vuz216 * vuz217) vuz216 (Neg vuz218)",fontsize=16,color="black",shape="box"];4384 -> 4392[label="",style="solid", color="black", weight=3]; 4385 -> 1024[label="",style="dashed", color="red", weight=0]; 4385[label="vuz216 * vuz217",fontsize=16,color="magenta"];4385 -> 4393[label="",style="dashed", color="magenta", weight=3]; 4385 -> 4394[label="",style="dashed", color="magenta", weight=3]; 4386 -> 1041[label="",style="dashed", color="red", weight=0]; 4386[label="vuz216 * vuz217",fontsize=16,color="magenta"];4386 -> 4395[label="",style="dashed", color="magenta", weight=3]; 4386 -> 4396[label="",style="dashed", color="magenta", weight=3]; 4387 -> 1051[label="",style="dashed", color="red", weight=0]; 4387[label="vuz216 * vuz217",fontsize=16,color="magenta"];4387 -> 4397[label="",style="dashed", color="magenta", weight=3]; 4387 -> 4398[label="",style="dashed", color="magenta", weight=3]; 4388 -> 1061[label="",style="dashed", color="red", weight=0]; 4388[label="vuz216 * vuz217",fontsize=16,color="magenta"];4388 -> 4399[label="",style="dashed", color="magenta", weight=3]; 4388 -> 4400[label="",style="dashed", color="magenta", weight=3]; 4389 -> 1073[label="",style="dashed", color="red", weight=0]; 4389[label="vuz216 * vuz217",fontsize=16,color="magenta"];4389 -> 4401[label="",style="dashed", color="magenta", weight=3]; 4389 -> 4402[label="",style="dashed", color="magenta", weight=3]; 1883[label="pr2F3 (Neg vuz113 - vuz116 == fromInt (Pos Zero)) (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1883 -> 1904[label="",style="solid", color="black", weight=3]; 1896 -> 1755[label="",style="dashed", color="red", weight=0]; 1896[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg (primDivNatS vuz113 (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS vuz113 (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1896 -> 1917[label="",style="dashed", color="magenta", weight=3]; 1896 -> 1918[label="",style="dashed", color="magenta", weight=3]; 1896 -> 1919[label="",style="dashed", color="magenta", weight=3]; 4380[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (primEvenInt (Pos vuz211))",fontsize=16,color="black",shape="box"];4380 -> 4383[label="",style="solid", color="black", weight=3]; 1057[label="primMulInt (Pos vuz700) vuz20",fontsize=16,color="burlywood",shape="box"];4819[label="vuz20/Pos vuz200",fontsize=10,color="white",style="solid",shape="box"];1057 -> 4819[label="",style="solid", color="burlywood", weight=9]; 4819 -> 1067[label="",style="solid", color="burlywood", weight=3]; 4820[label="vuz20/Neg vuz200",fontsize=10,color="white",style="solid",shape="box"];1057 -> 4820[label="",style="solid", color="burlywood", weight=9]; 4820 -> 1068[label="",style="solid", color="burlywood", weight=3]; 1058[label="primMulInt (Neg vuz700) vuz20",fontsize=16,color="burlywood",shape="box"];4821[label="vuz20/Pos vuz200",fontsize=10,color="white",style="solid",shape="box"];1058 -> 4821[label="",style="solid", color="burlywood", weight=9]; 4821 -> 1069[label="",style="solid", color="burlywood", weight=3]; 4822[label="vuz20/Neg vuz200",fontsize=10,color="white",style="solid",shape="box"];1058 -> 4822[label="",style="solid", color="burlywood", weight=9]; 4822 -> 1070[label="",style="solid", color="burlywood", weight=3]; 554[label="primDivNatS (Succ vuz1300) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];554 -> 562[label="",style="solid", color="black", weight=3]; 557[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];557 -> 566[label="",style="solid", color="black", weight=3]; 1897[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) (Pos vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) (Pos vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1897 -> 1920[label="",style="solid", color="black", weight=3]; 1898[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) (Neg vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) (Neg vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1898 -> 1921[label="",style="solid", color="black", weight=3]; 1899[label="vuz103",fontsize=16,color="green",shape="box"];397 -> 1024[label="",style="dashed", color="red", weight=0]; 397[label="vuz12 * vuz12",fontsize=16,color="magenta"];397 -> 1026[label="",style="dashed", color="magenta", weight=3]; 397 -> 1027[label="",style="dashed", color="magenta", weight=3]; 1900[label="vuz103",fontsize=16,color="green",shape="box"];398 -> 1041[label="",style="dashed", color="red", weight=0]; 398[label="vuz12 * vuz12",fontsize=16,color="magenta"];398 -> 1043[label="",style="dashed", color="magenta", weight=3]; 398 -> 1044[label="",style="dashed", color="magenta", weight=3]; 1901[label="vuz103",fontsize=16,color="green",shape="box"];399 -> 1051[label="",style="dashed", color="red", weight=0]; 399[label="vuz12 * vuz12",fontsize=16,color="magenta"];399 -> 1053[label="",style="dashed", color="magenta", weight=3]; 399 -> 1054[label="",style="dashed", color="magenta", weight=3]; 1902[label="vuz103",fontsize=16,color="green",shape="box"];400 -> 1061[label="",style="dashed", color="red", weight=0]; 400[label="vuz12 * vuz12",fontsize=16,color="magenta"];400 -> 1063[label="",style="dashed", color="magenta", weight=3]; 400 -> 1064[label="",style="dashed", color="magenta", weight=3]; 1903[label="vuz103",fontsize=16,color="green",shape="box"];401 -> 1073[label="",style="dashed", color="red", weight=0]; 401[label="vuz12 * vuz12",fontsize=16,color="magenta"];401 -> 1075[label="",style="dashed", color="magenta", weight=3]; 401 -> 1076[label="",style="dashed", color="magenta", weight=3]; 4392[label="pr2F0G2 (vuz216 * vuz217) vuz216 (Neg vuz218)",fontsize=16,color="black",shape="box"];4392 -> 4406[label="",style="solid", color="black", weight=3]; 4393[label="vuz216",fontsize=16,color="green",shape="box"];4394[label="vuz217",fontsize=16,color="green",shape="box"];4395[label="vuz216",fontsize=16,color="green",shape="box"];4396[label="vuz217",fontsize=16,color="green",shape="box"];4397[label="vuz217",fontsize=16,color="green",shape="box"];4398[label="vuz216",fontsize=16,color="green",shape="box"];4399[label="vuz216",fontsize=16,color="green",shape="box"];4400[label="vuz217",fontsize=16,color="green",shape="box"];4401[label="vuz217",fontsize=16,color="green",shape="box"];4402[label="vuz216",fontsize=16,color="green",shape="box"];1904[label="pr2F3 (primEqInt (Neg vuz113 - vuz116) (fromInt (Pos Zero))) (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1904 -> 1922[label="",style="solid", color="black", weight=3]; 1917[label="vuz111 * vuz111",fontsize=16,color="blue",shape="box"];4823[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4823[label="",style="solid", color="blue", weight=9]; 4823 -> 1930[label="",style="solid", color="blue", weight=3]; 4824[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4824[label="",style="solid", color="blue", weight=9]; 4824 -> 1931[label="",style="solid", color="blue", weight=3]; 4825[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4825[label="",style="solid", color="blue", weight=9]; 4825 -> 1932[label="",style="solid", color="blue", weight=3]; 4826[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4826[label="",style="solid", color="blue", weight=9]; 4826 -> 1933[label="",style="solid", color="blue", weight=3]; 4827[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4827[label="",style="solid", color="blue", weight=9]; 4827 -> 1934[label="",style="solid", color="blue", weight=3]; 1918 -> 1226[label="",style="dashed", color="red", weight=0]; 1918[label="primDivNatS vuz113 (Succ (Succ Zero))",fontsize=16,color="magenta"];1918 -> 1935[label="",style="dashed", color="magenta", weight=3]; 1919 -> 1226[label="",style="dashed", color="red", weight=0]; 1919[label="primDivNatS vuz113 (Succ (Succ Zero))",fontsize=16,color="magenta"];1919 -> 1936[label="",style="dashed", color="magenta", weight=3]; 4383[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (primEvenNat vuz211)",fontsize=16,color="burlywood",shape="box"];4828[label="vuz211/Succ vuz2110",fontsize=10,color="white",style="solid",shape="box"];4383 -> 4828[label="",style="solid", color="burlywood", weight=9]; 4828 -> 4390[label="",style="solid", color="burlywood", weight=3]; 4829[label="vuz211/Zero",fontsize=10,color="white",style="solid",shape="box"];4383 -> 4829[label="",style="solid", color="burlywood", weight=9]; 4829 -> 4391[label="",style="solid", color="burlywood", weight=3]; 1067[label="primMulInt (Pos vuz700) (Pos vuz200)",fontsize=16,color="black",shape="box"];1067 -> 1079[label="",style="solid", color="black", weight=3]; 1068[label="primMulInt (Pos vuz700) (Neg vuz200)",fontsize=16,color="black",shape="box"];1068 -> 1080[label="",style="solid", color="black", weight=3]; 1069[label="primMulInt (Neg vuz700) (Pos vuz200)",fontsize=16,color="black",shape="box"];1069 -> 1081[label="",style="solid", color="black", weight=3]; 1070[label="primMulInt (Neg vuz700) (Neg vuz200)",fontsize=16,color="black",shape="box"];1070 -> 1082[label="",style="solid", color="black", weight=3]; 566[label="Zero",fontsize=16,color="green",shape="box"];1920[label="pr2F3 (primEqInt (primMinusNat vuz105 vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat vuz105 vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="triangle"];4830[label="vuz105/Succ vuz1050",fontsize=10,color="white",style="solid",shape="box"];1920 -> 4830[label="",style="solid", color="burlywood", weight=9]; 4830 -> 1937[label="",style="solid", color="burlywood", weight=3]; 4831[label="vuz105/Zero",fontsize=10,color="white",style="solid",shape="box"];1920 -> 4831[label="",style="solid", color="burlywood", weight=9]; 4831 -> 1938[label="",style="solid", color="burlywood", weight=3]; 1921 -> 4007[label="",style="dashed", color="red", weight=0]; 1921[label="pr2F3 (primEqInt (Pos (primPlusNat vuz105 vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos (primPlusNat vuz105 vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1921 -> 4022[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4023[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4024[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4025[label="",style="dashed", color="magenta", weight=3]; 1026[label="vuz12",fontsize=16,color="green",shape="box"];1027[label="vuz12",fontsize=16,color="green",shape="box"];1043[label="vuz12",fontsize=16,color="green",shape="box"];1044[label="vuz12",fontsize=16,color="green",shape="box"];1053[label="vuz12",fontsize=16,color="green",shape="box"];1054[label="vuz12",fontsize=16,color="green",shape="box"];1063[label="vuz12",fontsize=16,color="green",shape="box"];1064[label="vuz12",fontsize=16,color="green",shape="box"];1075[label="vuz12",fontsize=16,color="green",shape="box"];1076[label="vuz12",fontsize=16,color="green",shape="box"];4406[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (even (Neg vuz218))",fontsize=16,color="black",shape="box"];4406 -> 4410[label="",style="solid", color="black", weight=3]; 1922[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) vuz116) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="burlywood",shape="box"];4832[label="vuz116/Pos vuz1160",fontsize=10,color="white",style="solid",shape="box"];1922 -> 4832[label="",style="solid", color="burlywood", weight=9]; 4832 -> 1942[label="",style="solid", color="burlywood", weight=3]; 4833[label="vuz116/Neg vuz1160",fontsize=10,color="white",style="solid",shape="box"];1922 -> 4833[label="",style="solid", color="burlywood", weight=9]; 4833 -> 1943[label="",style="solid", color="burlywood", weight=3]; 1930 -> 397[label="",style="dashed", color="red", weight=0]; 1930[label="vuz111 * vuz111",fontsize=16,color="magenta"];1930 -> 1944[label="",style="dashed", color="magenta", weight=3]; 1931 -> 398[label="",style="dashed", color="red", weight=0]; 1931[label="vuz111 * vuz111",fontsize=16,color="magenta"];1931 -> 1945[label="",style="dashed", color="magenta", weight=3]; 1932 -> 399[label="",style="dashed", color="red", weight=0]; 1932[label="vuz111 * vuz111",fontsize=16,color="magenta"];1932 -> 1946[label="",style="dashed", color="magenta", weight=3]; 1933 -> 400[label="",style="dashed", color="red", weight=0]; 1933[label="vuz111 * vuz111",fontsize=16,color="magenta"];1933 -> 1947[label="",style="dashed", color="magenta", weight=3]; 1934 -> 401[label="",style="dashed", color="red", weight=0]; 1934[label="vuz111 * vuz111",fontsize=16,color="magenta"];1934 -> 1948[label="",style="dashed", color="magenta", weight=3]; 1935[label="vuz113",fontsize=16,color="green",shape="box"];1936[label="vuz113",fontsize=16,color="green",shape="box"];4390[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ vuz2110)) (primEvenNat (Succ vuz2110))",fontsize=16,color="burlywood",shape="box"];4834[label="vuz2110/Succ vuz21100",fontsize=10,color="white",style="solid",shape="box"];4390 -> 4834[label="",style="solid", color="burlywood", weight=9]; 4834 -> 4403[label="",style="solid", color="burlywood", weight=3]; 4835[label="vuz2110/Zero",fontsize=10,color="white",style="solid",shape="box"];4390 -> 4835[label="",style="solid", color="burlywood", weight=9]; 4835 -> 4404[label="",style="solid", color="burlywood", weight=3]; 4391[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos Zero) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4391 -> 4405[label="",style="solid", color="black", weight=3]; 1079[label="Pos (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1079 -> 1085[label="",style="dashed", color="green", weight=3]; 1080[label="Neg (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1080 -> 1086[label="",style="dashed", color="green", weight=3]; 1081[label="Neg (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1081 -> 1087[label="",style="dashed", color="green", weight=3]; 1082[label="Pos (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1082 -> 1088[label="",style="dashed", color="green", weight=3]; 1937[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4836[label="vuz1150/Succ vuz11500",fontsize=10,color="white",style="solid",shape="box"];1937 -> 4836[label="",style="solid", color="burlywood", weight=9]; 4836 -> 1949[label="",style="solid", color="burlywood", weight=3]; 4837[label="vuz1150/Zero",fontsize=10,color="white",style="solid",shape="box"];1937 -> 4837[label="",style="solid", color="burlywood", weight=9]; 4837 -> 1950[label="",style="solid", color="burlywood", weight=3]; 1938[label="pr2F3 (primEqInt (primMinusNat Zero vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4838[label="vuz1150/Succ vuz11500",fontsize=10,color="white",style="solid",shape="box"];1938 -> 4838[label="",style="solid", color="burlywood", weight=9]; 4838 -> 1951[label="",style="solid", color="burlywood", weight=3]; 4839[label="vuz1150/Zero",fontsize=10,color="white",style="solid",shape="box"];1938 -> 4839[label="",style="solid", color="burlywood", weight=9]; 4839 -> 1952[label="",style="solid", color="burlywood", weight=3]; 4022[label="vuz102",fontsize=16,color="green",shape="box"];4023 -> 71[label="",style="dashed", color="red", weight=0]; 4023[label="primPlusNat vuz105 vuz1150",fontsize=16,color="magenta"];4023 -> 4094[label="",style="dashed", color="magenta", weight=3]; 4023 -> 4095[label="",style="dashed", color="magenta", weight=3]; 4024[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4840[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4840[label="",style="solid", color="blue", weight=9]; 4840 -> 4096[label="",style="solid", color="blue", weight=3]; 4841[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4841[label="",style="solid", color="blue", weight=9]; 4841 -> 4097[label="",style="solid", color="blue", weight=3]; 4842[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4842[label="",style="solid", color="blue", weight=9]; 4842 -> 4098[label="",style="solid", color="blue", weight=3]; 4843[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4843[label="",style="solid", color="blue", weight=9]; 4843 -> 4099[label="",style="solid", color="blue", weight=3]; 4844[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4844[label="",style="solid", color="blue", weight=9]; 4844 -> 4100[label="",style="solid", color="blue", weight=3]; 4025 -> 71[label="",style="dashed", color="red", weight=0]; 4025[label="primPlusNat vuz105 vuz1150",fontsize=16,color="magenta"];4025 -> 4101[label="",style="dashed", color="magenta", weight=3]; 4025 -> 4102[label="",style="dashed", color="magenta", weight=3]; 4410[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (primEvenInt (Neg vuz218))",fontsize=16,color="black",shape="box"];4410 -> 4415[label="",style="solid", color="black", weight=3]; 1942[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) (Pos vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) (Pos vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1942 -> 1969[label="",style="solid", color="black", weight=3]; 1943[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) (Neg vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) (Neg vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1943 -> 1970[label="",style="solid", color="black", weight=3]; 1944[label="vuz111",fontsize=16,color="green",shape="box"];1945[label="vuz111",fontsize=16,color="green",shape="box"];1946[label="vuz111",fontsize=16,color="green",shape="box"];1947[label="vuz111",fontsize=16,color="green",shape="box"];1948[label="vuz111",fontsize=16,color="green",shape="box"];4403[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ (Succ vuz21100))) (primEvenNat (Succ (Succ vuz21100)))",fontsize=16,color="black",shape="box"];4403 -> 4407[label="",style="solid", color="black", weight=3]; 4404[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4404 -> 4408[label="",style="solid", color="black", weight=3]; 4405[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos Zero) True",fontsize=16,color="black",shape="box"];4405 -> 4409[label="",style="solid", color="black", weight=3]; 1085[label="primMulNat vuz700 vuz200",fontsize=16,color="burlywood",shape="triangle"];4845[label="vuz700/Succ vuz7000",fontsize=10,color="white",style="solid",shape="box"];1085 -> 4845[label="",style="solid", color="burlywood", weight=9]; 4845 -> 1111[label="",style="solid", color="burlywood", weight=3]; 4846[label="vuz700/Zero",fontsize=10,color="white",style="solid",shape="box"];1085 -> 4846[label="",style="solid", color="burlywood", weight=9]; 4846 -> 1112[label="",style="solid", color="burlywood", weight=3]; 1086 -> 1085[label="",style="dashed", color="red", weight=0]; 1086[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1086 -> 1113[label="",style="dashed", color="magenta", weight=3]; 1087 -> 1085[label="",style="dashed", color="red", weight=0]; 1087[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1087 -> 1114[label="",style="dashed", color="magenta", weight=3]; 1088 -> 1085[label="",style="dashed", color="red", weight=0]; 1088[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1088 -> 1115[label="",style="dashed", color="magenta", weight=3]; 1088 -> 1116[label="",style="dashed", color="magenta", weight=3]; 1949[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1949 -> 1971[label="",style="solid", color="black", weight=3]; 1950[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1950 -> 1972[label="",style="solid", color="black", weight=3]; 1951[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1951 -> 1973[label="",style="solid", color="black", weight=3]; 1952[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1952 -> 1974[label="",style="solid", color="black", weight=3]; 4094[label="vuz105",fontsize=16,color="green",shape="box"];4095[label="vuz1150",fontsize=16,color="green",shape="box"];4096 -> 397[label="",style="dashed", color="red", weight=0]; 4096[label="vuz103 * vuz103",fontsize=16,color="magenta"];4096 -> 4171[label="",style="dashed", color="magenta", weight=3]; 4097 -> 398[label="",style="dashed", color="red", weight=0]; 4097[label="vuz103 * vuz103",fontsize=16,color="magenta"];4097 -> 4172[label="",style="dashed", color="magenta", weight=3]; 4098 -> 399[label="",style="dashed", color="red", weight=0]; 4098[label="vuz103 * vuz103",fontsize=16,color="magenta"];4098 -> 4173[label="",style="dashed", color="magenta", weight=3]; 4099 -> 400[label="",style="dashed", color="red", weight=0]; 4099[label="vuz103 * vuz103",fontsize=16,color="magenta"];4099 -> 4174[label="",style="dashed", color="magenta", weight=3]; 4100 -> 401[label="",style="dashed", color="red", weight=0]; 4100[label="vuz103 * vuz103",fontsize=16,color="magenta"];4100 -> 4175[label="",style="dashed", color="magenta", weight=3]; 4101[label="vuz105",fontsize=16,color="green",shape="box"];4102[label="vuz1150",fontsize=16,color="green",shape="box"];4415[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (primEvenNat vuz218)",fontsize=16,color="burlywood",shape="box"];4847[label="vuz218/Succ vuz2180",fontsize=10,color="white",style="solid",shape="box"];4415 -> 4847[label="",style="solid", color="burlywood", weight=9]; 4847 -> 4421[label="",style="solid", color="burlywood", weight=3]; 4848[label="vuz218/Zero",fontsize=10,color="white",style="solid",shape="box"];4415 -> 4848[label="",style="solid", color="burlywood", weight=9]; 4848 -> 4422[label="",style="solid", color="burlywood", weight=3]; 1969 -> 4311[label="",style="dashed", color="red", weight=0]; 1969[label="pr2F3 (primEqInt (Neg (primPlusNat vuz113 vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (Neg (primPlusNat vuz113 vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1969 -> 4318[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4319[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4320[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4321[label="",style="dashed", color="magenta", weight=3]; 1970 -> 1920[label="",style="dashed", color="red", weight=0]; 1970[label="pr2F3 (primEqInt (primMinusNat vuz1160 vuz113) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusNat vuz1160 vuz113) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1970 -> 2001[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2002[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2003[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2004[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4464[label="",style="dashed", color="red", weight=0]; 4407[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ (Succ vuz21100))) (primEvenNat vuz21100)",fontsize=16,color="magenta"];4407 -> 4465[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4466[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4467[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4468[label="",style="dashed", color="magenta", weight=3]; 4408[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) False",fontsize=16,color="black",shape="box"];4408 -> 4413[label="",style="solid", color="black", weight=3]; 4409[label="pr2F0G (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4409 -> 4414[label="",style="solid", color="black", weight=3]; 1111[label="primMulNat (Succ vuz7000) vuz200",fontsize=16,color="burlywood",shape="box"];4849[label="vuz200/Succ vuz2000",fontsize=10,color="white",style="solid",shape="box"];1111 -> 4849[label="",style="solid", color="burlywood", weight=9]; 4849 -> 1179[label="",style="solid", color="burlywood", weight=3]; 4850[label="vuz200/Zero",fontsize=10,color="white",style="solid",shape="box"];1111 -> 4850[label="",style="solid", color="burlywood", weight=9]; 4850 -> 1180[label="",style="solid", color="burlywood", weight=3]; 1112[label="primMulNat Zero vuz200",fontsize=16,color="burlywood",shape="box"];4851[label="vuz200/Succ vuz2000",fontsize=10,color="white",style="solid",shape="box"];1112 -> 4851[label="",style="solid", color="burlywood", weight=9]; 4851 -> 1181[label="",style="solid", color="burlywood", weight=3]; 4852[label="vuz200/Zero",fontsize=10,color="white",style="solid",shape="box"];1112 -> 4852[label="",style="solid", color="burlywood", weight=9]; 4852 -> 1182[label="",style="solid", color="burlywood", weight=3]; 1113[label="vuz200",fontsize=16,color="green",shape="box"];1114[label="vuz700",fontsize=16,color="green",shape="box"];1115[label="vuz200",fontsize=16,color="green",shape="box"];1116[label="vuz700",fontsize=16,color="green",shape="box"];1971 -> 1920[label="",style="dashed", color="red", weight=0]; 1971[label="pr2F3 (primEqInt (primMinusNat vuz1050 vuz11500) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat vuz1050 vuz11500) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1971 -> 2005[label="",style="dashed", color="magenta", weight=3]; 1971 -> 2006[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4007[label="",style="dashed", color="red", weight=0]; 1972[label="pr2F3 (primEqInt (Pos (Succ vuz1050)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos (Succ vuz1050)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1972 -> 4030[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4031[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4032[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4033[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4311[label="",style="dashed", color="red", weight=0]; 1973[label="pr2F3 (primEqInt (Neg (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Neg (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1973 -> 4322[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4323[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4324[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4325[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4007[label="",style="dashed", color="red", weight=0]; 1974[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1974 -> 4034[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4035[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4036[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4037[label="",style="dashed", color="magenta", weight=3]; 4171[label="vuz103",fontsize=16,color="green",shape="box"];4172[label="vuz103",fontsize=16,color="green",shape="box"];4173[label="vuz103",fontsize=16,color="green",shape="box"];4174[label="vuz103",fontsize=16,color="green",shape="box"];4175[label="vuz103",fontsize=16,color="green",shape="box"];4421[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ vuz2180)) (primEvenNat (Succ vuz2180))",fontsize=16,color="burlywood",shape="box"];4853[label="vuz2180/Succ vuz21800",fontsize=10,color="white",style="solid",shape="box"];4421 -> 4853[label="",style="solid", color="burlywood", weight=9]; 4853 -> 4428[label="",style="solid", color="burlywood", weight=3]; 4854[label="vuz2180/Zero",fontsize=10,color="white",style="solid",shape="box"];4421 -> 4854[label="",style="solid", color="burlywood", weight=9]; 4854 -> 4429[label="",style="solid", color="burlywood", weight=3]; 4422[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg Zero) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4422 -> 4430[label="",style="solid", color="black", weight=3]; 4318 -> 71[label="",style="dashed", color="red", weight=0]; 4318[label="primPlusNat vuz113 vuz1160",fontsize=16,color="magenta"];4318 -> 4351[label="",style="dashed", color="magenta", weight=3]; 4318 -> 4352[label="",style="dashed", color="magenta", weight=3]; 4319[label="vuz111 * vuz111",fontsize=16,color="blue",shape="box"];4855[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4855[label="",style="solid", color="blue", weight=9]; 4855 -> 4353[label="",style="solid", color="blue", weight=3]; 4856[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4856[label="",style="solid", color="blue", weight=9]; 4856 -> 4354[label="",style="solid", color="blue", weight=3]; 4857[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4857[label="",style="solid", color="blue", weight=9]; 4857 -> 4355[label="",style="solid", color="blue", weight=3]; 4858[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4858[label="",style="solid", color="blue", weight=9]; 4858 -> 4356[label="",style="solid", color="blue", weight=3]; 4859[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4859[label="",style="solid", color="blue", weight=9]; 4859 -> 4357[label="",style="solid", color="blue", weight=3]; 4320 -> 71[label="",style="dashed", color="red", weight=0]; 4320[label="primPlusNat vuz113 vuz1160",fontsize=16,color="magenta"];4320 -> 4358[label="",style="dashed", color="magenta", weight=3]; 4320 -> 4359[label="",style="dashed", color="magenta", weight=3]; 4321[label="vuz110",fontsize=16,color="green",shape="box"];2001[label="vuz1160",fontsize=16,color="green",shape="box"];2002[label="vuz113",fontsize=16,color="green",shape="box"];2003[label="vuz111",fontsize=16,color="green",shape="box"];2004[label="vuz110",fontsize=16,color="green",shape="box"];4465[label="vuz205",fontsize=16,color="green",shape="box"];4466[label="vuz21100",fontsize=16,color="green",shape="box"];4467[label="vuz204",fontsize=16,color="green",shape="box"];4468[label="Succ vuz21100",fontsize=16,color="green",shape="box"];4464[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat vuz225)",fontsize=16,color="burlywood",shape="triangle"];4860[label="vuz225/Succ vuz2250",fontsize=10,color="white",style="solid",shape="box"];4464 -> 4860[label="",style="solid", color="burlywood", weight=9]; 4860 -> 4477[label="",style="solid", color="burlywood", weight=3]; 4861[label="vuz225/Zero",fontsize=10,color="white",style="solid",shape="box"];4464 -> 4861[label="",style="solid", color="burlywood", weight=9]; 4861 -> 4478[label="",style="solid", color="burlywood", weight=3]; 4413[label="pr2F0G0 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) otherwise",fontsize=16,color="black",shape="box"];4413 -> 4419[label="",style="solid", color="black", weight=3]; 4414[label="pr2F0G2 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4414 -> 4420[label="",style="solid", color="black", weight=3]; 1179[label="primMulNat (Succ vuz7000) (Succ vuz2000)",fontsize=16,color="black",shape="box"];1179 -> 1232[label="",style="solid", color="black", weight=3]; 1180[label="primMulNat (Succ vuz7000) Zero",fontsize=16,color="black",shape="box"];1180 -> 1233[label="",style="solid", color="black", weight=3]; 1181[label="primMulNat Zero (Succ vuz2000)",fontsize=16,color="black",shape="box"];1181 -> 1234[label="",style="solid", color="black", weight=3]; 1182[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];1182 -> 1235[label="",style="solid", color="black", weight=3]; 2005[label="vuz1050",fontsize=16,color="green",shape="box"];2006[label="vuz11500",fontsize=16,color="green",shape="box"];4030[label="vuz102",fontsize=16,color="green",shape="box"];4031[label="Succ vuz1050",fontsize=16,color="green",shape="box"];4032[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4862[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4862[label="",style="solid", color="blue", weight=9]; 4862 -> 4103[label="",style="solid", color="blue", weight=3]; 4863[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4863[label="",style="solid", color="blue", weight=9]; 4863 -> 4104[label="",style="solid", color="blue", weight=3]; 4864[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4864[label="",style="solid", color="blue", weight=9]; 4864 -> 4105[label="",style="solid", color="blue", weight=3]; 4865[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4865[label="",style="solid", color="blue", weight=9]; 4865 -> 4106[label="",style="solid", color="blue", weight=3]; 4866[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4866[label="",style="solid", color="blue", weight=9]; 4866 -> 4107[label="",style="solid", color="blue", weight=3]; 4033[label="Succ vuz1050",fontsize=16,color="green",shape="box"];4322[label="Succ vuz11500",fontsize=16,color="green",shape="box"];4323[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4867[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4867[label="",style="solid", color="blue", weight=9]; 4867 -> 4360[label="",style="solid", color="blue", weight=3]; 4868[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4868[label="",style="solid", color="blue", weight=9]; 4868 -> 4361[label="",style="solid", color="blue", weight=3]; 4869[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4869[label="",style="solid", color="blue", weight=9]; 4869 -> 4362[label="",style="solid", color="blue", weight=3]; 4870[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4870[label="",style="solid", color="blue", weight=9]; 4870 -> 4363[label="",style="solid", color="blue", weight=3]; 4871[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4871[label="",style="solid", color="blue", weight=9]; 4871 -> 4364[label="",style="solid", color="blue", weight=3]; 4324[label="Succ vuz11500",fontsize=16,color="green",shape="box"];4325[label="vuz102",fontsize=16,color="green",shape="box"];4034[label="vuz102",fontsize=16,color="green",shape="box"];4035[label="Zero",fontsize=16,color="green",shape="box"];4036[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4872[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4872[label="",style="solid", color="blue", weight=9]; 4872 -> 4108[label="",style="solid", color="blue", weight=3]; 4873[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4873[label="",style="solid", color="blue", weight=9]; 4873 -> 4109[label="",style="solid", color="blue", weight=3]; 4874[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4874[label="",style="solid", color="blue", weight=9]; 4874 -> 4110[label="",style="solid", color="blue", weight=3]; 4875[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4875[label="",style="solid", color="blue", weight=9]; 4875 -> 4111[label="",style="solid", color="blue", weight=3]; 4876[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4876[label="",style="solid", color="blue", weight=9]; 4876 -> 4112[label="",style="solid", color="blue", weight=3]; 4037[label="Zero",fontsize=16,color="green",shape="box"];4428[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ (Succ vuz21800))) (primEvenNat (Succ (Succ vuz21800)))",fontsize=16,color="black",shape="box"];4428 -> 4437[label="",style="solid", color="black", weight=3]; 4429[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4429 -> 4438[label="",style="solid", color="black", weight=3]; 4430[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg Zero) True",fontsize=16,color="black",shape="box"];4430 -> 4439[label="",style="solid", color="black", weight=3]; 4351[label="vuz113",fontsize=16,color="green",shape="box"];4352[label="vuz1160",fontsize=16,color="green",shape="box"];4353 -> 397[label="",style="dashed", color="red", weight=0]; 4353[label="vuz111 * vuz111",fontsize=16,color="magenta"];4353 -> 4368[label="",style="dashed", color="magenta", weight=3]; 4354 -> 398[label="",style="dashed", color="red", weight=0]; 4354[label="vuz111 * vuz111",fontsize=16,color="magenta"];4354 -> 4369[label="",style="dashed", color="magenta", weight=3]; 4355 -> 399[label="",style="dashed", color="red", weight=0]; 4355[label="vuz111 * vuz111",fontsize=16,color="magenta"];4355 -> 4370[label="",style="dashed", color="magenta", weight=3]; 4356 -> 400[label="",style="dashed", color="red", weight=0]; 4356[label="vuz111 * vuz111",fontsize=16,color="magenta"];4356 -> 4371[label="",style="dashed", color="magenta", weight=3]; 4357 -> 401[label="",style="dashed", color="red", weight=0]; 4357[label="vuz111 * vuz111",fontsize=16,color="magenta"];4357 -> 4372[label="",style="dashed", color="magenta", weight=3]; 4358[label="vuz113",fontsize=16,color="green",shape="box"];4359[label="vuz1160",fontsize=16,color="green",shape="box"];4477[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ vuz2250))",fontsize=16,color="burlywood",shape="box"];4877[label="vuz2250/Succ vuz22500",fontsize=10,color="white",style="solid",shape="box"];4477 -> 4877[label="",style="solid", color="burlywood", weight=9]; 4877 -> 4489[label="",style="solid", color="burlywood", weight=3]; 4878[label="vuz2250/Zero",fontsize=10,color="white",style="solid",shape="box"];4477 -> 4878[label="",style="solid", color="burlywood", weight=9]; 4878 -> 4490[label="",style="solid", color="burlywood", weight=3]; 4478[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4478 -> 4491[label="",style="solid", color="black", weight=3]; 4419[label="pr2F0G0 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) True",fontsize=16,color="black",shape="box"];4419 -> 4426[label="",style="solid", color="black", weight=3]; 4420[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4420 -> 4427[label="",style="solid", color="black", weight=3]; 1232 -> 71[label="",style="dashed", color="red", weight=0]; 1232[label="primPlusNat (primMulNat vuz7000 (Succ vuz2000)) (Succ vuz2000)",fontsize=16,color="magenta"];1232 -> 1267[label="",style="dashed", color="magenta", weight=3]; 1232 -> 1268[label="",style="dashed", color="magenta", weight=3]; 1233[label="Zero",fontsize=16,color="green",shape="box"];1234[label="Zero",fontsize=16,color="green",shape="box"];1235[label="Zero",fontsize=16,color="green",shape="box"];4103 -> 397[label="",style="dashed", color="red", weight=0]; 4103[label="vuz103 * vuz103",fontsize=16,color="magenta"];4103 -> 4176[label="",style="dashed", color="magenta", weight=3]; 4104 -> 398[label="",style="dashed", color="red", weight=0]; 4104[label="vuz103 * vuz103",fontsize=16,color="magenta"];4104 -> 4177[label="",style="dashed", color="magenta", weight=3]; 4105 -> 399[label="",style="dashed", color="red", weight=0]; 4105[label="vuz103 * vuz103",fontsize=16,color="magenta"];4105 -> 4178[label="",style="dashed", color="magenta", weight=3]; 4106 -> 400[label="",style="dashed", color="red", weight=0]; 4106[label="vuz103 * vuz103",fontsize=16,color="magenta"];4106 -> 4179[label="",style="dashed", color="magenta", weight=3]; 4107 -> 401[label="",style="dashed", color="red", weight=0]; 4107[label="vuz103 * vuz103",fontsize=16,color="magenta"];4107 -> 4180[label="",style="dashed", color="magenta", weight=3]; 4360 -> 397[label="",style="dashed", color="red", weight=0]; 4360[label="vuz103 * vuz103",fontsize=16,color="magenta"];4360 -> 4373[label="",style="dashed", color="magenta", weight=3]; 4361 -> 398[label="",style="dashed", color="red", weight=0]; 4361[label="vuz103 * vuz103",fontsize=16,color="magenta"];4361 -> 4374[label="",style="dashed", color="magenta", weight=3]; 4362 -> 399[label="",style="dashed", color="red", weight=0]; 4362[label="vuz103 * vuz103",fontsize=16,color="magenta"];4362 -> 4375[label="",style="dashed", color="magenta", weight=3]; 4363 -> 400[label="",style="dashed", color="red", weight=0]; 4363[label="vuz103 * vuz103",fontsize=16,color="magenta"];4363 -> 4376[label="",style="dashed", color="magenta", weight=3]; 4364 -> 401[label="",style="dashed", color="red", weight=0]; 4364[label="vuz103 * vuz103",fontsize=16,color="magenta"];4364 -> 4377[label="",style="dashed", color="magenta", weight=3]; 4108 -> 397[label="",style="dashed", color="red", weight=0]; 4108[label="vuz103 * vuz103",fontsize=16,color="magenta"];4108 -> 4181[label="",style="dashed", color="magenta", weight=3]; 4109 -> 398[label="",style="dashed", color="red", weight=0]; 4109[label="vuz103 * vuz103",fontsize=16,color="magenta"];4109 -> 4182[label="",style="dashed", color="magenta", weight=3]; 4110 -> 399[label="",style="dashed", color="red", weight=0]; 4110[label="vuz103 * vuz103",fontsize=16,color="magenta"];4110 -> 4183[label="",style="dashed", color="magenta", weight=3]; 4111 -> 400[label="",style="dashed", color="red", weight=0]; 4111[label="vuz103 * vuz103",fontsize=16,color="magenta"];4111 -> 4184[label="",style="dashed", color="magenta", weight=3]; 4112 -> 401[label="",style="dashed", color="red", weight=0]; 4112[label="vuz103 * vuz103",fontsize=16,color="magenta"];4112 -> 4185[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4546[label="",style="dashed", color="red", weight=0]; 4437[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ (Succ vuz21800))) (primEvenNat vuz21800)",fontsize=16,color="magenta"];4437 -> 4547[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4548[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4549[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4550[label="",style="dashed", color="magenta", weight=3]; 4438[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) False",fontsize=16,color="black",shape="box"];4438 -> 4449[label="",style="solid", color="black", weight=3]; 4439[label="pr2F0G (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4439 -> 4450[label="",style="solid", color="black", weight=3]; 4368[label="vuz111",fontsize=16,color="green",shape="box"];4369[label="vuz111",fontsize=16,color="green",shape="box"];4370[label="vuz111",fontsize=16,color="green",shape="box"];4371[label="vuz111",fontsize=16,color="green",shape="box"];4372[label="vuz111",fontsize=16,color="green",shape="box"];4489[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ (Succ vuz22500)))",fontsize=16,color="black",shape="box"];4489 -> 4498[label="",style="solid", color="black", weight=3]; 4490[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4490 -> 4499[label="",style="solid", color="black", weight=3]; 4491[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) True",fontsize=16,color="black",shape="box"];4491 -> 4500[label="",style="solid", color="black", weight=3]; 4426 -> 4581[label="",style="dashed", color="red", weight=0]; 4426[label="pr2F vuz204 (Pos (Succ Zero) - fromInt (Pos (Succ Zero))) (vuz204 * (vuz204 * vuz205))",fontsize=16,color="magenta"];4426 -> 4582[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4583[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4584[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4585[label="",style="dashed", color="magenta", weight=3]; 4427[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4427 -> 4440[label="",style="solid", color="black", weight=3]; 1267 -> 1085[label="",style="dashed", color="red", weight=0]; 1267[label="primMulNat vuz7000 (Succ vuz2000)",fontsize=16,color="magenta"];1267 -> 1275[label="",style="dashed", color="magenta", weight=3]; 1267 -> 1276[label="",style="dashed", color="magenta", weight=3]; 1268[label="Succ vuz2000",fontsize=16,color="green",shape="box"];4176[label="vuz103",fontsize=16,color="green",shape="box"];4177[label="vuz103",fontsize=16,color="green",shape="box"];4178[label="vuz103",fontsize=16,color="green",shape="box"];4179[label="vuz103",fontsize=16,color="green",shape="box"];4180[label="vuz103",fontsize=16,color="green",shape="box"];4373[label="vuz103",fontsize=16,color="green",shape="box"];4374[label="vuz103",fontsize=16,color="green",shape="box"];4375[label="vuz103",fontsize=16,color="green",shape="box"];4376[label="vuz103",fontsize=16,color="green",shape="box"];4377[label="vuz103",fontsize=16,color="green",shape="box"];4181[label="vuz103",fontsize=16,color="green",shape="box"];4182[label="vuz103",fontsize=16,color="green",shape="box"];4183[label="vuz103",fontsize=16,color="green",shape="box"];4184[label="vuz103",fontsize=16,color="green",shape="box"];4185[label="vuz103",fontsize=16,color="green",shape="box"];4547[label="vuz216",fontsize=16,color="green",shape="box"];4548[label="vuz21800",fontsize=16,color="green",shape="box"];4549[label="Succ vuz21800",fontsize=16,color="green",shape="box"];4550[label="vuz217",fontsize=16,color="green",shape="box"];4546[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat vuz231)",fontsize=16,color="burlywood",shape="triangle"];4879[label="vuz231/Succ vuz2310",fontsize=10,color="white",style="solid",shape="box"];4546 -> 4879[label="",style="solid", color="burlywood", weight=9]; 4879 -> 4559[label="",style="solid", color="burlywood", weight=3]; 4880[label="vuz231/Zero",fontsize=10,color="white",style="solid",shape="box"];4546 -> 4880[label="",style="solid", color="burlywood", weight=9]; 4880 -> 4560[label="",style="solid", color="burlywood", weight=3]; 4449[label="pr2F0G0 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) otherwise",fontsize=16,color="black",shape="box"];4449 -> 4461[label="",style="solid", color="black", weight=3]; 4450[label="pr2F0G2 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4450 -> 4462[label="",style="solid", color="black", weight=3]; 4498 -> 4464[label="",style="dashed", color="red", weight=0]; 4498[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat vuz22500)",fontsize=16,color="magenta"];4498 -> 4518[label="",style="dashed", color="magenta", weight=3]; 4499[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) False",fontsize=16,color="black",shape="box"];4499 -> 4519[label="",style="solid", color="black", weight=3]; 4500[label="pr2F0G (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4500 -> 4520[label="",style="solid", color="black", weight=3]; 4582[label="vuz205",fontsize=16,color="green",shape="box"];4583 -> 23[label="",style="dashed", color="red", weight=0]; 4583[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4584[label="vuz204",fontsize=16,color="green",shape="box"];4585[label="Zero",fontsize=16,color="green",shape="box"];4581[label="pr2F vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="triangle"];4581 -> 4591[label="",style="solid", color="black", weight=3]; 4440[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (primQuotInt (Pos Zero) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos Zero) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4440 -> 4451[label="",style="solid", color="black", weight=3]; 1275[label="Succ vuz2000",fontsize=16,color="green",shape="box"];1276[label="vuz7000",fontsize=16,color="green",shape="box"];4559[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ vuz2310))",fontsize=16,color="burlywood",shape="box"];4881[label="vuz2310/Succ vuz23100",fontsize=10,color="white",style="solid",shape="box"];4559 -> 4881[label="",style="solid", color="burlywood", weight=9]; 4881 -> 4578[label="",style="solid", color="burlywood", weight=3]; 4882[label="vuz2310/Zero",fontsize=10,color="white",style="solid",shape="box"];4559 -> 4882[label="",style="solid", color="burlywood", weight=9]; 4882 -> 4579[label="",style="solid", color="burlywood", weight=3]; 4560[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4560 -> 4580[label="",style="solid", color="black", weight=3]; 4461[label="pr2F0G0 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) True",fontsize=16,color="black",shape="box"];4461 -> 4482[label="",style="solid", color="black", weight=3]; 4462[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4462 -> 4483[label="",style="solid", color="black", weight=3]; 4518[label="vuz22500",fontsize=16,color="green",shape="box"];4519[label="pr2F0G0 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) otherwise",fontsize=16,color="black",shape="box"];4519 -> 4543[label="",style="solid", color="black", weight=3]; 4520[label="pr2F0G2 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4520 -> 4544[label="",style="solid", color="black", weight=3]; 4591[label="pr2F4 vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="box"];4591 -> 4606[label="",style="solid", color="black", weight=3]; 4451[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (primQuotInt (Pos Zero) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos Zero) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4451 -> 4463[label="",style="solid", color="black", weight=3]; 4578[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ (Succ vuz23100)))",fontsize=16,color="black",shape="box"];4578 -> 4592[label="",style="solid", color="black", weight=3]; 4579[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4579 -> 4593[label="",style="solid", color="black", weight=3]; 4580[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) True",fontsize=16,color="black",shape="box"];4580 -> 4594[label="",style="solid", color="black", weight=3]; 4482 -> 4655[label="",style="dashed", color="red", weight=0]; 4482[label="pr2F vuz216 (Neg (Succ Zero) - fromInt (Pos (Succ Zero))) (vuz216 * (vuz216 * vuz217))",fontsize=16,color="magenta"];4482 -> 4656[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4657[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4658[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4659[label="",style="dashed", color="magenta", weight=3]; 4483[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4483 -> 4501[label="",style="solid", color="black", weight=3]; 4543[label="pr2F0G0 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) True",fontsize=16,color="black",shape="box"];4543 -> 4561[label="",style="solid", color="black", weight=3]; 4544[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4544 -> 4562[label="",style="solid", color="black", weight=3]; 4606[label="pr2F3 (Pos (Succ vuz224) - vuz232 == fromInt (Pos Zero)) vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="box"];4606 -> 4626[label="",style="solid", color="black", weight=3]; 4463 -> 1605[label="",style="dashed", color="red", weight=0]; 4463[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos (primDivNatS Zero (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS Zero (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4463 -> 4485[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4486[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4487[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4488[label="",style="dashed", color="magenta", weight=3]; 4592 -> 4546[label="",style="dashed", color="red", weight=0]; 4592[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat vuz23100)",fontsize=16,color="magenta"];4592 -> 4607[label="",style="dashed", color="magenta", weight=3]; 4593[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) False",fontsize=16,color="black",shape="box"];4593 -> 4608[label="",style="solid", color="black", weight=3]; 4594[label="pr2F0G (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4594 -> 4609[label="",style="solid", color="black", weight=3]; 4656[label="vuz216",fontsize=16,color="green",shape="box"];4657[label="Zero",fontsize=16,color="green",shape="box"];4658 -> 23[label="",style="dashed", color="red", weight=0]; 4658[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4659[label="vuz217",fontsize=16,color="green",shape="box"];4655[label="pr2F vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="triangle"];4655 -> 4665[label="",style="solid", color="black", weight=3]; 4501[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (primQuotInt (Neg Zero) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg Zero) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4501 -> 4521[label="",style="solid", color="black", weight=3]; 4561 -> 4581[label="",style="dashed", color="red", weight=0]; 4561[label="pr2F vuz222 (Pos (Succ vuz224) - fromInt (Pos (Succ Zero))) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="magenta"];4561 -> 4590[label="",style="dashed", color="magenta", weight=3]; 4562[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4562 -> 4595[label="",style="solid", color="black", weight=3]; 4626 -> 3789[label="",style="dashed", color="red", weight=0]; 4626[label="pr2F3 (primEqInt (Pos (Succ vuz224) - vuz232) (fromInt (Pos Zero))) vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="magenta"];4626 -> 4640[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4641[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4642[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4643[label="",style="dashed", color="magenta", weight=3]; 4485 -> 1226[label="",style="dashed", color="red", weight=0]; 4485[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4485 -> 4511[label="",style="dashed", color="magenta", weight=3]; 4486[label="vuz204",fontsize=16,color="green",shape="box"];4487[label="vuz204 * vuz205",fontsize=16,color="blue",shape="box"];4883[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4883[label="",style="solid", color="blue", weight=9]; 4883 -> 4512[label="",style="solid", color="blue", weight=3]; 4884[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4884[label="",style="solid", color="blue", weight=9]; 4884 -> 4513[label="",style="solid", color="blue", weight=3]; 4885[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4885[label="",style="solid", color="blue", weight=9]; 4885 -> 4514[label="",style="solid", color="blue", weight=3]; 4886[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4886[label="",style="solid", color="blue", weight=9]; 4886 -> 4515[label="",style="solid", color="blue", weight=3]; 4887[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4887[label="",style="solid", color="blue", weight=9]; 4887 -> 4516[label="",style="solid", color="blue", weight=3]; 4488 -> 1226[label="",style="dashed", color="red", weight=0]; 4488[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4488 -> 4517[label="",style="dashed", color="magenta", weight=3]; 4607[label="vuz23100",fontsize=16,color="green",shape="box"];4608[label="pr2F0G0 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) otherwise",fontsize=16,color="black",shape="box"];4608 -> 4627[label="",style="solid", color="black", weight=3]; 4609[label="pr2F0G2 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4609 -> 4628[label="",style="solid", color="black", weight=3]; 4665[label="pr2F4 vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="box"];4665 -> 4684[label="",style="solid", color="black", weight=3]; 4521[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (primQuotInt (Neg Zero) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg Zero) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4521 -> 4545[label="",style="solid", color="black", weight=3]; 4590 -> 23[label="",style="dashed", color="red", weight=0]; 4590[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4595[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (primQuotInt (Pos (Succ vuz224)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos (Succ vuz224)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4595 -> 4610[label="",style="solid", color="black", weight=3]; 4640[label="vuz232",fontsize=16,color="green",shape="box"];4641[label="vuz224",fontsize=16,color="green",shape="box"];4642[label="vuz222 * vuz223",fontsize=16,color="blue",shape="box"];4888[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4888[label="",style="solid", color="blue", weight=9]; 4888 -> 4650[label="",style="solid", color="blue", weight=3]; 4889[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4889[label="",style="solid", color="blue", weight=9]; 4889 -> 4651[label="",style="solid", color="blue", weight=3]; 4890[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4890[label="",style="solid", color="blue", weight=9]; 4890 -> 4652[label="",style="solid", color="blue", weight=3]; 4891[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4891[label="",style="solid", color="blue", weight=9]; 4891 -> 4653[label="",style="solid", color="blue", weight=3]; 4892[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4892[label="",style="solid", color="blue", weight=9]; 4892 -> 4654[label="",style="solid", color="blue", weight=3]; 4643[label="vuz222",fontsize=16,color="green",shape="box"];4511[label="Zero",fontsize=16,color="green",shape="box"];4512 -> 1024[label="",style="dashed", color="red", weight=0]; 4512[label="vuz204 * vuz205",fontsize=16,color="magenta"];4512 -> 4533[label="",style="dashed", color="magenta", weight=3]; 4512 -> 4534[label="",style="dashed", color="magenta", weight=3]; 4513 -> 1041[label="",style="dashed", color="red", weight=0]; 4513[label="vuz204 * vuz205",fontsize=16,color="magenta"];4513 -> 4535[label="",style="dashed", color="magenta", weight=3]; 4513 -> 4536[label="",style="dashed", color="magenta", weight=3]; 4514 -> 1051[label="",style="dashed", color="red", weight=0]; 4514[label="vuz204 * vuz205",fontsize=16,color="magenta"];4514 -> 4537[label="",style="dashed", color="magenta", weight=3]; 4514 -> 4538[label="",style="dashed", color="magenta", weight=3]; 4515 -> 1061[label="",style="dashed", color="red", weight=0]; 4515[label="vuz204 * vuz205",fontsize=16,color="magenta"];4515 -> 4539[label="",style="dashed", color="magenta", weight=3]; 4515 -> 4540[label="",style="dashed", color="magenta", weight=3]; 4516 -> 1073[label="",style="dashed", color="red", weight=0]; 4516[label="vuz204 * vuz205",fontsize=16,color="magenta"];4516 -> 4541[label="",style="dashed", color="magenta", weight=3]; 4516 -> 4542[label="",style="dashed", color="magenta", weight=3]; 4517[label="Zero",fontsize=16,color="green",shape="box"];4627[label="pr2F0G0 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) True",fontsize=16,color="black",shape="box"];4627 -> 4644[label="",style="solid", color="black", weight=3]; 4628[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4628 -> 4645[label="",style="solid", color="black", weight=3]; 4684[label="pr2F3 (Neg (Succ vuz230) - vuz233 == fromInt (Pos Zero)) vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="box"];4684 -> 4696[label="",style="solid", color="black", weight=3]; 4545 -> 1755[label="",style="dashed", color="red", weight=0]; 4545[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg (primDivNatS Zero (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS Zero (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4545 -> 4564[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4565[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4566[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4567[label="",style="dashed", color="magenta", weight=3]; 4610[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (primQuotInt (Pos (Succ vuz224)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos (Succ vuz224)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4610 -> 4629[label="",style="solid", color="black", weight=3]; 4650 -> 1024[label="",style="dashed", color="red", weight=0]; 4650[label="vuz222 * vuz223",fontsize=16,color="magenta"];4650 -> 4666[label="",style="dashed", color="magenta", weight=3]; 4650 -> 4667[label="",style="dashed", color="magenta", weight=3]; 4651 -> 1041[label="",style="dashed", color="red", weight=0]; 4651[label="vuz222 * vuz223",fontsize=16,color="magenta"];4651 -> 4668[label="",style="dashed", color="magenta", weight=3]; 4651 -> 4669[label="",style="dashed", color="magenta", weight=3]; 4652 -> 1051[label="",style="dashed", color="red", weight=0]; 4652[label="vuz222 * vuz223",fontsize=16,color="magenta"];4652 -> 4670[label="",style="dashed", color="magenta", weight=3]; 4652 -> 4671[label="",style="dashed", color="magenta", weight=3]; 4653 -> 1061[label="",style="dashed", color="red", weight=0]; 4653[label="vuz222 * vuz223",fontsize=16,color="magenta"];4653 -> 4672[label="",style="dashed", color="magenta", weight=3]; 4653 -> 4673[label="",style="dashed", color="magenta", weight=3]; 4654 -> 1073[label="",style="dashed", color="red", weight=0]; 4654[label="vuz222 * vuz223",fontsize=16,color="magenta"];4654 -> 4674[label="",style="dashed", color="magenta", weight=3]; 4654 -> 4675[label="",style="dashed", color="magenta", weight=3]; 4533[label="vuz204",fontsize=16,color="green",shape="box"];4534[label="vuz205",fontsize=16,color="green",shape="box"];4535[label="vuz204",fontsize=16,color="green",shape="box"];4536[label="vuz205",fontsize=16,color="green",shape="box"];4537[label="vuz205",fontsize=16,color="green",shape="box"];4538[label="vuz204",fontsize=16,color="green",shape="box"];4539[label="vuz204",fontsize=16,color="green",shape="box"];4540[label="vuz205",fontsize=16,color="green",shape="box"];4541[label="vuz205",fontsize=16,color="green",shape="box"];4542[label="vuz204",fontsize=16,color="green",shape="box"];4644 -> 4655[label="",style="dashed", color="red", weight=0]; 4644[label="pr2F vuz228 (Neg (Succ vuz230) - fromInt (Pos (Succ Zero))) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="magenta"];4644 -> 4664[label="",style="dashed", color="magenta", weight=3]; 4645[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4645 -> 4676[label="",style="solid", color="black", weight=3]; 4696 -> 4256[label="",style="dashed", color="red", weight=0]; 4696[label="pr2F3 (primEqInt (Neg (Succ vuz230) - vuz233) (fromInt (Pos Zero))) vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="magenta"];4696 -> 4698[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4699[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4700[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4701[label="",style="dashed", color="magenta", weight=3]; 4564[label="vuz216",fontsize=16,color="green",shape="box"];4565 -> 1226[label="",style="dashed", color="red", weight=0]; 4565[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4565 -> 4599[label="",style="dashed", color="magenta", weight=3]; 4566[label="vuz216 * vuz217",fontsize=16,color="blue",shape="box"];4893[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4893[label="",style="solid", color="blue", weight=9]; 4893 -> 4600[label="",style="solid", color="blue", weight=3]; 4894[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4894[label="",style="solid", color="blue", weight=9]; 4894 -> 4601[label="",style="solid", color="blue", weight=3]; 4895[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4895[label="",style="solid", color="blue", weight=9]; 4895 -> 4602[label="",style="solid", color="blue", weight=3]; 4896[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4896[label="",style="solid", color="blue", weight=9]; 4896 -> 4603[label="",style="solid", color="blue", weight=3]; 4897[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4897[label="",style="solid", color="blue", weight=9]; 4897 -> 4604[label="",style="solid", color="blue", weight=3]; 4567 -> 1226[label="",style="dashed", color="red", weight=0]; 4567[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4567 -> 4605[label="",style="dashed", color="magenta", weight=3]; 4629 -> 1605[label="",style="dashed", color="red", weight=0]; 4629[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (primDivNatS (Succ vuz224) (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS (Succ vuz224) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4629 -> 4646[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4647[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4648[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4649[label="",style="dashed", color="magenta", weight=3]; 4666[label="vuz222",fontsize=16,color="green",shape="box"];4667[label="vuz223",fontsize=16,color="green",shape="box"];4668[label="vuz222",fontsize=16,color="green",shape="box"];4669[label="vuz223",fontsize=16,color="green",shape="box"];4670[label="vuz223",fontsize=16,color="green",shape="box"];4671[label="vuz222",fontsize=16,color="green",shape="box"];4672[label="vuz222",fontsize=16,color="green",shape="box"];4673[label="vuz223",fontsize=16,color="green",shape="box"];4674[label="vuz223",fontsize=16,color="green",shape="box"];4675[label="vuz222",fontsize=16,color="green",shape="box"];4664 -> 23[label="",style="dashed", color="red", weight=0]; 4664[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4676[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (primQuotInt (Neg (Succ vuz230)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg (Succ vuz230)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4676 -> 4685[label="",style="solid", color="black", weight=3]; 4698[label="vuz230",fontsize=16,color="green",shape="box"];4699[label="vuz228",fontsize=16,color="green",shape="box"];4700[label="vuz233",fontsize=16,color="green",shape="box"];4701[label="vuz228 * vuz229",fontsize=16,color="blue",shape="box"];4898[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4898[label="",style="solid", color="blue", weight=9]; 4898 -> 4706[label="",style="solid", color="blue", weight=3]; 4899[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4899[label="",style="solid", color="blue", weight=9]; 4899 -> 4707[label="",style="solid", color="blue", weight=3]; 4900[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4900[label="",style="solid", color="blue", weight=9]; 4900 -> 4708[label="",style="solid", color="blue", weight=3]; 4901[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4901[label="",style="solid", color="blue", weight=9]; 4901 -> 4709[label="",style="solid", color="blue", weight=3]; 4902[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4902[label="",style="solid", color="blue", weight=9]; 4902 -> 4710[label="",style="solid", color="blue", weight=3]; 4599[label="Zero",fontsize=16,color="green",shape="box"];4600 -> 1024[label="",style="dashed", color="red", weight=0]; 4600[label="vuz216 * vuz217",fontsize=16,color="magenta"];4600 -> 4616[label="",style="dashed", color="magenta", weight=3]; 4600 -> 4617[label="",style="dashed", color="magenta", weight=3]; 4601 -> 1041[label="",style="dashed", color="red", weight=0]; 4601[label="vuz216 * vuz217",fontsize=16,color="magenta"];4601 -> 4618[label="",style="dashed", color="magenta", weight=3]; 4601 -> 4619[label="",style="dashed", color="magenta", weight=3]; 4602 -> 1051[label="",style="dashed", color="red", weight=0]; 4602[label="vuz216 * vuz217",fontsize=16,color="magenta"];4602 -> 4620[label="",style="dashed", color="magenta", weight=3]; 4602 -> 4621[label="",style="dashed", color="magenta", weight=3]; 4603 -> 1061[label="",style="dashed", color="red", weight=0]; 4603[label="vuz216 * vuz217",fontsize=16,color="magenta"];4603 -> 4622[label="",style="dashed", color="magenta", weight=3]; 4603 -> 4623[label="",style="dashed", color="magenta", weight=3]; 4604 -> 1073[label="",style="dashed", color="red", weight=0]; 4604[label="vuz216 * vuz217",fontsize=16,color="magenta"];4604 -> 4624[label="",style="dashed", color="magenta", weight=3]; 4604 -> 4625[label="",style="dashed", color="magenta", weight=3]; 4605[label="Zero",fontsize=16,color="green",shape="box"];4646 -> 1226[label="",style="dashed", color="red", weight=0]; 4646[label="primDivNatS (Succ vuz224) (Succ (Succ Zero))",fontsize=16,color="magenta"];4646 -> 4677[label="",style="dashed", color="magenta", weight=3]; 4647[label="vuz222",fontsize=16,color="green",shape="box"];4648[label="vuz222 * vuz223",fontsize=16,color="blue",shape="box"];4903[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4903[label="",style="solid", color="blue", weight=9]; 4903 -> 4678[label="",style="solid", color="blue", weight=3]; 4904[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4904[label="",style="solid", color="blue", weight=9]; 4904 -> 4679[label="",style="solid", color="blue", weight=3]; 4905[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4905[label="",style="solid", color="blue", weight=9]; 4905 -> 4680[label="",style="solid", color="blue", weight=3]; 4906[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4906[label="",style="solid", color="blue", weight=9]; 4906 -> 4681[label="",style="solid", color="blue", weight=3]; 4907[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4907[label="",style="solid", color="blue", weight=9]; 4907 -> 4682[label="",style="solid", color="blue", weight=3]; 4649 -> 1226[label="",style="dashed", color="red", weight=0]; 4649[label="primDivNatS (Succ vuz224) (Succ (Succ Zero))",fontsize=16,color="magenta"];4649 -> 4683[label="",style="dashed", color="magenta", weight=3]; 4685[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (primQuotInt (Neg (Succ vuz230)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg (Succ vuz230)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4685 -> 4697[label="",style="solid", color="black", weight=3]; 4706 -> 1024[label="",style="dashed", color="red", weight=0]; 4706[label="vuz228 * vuz229",fontsize=16,color="magenta"];4706 -> 4718[label="",style="dashed", color="magenta", weight=3]; 4706 -> 4719[label="",style="dashed", color="magenta", weight=3]; 4707 -> 1041[label="",style="dashed", color="red", weight=0]; 4707[label="vuz228 * vuz229",fontsize=16,color="magenta"];4707 -> 4720[label="",style="dashed", color="magenta", weight=3]; 4707 -> 4721[label="",style="dashed", color="magenta", weight=3]; 4708 -> 1051[label="",style="dashed", color="red", weight=0]; 4708[label="vuz228 * vuz229",fontsize=16,color="magenta"];4708 -> 4722[label="",style="dashed", color="magenta", weight=3]; 4708 -> 4723[label="",style="dashed", color="magenta", weight=3]; 4709 -> 1061[label="",style="dashed", color="red", weight=0]; 4709[label="vuz228 * vuz229",fontsize=16,color="magenta"];4709 -> 4724[label="",style="dashed", color="magenta", weight=3]; 4709 -> 4725[label="",style="dashed", color="magenta", weight=3]; 4710 -> 1073[label="",style="dashed", color="red", weight=0]; 4710[label="vuz228 * vuz229",fontsize=16,color="magenta"];4710 -> 4726[label="",style="dashed", color="magenta", weight=3]; 4710 -> 4727[label="",style="dashed", color="magenta", weight=3]; 4616[label="vuz216",fontsize=16,color="green",shape="box"];4617[label="vuz217",fontsize=16,color="green",shape="box"];4618[label="vuz216",fontsize=16,color="green",shape="box"];4619[label="vuz217",fontsize=16,color="green",shape="box"];4620[label="vuz217",fontsize=16,color="green",shape="box"];4621[label="vuz216",fontsize=16,color="green",shape="box"];4622[label="vuz216",fontsize=16,color="green",shape="box"];4623[label="vuz217",fontsize=16,color="green",shape="box"];4624[label="vuz217",fontsize=16,color="green",shape="box"];4625[label="vuz216",fontsize=16,color="green",shape="box"];4677[label="Succ vuz224",fontsize=16,color="green",shape="box"];4678 -> 1024[label="",style="dashed", color="red", weight=0]; 4678[label="vuz222 * vuz223",fontsize=16,color="magenta"];4678 -> 4686[label="",style="dashed", color="magenta", weight=3]; 4678 -> 4687[label="",style="dashed", color="magenta", weight=3]; 4679 -> 1041[label="",style="dashed", color="red", weight=0]; 4679[label="vuz222 * vuz223",fontsize=16,color="magenta"];4679 -> 4688[label="",style="dashed", color="magenta", weight=3]; 4679 -> 4689[label="",style="dashed", color="magenta", weight=3]; 4680 -> 1051[label="",style="dashed", color="red", weight=0]; 4680[label="vuz222 * vuz223",fontsize=16,color="magenta"];4680 -> 4690[label="",style="dashed", color="magenta", weight=3]; 4680 -> 4691[label="",style="dashed", color="magenta", weight=3]; 4681 -> 1061[label="",style="dashed", color="red", weight=0]; 4681[label="vuz222 * vuz223",fontsize=16,color="magenta"];4681 -> 4692[label="",style="dashed", color="magenta", weight=3]; 4681 -> 4693[label="",style="dashed", color="magenta", weight=3]; 4682 -> 1073[label="",style="dashed", color="red", weight=0]; 4682[label="vuz222 * vuz223",fontsize=16,color="magenta"];4682 -> 4694[label="",style="dashed", color="magenta", weight=3]; 4682 -> 4695[label="",style="dashed", color="magenta", weight=3]; 4683[label="Succ vuz224",fontsize=16,color="green",shape="box"];4697 -> 1755[label="",style="dashed", color="red", weight=0]; 4697[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (primDivNatS (Succ vuz230) (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS (Succ vuz230) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4697 -> 4702[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4703[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4704[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4705[label="",style="dashed", color="magenta", weight=3]; 4718[label="vuz228",fontsize=16,color="green",shape="box"];4719[label="vuz229",fontsize=16,color="green",shape="box"];4720[label="vuz228",fontsize=16,color="green",shape="box"];4721[label="vuz229",fontsize=16,color="green",shape="box"];4722[label="vuz229",fontsize=16,color="green",shape="box"];4723[label="vuz228",fontsize=16,color="green",shape="box"];4724[label="vuz228",fontsize=16,color="green",shape="box"];4725[label="vuz229",fontsize=16,color="green",shape="box"];4726[label="vuz229",fontsize=16,color="green",shape="box"];4727[label="vuz228",fontsize=16,color="green",shape="box"];4686[label="vuz222",fontsize=16,color="green",shape="box"];4687[label="vuz223",fontsize=16,color="green",shape="box"];4688[label="vuz222",fontsize=16,color="green",shape="box"];4689[label="vuz223",fontsize=16,color="green",shape="box"];4690[label="vuz223",fontsize=16,color="green",shape="box"];4691[label="vuz222",fontsize=16,color="green",shape="box"];4692[label="vuz222",fontsize=16,color="green",shape="box"];4693[label="vuz223",fontsize=16,color="green",shape="box"];4694[label="vuz223",fontsize=16,color="green",shape="box"];4695[label="vuz222",fontsize=16,color="green",shape="box"];4702[label="vuz228",fontsize=16,color="green",shape="box"];4703 -> 1226[label="",style="dashed", color="red", weight=0]; 4703[label="primDivNatS (Succ vuz230) (Succ (Succ Zero))",fontsize=16,color="magenta"];4703 -> 4711[label="",style="dashed", color="magenta", weight=3]; 4704[label="vuz228 * vuz229",fontsize=16,color="blue",shape="box"];4908[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4908[label="",style="solid", color="blue", weight=9]; 4908 -> 4712[label="",style="solid", color="blue", weight=3]; 4909[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4909[label="",style="solid", color="blue", weight=9]; 4909 -> 4713[label="",style="solid", color="blue", weight=3]; 4910[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4910[label="",style="solid", color="blue", weight=9]; 4910 -> 4714[label="",style="solid", color="blue", weight=3]; 4911[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4911[label="",style="solid", color="blue", weight=9]; 4911 -> 4715[label="",style="solid", color="blue", weight=3]; 4912[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4912[label="",style="solid", color="blue", weight=9]; 4912 -> 4716[label="",style="solid", color="blue", weight=3]; 4705 -> 1226[label="",style="dashed", color="red", weight=0]; 4705[label="primDivNatS (Succ vuz230) (Succ (Succ Zero))",fontsize=16,color="magenta"];4705 -> 4717[label="",style="dashed", color="magenta", weight=3]; 4711[label="Succ vuz230",fontsize=16,color="green",shape="box"];4712 -> 1024[label="",style="dashed", color="red", weight=0]; 4712[label="vuz228 * vuz229",fontsize=16,color="magenta"];4712 -> 4728[label="",style="dashed", color="magenta", weight=3]; 4712 -> 4729[label="",style="dashed", color="magenta", weight=3]; 4713 -> 1041[label="",style="dashed", color="red", weight=0]; 4713[label="vuz228 * vuz229",fontsize=16,color="magenta"];4713 -> 4730[label="",style="dashed", color="magenta", weight=3]; 4713 -> 4731[label="",style="dashed", color="magenta", weight=3]; 4714 -> 1051[label="",style="dashed", color="red", weight=0]; 4714[label="vuz228 * vuz229",fontsize=16,color="magenta"];4714 -> 4732[label="",style="dashed", color="magenta", weight=3]; 4714 -> 4733[label="",style="dashed", color="magenta", weight=3]; 4715 -> 1061[label="",style="dashed", color="red", weight=0]; 4715[label="vuz228 * vuz229",fontsize=16,color="magenta"];4715 -> 4734[label="",style="dashed", color="magenta", weight=3]; 4715 -> 4735[label="",style="dashed", color="magenta", weight=3]; 4716 -> 1073[label="",style="dashed", color="red", weight=0]; 4716[label="vuz228 * vuz229",fontsize=16,color="magenta"];4716 -> 4736[label="",style="dashed", color="magenta", weight=3]; 4716 -> 4737[label="",style="dashed", color="magenta", weight=3]; 4717[label="Succ vuz230",fontsize=16,color="green",shape="box"];4728[label="vuz228",fontsize=16,color="green",shape="box"];4729[label="vuz229",fontsize=16,color="green",shape="box"];4730[label="vuz228",fontsize=16,color="green",shape="box"];4731[label="vuz229",fontsize=16,color="green",shape="box"];4732[label="vuz229",fontsize=16,color="green",shape="box"];4733[label="vuz228",fontsize=16,color="green",shape="box"];4734[label="vuz228",fontsize=16,color="green",shape="box"];4735[label="vuz229",fontsize=16,color="green",shape="box"];4736[label="vuz229",fontsize=16,color="green",shape="box"];4737[label="vuz228",fontsize=16,color="green",shape="box"];} ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F35(Succ(vuz4000), Succ(vuz5000), vuz3) -> new_pr2F35(vuz4000, vuz5000, vuz3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F35(Succ(vuz4000), Succ(vuz5000), vuz3) -> new_pr2F35(vuz4000, vuz5000, vuz3) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 ---------------------------------------- (15) YES ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G16(vuz20, vuz21, Succ(Succ(vuz2200)), h) -> new_pr2F0G16(vuz20, vuz21, vuz2200, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G16(vuz20, vuz21, Succ(Succ(vuz2200)), h) -> new_pr2F0G16(vuz20, vuz21, vuz2200, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 ---------------------------------------- (18) YES ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(vuz130000))) -> new_primDivNatS(vuz130000) new_primDivNatS00(Succ(Succ(vuz130000))) -> new_primDivNatS(vuz130000) new_primDivNatS0(Succ(Succ(vuz130000))) -> new_primDivNatS(vuz130000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(vuz130000))) -> new_primDivNatS(vuz130000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primDivNatS(Succ(Succ(vuz130000))) -> new_primDivNatS(vuz130000) The graph contains the following edges 1 > 1 ---------------------------------------- (23) YES ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F30(Succ(vuz2120), vuz204, Zero, vuz205, h) -> new_pr2F0G10(new_sr(vuz204, vuz205, h), vuz204, new_primDivNatS1(Zero), new_primDivNatS1(Zero), h) new_pr2F0(vuz103, Zero, Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) new_pr2F32(vuz202, Neg(vuz2030), vuz204, vuz205, h) -> new_pr2F30(new_primPlusNat0(Succ(vuz202), vuz2030), vuz204, new_primPlusNat0(Succ(vuz202), vuz2030), vuz205, h) new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F32(Zero, Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F30(Succ(vuz2120), vuz204, Succ(Zero), vuz205, h) -> new_pr2F(vuz204, Zero, new_fromInt, vuz205, h) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F2(vuz111, vuz113, Neg(vuz1160), vuz110, be) -> new_pr2F33(vuz1160, vuz113, vuz111, vuz110, be) new_pr2F0(vuz103, Zero, Pos(Zero), vuz102, bb) -> new_pr2F30(Zero, new_sr5(vuz103, bb), Zero, vuz102, bb) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F3(Zero, Zero, vuz204, vuz205, h) -> new_pr2F30(Zero, vuz204, Zero, vuz205, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F33(Zero, Zero, vuz103, vuz102, bb) -> new_pr2F30(Zero, new_sr5(vuz103, bb), Zero, vuz102, bb) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F34(vuz214, Neg(vuz2150), vuz216, vuz217, bc) -> new_pr2F3(vuz2150, Succ(vuz214), vuz216, vuz217, bc) new_pr2F3(Zero, Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F31(Succ(vuz2190), vuz216, Zero, vuz217, bc) -> new_pr2F0G13(new_sr7(vuz216, vuz217, bc), vuz216, new_primDivNatS1(Zero), new_primDivNatS1(Zero), bc) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F32(Zero, Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Zero, vuz204, Zero, vuz205, h) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F0(vuz103, vuz105, Neg(vuz1150), vuz102, bb) -> new_pr2F30(new_primPlusNat0(vuz105, vuz1150), new_sr6(vuz103, bb), new_primPlusNat0(vuz105, vuz1150), vuz102, bb) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0(vuz103, Succ(vuz1050), Pos(Zero), vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F33(Zero, Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 7 less nodes. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Zero, Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Zero, vuz217, bc) -> new_pr2F0G13(new_sr7(vuz216, vuz217, bc), vuz216, new_primDivNatS1(Zero), new_primDivNatS1(Zero), bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F34(vuz214, Neg(vuz2150), vuz216, vuz217, bc) -> new_pr2F3(vuz2150, Succ(vuz214), vuz216, vuz217, bc) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Zero), vuz205, h) -> new_pr2F(vuz204, Zero, new_fromInt, vuz205, h) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(vuz202, Neg(vuz2030), vuz204, vuz205, h) -> new_pr2F30(new_primPlusNat0(Succ(vuz202), vuz2030), vuz204, new_primPlusNat0(Succ(vuz202), vuz2030), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Zero, vuz205, h) -> new_pr2F0G10(new_sr(vuz204, vuz205, h), vuz204, new_primDivNatS1(Zero), new_primDivNatS1(Zero), h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Zero, Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) new_pr2F32(Zero, Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Zero, Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_pr2F34(vuz214, Neg(vuz2150), vuz216, vuz217, bc) -> new_pr2F3(vuz2150, Succ(vuz214), vuz216, vuz217, bc) new_pr2F32(vuz202, Neg(vuz2030), vuz204, vuz205, h) -> new_pr2F30(new_primPlusNat0(Succ(vuz202), vuz2030), vuz204, new_primPlusNat0(Succ(vuz202), vuz2030), vuz205, h) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 1 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 0 POL(Zero) = 0 POL([]) = 1 POL(app(x_1, x_2)) = 1 + x_1 + x_2 POL(error(x_1)) = 1 + x_1 POL(new_fromInt) = 0 POL(new_pr2F(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(new_pr2F0(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G1(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G10(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G11(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G12(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G13(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G14(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F1(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(new_pr2F2(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F3(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F30(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F31(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F32(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5 POL(new_pr2F33(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F34(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5 POL(new_primDivNatS01(x_1)) = 0 POL(new_primDivNatS1(x_1)) = 0 POL(new_primDivNatS2) = 1 POL(new_primDivNatS3) = 1 POL(new_primDivNatS4(x_1)) = 1 + x_1 POL(new_primDivNatS5(x_1)) = 1 + x_1 POL(new_primMulNat0(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = 0 POL(new_sr(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr0(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr1(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr10(x_1, x_2)) = x_1 + x_2 POL(new_sr11(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr12(x_1)) = 1 + x_1 POL(new_sr13(x_1, x_2)) = 1 + x_1 POL(new_sr14(x_1, x_2, x_3)) = 1 + x_1 POL(new_sr15(x_1, x_2)) = 0 POL(new_sr16(x_1, x_2)) = 1 + x_1 POL(new_sr17(x_1, x_2)) = 1 + x_1 POL(new_sr18(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr19(x_1)) = 1 + x_1 POL(new_sr2(x_1, x_2)) = x_1 + x_2 POL(new_sr20(x_1)) = 1 + x_1 POL(new_sr21(x_1)) = 1 + x_1 POL(new_sr3(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr4(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr7(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr8(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr9(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(ty_Double) = 1 POL(ty_Float) = 1 POL(ty_Int) = 1 POL(ty_Integer) = 1 POL(ty_Ratio) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_fromInt -> Pos(Succ(Zero)) ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Zero, Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Zero, vuz217, bc) -> new_pr2F0G13(new_sr7(vuz216, vuz217, bc), vuz216, new_primDivNatS1(Zero), new_primDivNatS1(Zero), bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Zero), vuz205, h) -> new_pr2F(vuz204, Zero, new_fromInt, vuz205, h) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F30(Succ(vuz2120), vuz204, Zero, vuz205, h) -> new_pr2F0G10(new_sr(vuz204, vuz205, h), vuz204, new_primDivNatS1(Zero), new_primDivNatS1(Zero), h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Zero, Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F31(Succ(vuz11500), new_sr4(vuz103, bb), Succ(vuz11500), vuz102, bb) new_pr2F32(Zero, Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Zero, Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F31(Succ(vuz203000), vuz204, Succ(vuz203000), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 5 less nodes. ---------------------------------------- (30) Complex Obligation (AND) ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Zero, vuz217, bc) -> new_pr2F0G13(new_sr7(vuz216, vuz217, bc), vuz216, new_primDivNatS1(Zero), new_primDivNatS1(Zero), bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_pr2F31(Succ(vuz2190), vuz216, Zero, vuz217, bc) -> new_pr2F0G13(new_sr7(vuz216, vuz217, bc), vuz216, new_primDivNatS1(Zero), new_primDivNatS1(Zero), bc) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 0 POL(Pos(x_1)) = x_1 POL(Succ(x_1)) = 0 POL(Zero) = 1 POL([]) = 1 POL(app(x_1, x_2)) = 1 + x_1 + x_2 POL(error(x_1)) = 1 + x_1 POL(new_fromInt) = 0 POL(new_pr2F0G12(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G13(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F0G14(x_1, x_2, x_3, x_4, x_5)) = x_5 POL(new_pr2F1(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(new_pr2F2(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(new_pr2F31(x_1, x_2, x_3, x_4, x_5)) = x_3 + x_5 POL(new_pr2F34(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_5 POL(new_primDivNatS01(x_1)) = 0 POL(new_primDivNatS1(x_1)) = 0 POL(new_primDivNatS2) = 1 POL(new_primDivNatS3) = 1 POL(new_primDivNatS4(x_1)) = 1 + x_1 POL(new_primDivNatS5(x_1)) = 1 + x_1 POL(new_primMulNat0(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = x_2 POL(new_sr10(x_1, x_2)) = x_1 + x_2 POL(new_sr11(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr12(x_1)) = 1 + x_1 POL(new_sr13(x_1, x_2)) = 1 + x_2 POL(new_sr14(x_1, x_2, x_3)) = 1 + x_2 POL(new_sr15(x_1, x_2)) = 0 POL(new_sr16(x_1, x_2)) = 1 + x_2 POL(new_sr17(x_1, x_2)) = 1 + x_2 POL(new_sr18(x_1, x_2)) = 1 + x_1 POL(new_sr19(x_1)) = 1 + x_1 POL(new_sr20(x_1)) = 1 + x_1 POL(new_sr21(x_1)) = 1 + x_1 POL(new_sr7(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr8(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr9(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(ty_Double) = 1 POL(ty_Float) = 1 POL(ty_Int) = 1 POL(ty_Integer) = 1 POL(ty_Ratio) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_fromInt -> Pos(Succ(Zero)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) ---------------------------------------- (33) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (34) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (35) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (36) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) the following chains were created: *We consider the chain new_pr2F0G12(x0, x1, x2, Succ(Succ(x3)), x4) -> new_pr2F0G12(x0, x1, x2, x3, x4), new_pr2F0G12(x5, x6, x7, Succ(Succ(x8)), x9) -> new_pr2F0G12(x5, x6, x7, x8, x9) which results in the following constraint: (1) (new_pr2F0G12(x0, x1, x2, x3, x4)=new_pr2F0G12(x5, x6, x7, Succ(Succ(x8)), x9) ==> new_pr2F0G12(x0, x1, x2, Succ(Succ(x3)), x4)_>=_new_pr2F0G12(x0, x1, x2, x3, x4)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x0, x1, x2, Succ(Succ(Succ(Succ(x8)))), x4)_>=_new_pr2F0G12(x0, x1, x2, Succ(Succ(x8)), x4)) *We consider the chain new_pr2F0G12(x10, x11, x12, Succ(Succ(x13)), x14) -> new_pr2F0G12(x10, x11, x12, x13, x14), new_pr2F0G12(x15, x16, x17, Succ(Zero), x18) -> new_pr2F1(x15, x17, new_fromInt, x16, x18) which results in the following constraint: (1) (new_pr2F0G12(x10, x11, x12, x13, x14)=new_pr2F0G12(x15, x16, x17, Succ(Zero), x18) ==> new_pr2F0G12(x10, x11, x12, Succ(Succ(x13)), x14)_>=_new_pr2F0G12(x10, x11, x12, x13, x14)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x10, x11, x12, Succ(Succ(Succ(Zero))), x14)_>=_new_pr2F0G12(x10, x11, x12, Succ(Zero), x14)) *We consider the chain new_pr2F0G12(x34, x35, x36, Succ(Succ(x37)), x38) -> new_pr2F0G12(x34, x35, x36, x37, x38), new_pr2F0G12(x39, x40, x41, Zero, x42) -> new_pr2F0G13(new_sr8(x39, x40, x42), x39, new_primDivNatS1(Succ(x41)), new_primDivNatS1(Succ(x41)), x42) which results in the following constraint: (1) (new_pr2F0G12(x34, x35, x36, x37, x38)=new_pr2F0G12(x39, x40, x41, Zero, x42) ==> new_pr2F0G12(x34, x35, x36, Succ(Succ(x37)), x38)_>=_new_pr2F0G12(x34, x35, x36, x37, x38)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x34, x35, x36, Succ(Succ(Zero)), x38)_>=_new_pr2F0G12(x34, x35, x36, Zero, x38)) For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) the following chains were created: *We consider the chain new_pr2F0G12(x91, x92, x93, Succ(Zero), x94) -> new_pr2F1(x91, x93, new_fromInt, x92, x94), new_pr2F1(x95, x96, x97, x98, x99) -> new_pr2F34(x96, x97, x95, new_sr9(x95, x98, x99), x99) which results in the following constraint: (1) (new_pr2F1(x91, x93, new_fromInt, x92, x94)=new_pr2F1(x95, x96, x97, x98, x99) ==> new_pr2F0G12(x91, x92, x93, Succ(Zero), x94)_>=_new_pr2F1(x91, x93, new_fromInt, x92, x94)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x91, x92, x93, Succ(Zero), x94)_>=_new_pr2F1(x91, x93, new_fromInt, x92, x94)) For Pair new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) the following chains were created: *We consider the chain new_pr2F1(x159, x160, x161, x162, x163) -> new_pr2F34(x160, x161, x159, new_sr9(x159, x162, x163), x163), new_pr2F34(x164, Pos(x165), x166, x167, x168) -> new_pr2F31(new_primPlusNat0(Succ(x164), x165), x166, new_primPlusNat0(Succ(x164), x165), x167, x168) which results in the following constraint: (1) (new_pr2F34(x160, x161, x159, new_sr9(x159, x162, x163), x163)=new_pr2F34(x164, Pos(x165), x166, x167, x168) ==> new_pr2F1(x159, x160, x161, x162, x163)_>=_new_pr2F34(x160, x161, x159, new_sr9(x159, x162, x163), x163)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F1(x159, x160, Pos(x165), x162, x163)_>=_new_pr2F34(x160, Pos(x165), x159, new_sr9(x159, x162, x163), x163)) For Pair new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) the following chains were created: *We consider the chain new_pr2F34(x239, Pos(x240), x241, x242, x243) -> new_pr2F31(new_primPlusNat0(Succ(x239), x240), x241, new_primPlusNat0(Succ(x239), x240), x242, x243), new_pr2F31(Succ(x244), x245, Succ(Succ(x246)), x247, x248) -> new_pr2F0G12(x245, x247, Succ(x246), x246, x248) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x239), x240), x241, new_primPlusNat0(Succ(x239), x240), x242, x243)=new_pr2F31(Succ(x244), x245, Succ(Succ(x246)), x247, x248) ==> new_pr2F34(x239, Pos(x240), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), x240), x241, new_primPlusNat0(Succ(x239), x240), x242, x243)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x239)=x1016 & new_primPlusNat0(x1016, x240)=Succ(x244) & Succ(x239)=x1017 & new_primPlusNat0(x1017, x240)=Succ(Succ(x246)) ==> new_pr2F34(x239, Pos(x240), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), x240), x241, new_primPlusNat0(Succ(x239), x240), x242, x243)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1016, x240)=Succ(x244) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1019, x1018)))=Succ(x244) & Succ(x239)=Succ(x1019) & Succ(x239)=x1017 & new_primPlusNat0(x1017, Succ(x1018))=Succ(Succ(x246)) & (\/x1020,x1021,x1022,x1023,x1024,x1025,x1026:new_primPlusNat0(x1019, x1018)=Succ(x1020) & Succ(x1021)=x1019 & Succ(x1021)=x1022 & new_primPlusNat0(x1022, x1018)=Succ(Succ(x1023)) ==> new_pr2F34(x1021, Pos(x1018), x1024, x1025, x1026)_>=_new_pr2F31(new_primPlusNat0(Succ(x1021), x1018), x1024, new_primPlusNat0(Succ(x1021), x1018), x1025, x1026)) ==> new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) (4) (Succ(x1027)=Succ(x244) & Succ(x239)=Succ(x1027) & Succ(x239)=x1017 & new_primPlusNat0(x1017, Zero)=Succ(Succ(x246)) ==> new_pr2F34(x239, Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Zero), x241, new_primPlusNat0(Succ(x239), Zero), x242, x243)) (5) (Succ(x1028)=Succ(x244) & Succ(x239)=Zero & Succ(x239)=x1017 & new_primPlusNat0(x1017, Succ(x1028))=Succ(Succ(x246)) ==> new_pr2F34(x239, Pos(Succ(x1028)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1028)), x241, new_primPlusNat0(Succ(x239), Succ(x1028)), x242, x243)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x239)=x1017 & Succ(x1018)=x1029 & new_primPlusNat0(x1017, x1029)=Succ(Succ(x246)) & (\/x1020,x1021,x1022,x1023,x1024,x1025,x1026:new_primPlusNat0(x239, x1018)=Succ(x1020) & Succ(x1021)=x239 & Succ(x1021)=x1022 & new_primPlusNat0(x1022, x1018)=Succ(Succ(x1023)) ==> new_pr2F34(x1021, Pos(x1018), x1024, x1025, x1026)_>=_new_pr2F31(new_primPlusNat0(Succ(x1021), x1018), x1024, new_primPlusNat0(Succ(x1021), x1018), x1025, x1026)) ==> new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x239)=x1017 & Zero=x1047 & new_primPlusNat0(x1017, x1047)=Succ(Succ(x246)) ==> new_pr2F34(x239, Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Zero), x241, new_primPlusNat0(Succ(x239), Zero), x242, x243)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1017, x1029)=Succ(Succ(x246)) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1031, x1030)))=Succ(Succ(x246)) & Succ(x239)=Succ(x1031) & Succ(x1018)=Succ(x1030) & (\/x1020,x1021,x1022,x1023,x1024,x1025,x1026:new_primPlusNat0(x239, x1018)=Succ(x1020) & Succ(x1021)=x239 & Succ(x1021)=x1022 & new_primPlusNat0(x1022, x1018)=Succ(Succ(x1023)) ==> new_pr2F34(x1021, Pos(x1018), x1024, x1025, x1026)_>=_new_pr2F31(new_primPlusNat0(Succ(x1021), x1018), x1024, new_primPlusNat0(Succ(x1021), x1018), x1025, x1026)) & (\/x1032,x1033,x1034,x1035,x1036,x1037,x1038,x1039,x1040,x1041,x1042,x1043,x1044:new_primPlusNat0(x1031, x1030)=Succ(Succ(x1032)) & Succ(x1033)=x1031 & Succ(x1034)=x1030 & (\/x1035,x1036,x1037,x1038,x1039,x1040,x1041:new_primPlusNat0(x1033, x1034)=Succ(x1035) & Succ(x1036)=x1033 & Succ(x1036)=x1037 & new_primPlusNat0(x1037, x1034)=Succ(Succ(x1038)) ==> new_pr2F34(x1036, Pos(x1034), x1039, x1040, x1041)_>=_new_pr2F31(new_primPlusNat0(Succ(x1036), x1034), x1039, new_primPlusNat0(Succ(x1036), x1034), x1040, x1041)) ==> new_pr2F34(x1033, Pos(Succ(x1034)), x1042, x1043, x1044)_>=_new_pr2F31(new_primPlusNat0(Succ(x1033), Succ(x1034)), x1042, new_primPlusNat0(Succ(x1033), Succ(x1034)), x1043, x1044)) ==> new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) (9) (Succ(x1045)=Succ(Succ(x246)) & Succ(x239)=Succ(x1045) & Succ(x1018)=Zero & (\/x1020,x1021,x1022,x1023,x1024,x1025,x1026:new_primPlusNat0(x239, x1018)=Succ(x1020) & Succ(x1021)=x239 & Succ(x1021)=x1022 & new_primPlusNat0(x1022, x1018)=Succ(Succ(x1023)) ==> new_pr2F34(x1021, Pos(x1018), x1024, x1025, x1026)_>=_new_pr2F31(new_primPlusNat0(Succ(x1021), x1018), x1024, new_primPlusNat0(Succ(x1021), x1018), x1025, x1026)) ==> new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) (10) (Succ(x1046)=Succ(Succ(x246)) & Succ(x239)=Zero & Succ(x1018)=Succ(x1046) & (\/x1020,x1021,x1022,x1023,x1024,x1025,x1026:new_primPlusNat0(x239, x1018)=Succ(x1020) & Succ(x1021)=x239 & Succ(x1021)=x1022 & new_primPlusNat0(x1022, x1018)=Succ(Succ(x1023)) ==> new_pr2F34(x1021, Pos(x1018), x1024, x1025, x1026)_>=_new_pr2F31(new_primPlusNat0(Succ(x1021), x1018), x1024, new_primPlusNat0(Succ(x1021), x1018), x1025, x1026)) ==> new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint: (11) (new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1017, x1047)=Succ(Succ(x246)) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1049, x1048)))=Succ(Succ(x246)) & Succ(x239)=Succ(x1049) & Zero=Succ(x1048) & (\/x1050,x1051,x1052,x1053,x1054:new_primPlusNat0(x1049, x1048)=Succ(Succ(x1050)) & Succ(x1051)=x1049 & Zero=x1048 ==> new_pr2F34(x1051, Pos(Zero), x1052, x1053, x1054)_>=_new_pr2F31(new_primPlusNat0(Succ(x1051), Zero), x1052, new_primPlusNat0(Succ(x1051), Zero), x1053, x1054)) ==> new_pr2F34(x239, Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Zero), x241, new_primPlusNat0(Succ(x239), Zero), x242, x243)) (13) (Succ(x1055)=Succ(Succ(x246)) & Succ(x239)=Succ(x1055) & Zero=Zero ==> new_pr2F34(x239, Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Zero), x241, new_primPlusNat0(Succ(x239), Zero), x242, x243)) (14) (Succ(x1056)=Succ(Succ(x246)) & Succ(x239)=Zero & Zero=Succ(x1056) ==> new_pr2F34(x239, Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Zero), x241, new_primPlusNat0(Succ(x239), Zero), x242, x243)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F34(Succ(x246), Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x246)), Zero), x241, new_primPlusNat0(Succ(Succ(x246)), Zero), x242, x243)) We solved constraint (14) using rules (I), (II). *We consider the chain new_pr2F34(x264, Pos(x265), x266, x267, x268) -> new_pr2F31(new_primPlusNat0(Succ(x264), x265), x266, new_primPlusNat0(Succ(x264), x265), x267, x268), new_pr2F31(Succ(x269), x270, Succ(Zero), x271, x272) -> new_pr2F1(x270, Zero, new_fromInt, x271, x272) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x264), x265), x266, new_primPlusNat0(Succ(x264), x265), x267, x268)=new_pr2F31(Succ(x269), x270, Succ(Zero), x271, x272) ==> new_pr2F34(x264, Pos(x265), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), x265), x266, new_primPlusNat0(Succ(x264), x265), x267, x268)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x264)=x1057 & new_primPlusNat0(x1057, x265)=Succ(x269) & Succ(x264)=x1058 & new_primPlusNat0(x1058, x265)=Succ(Zero) ==> new_pr2F34(x264, Pos(x265), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), x265), x266, new_primPlusNat0(Succ(x264), x265), x267, x268)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1057, x265)=Succ(x269) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1060, x1059)))=Succ(x269) & Succ(x264)=Succ(x1060) & Succ(x264)=x1058 & new_primPlusNat0(x1058, Succ(x1059))=Succ(Zero) & (\/x1061,x1062,x1063,x1064,x1065,x1066:new_primPlusNat0(x1060, x1059)=Succ(x1061) & Succ(x1062)=x1060 & Succ(x1062)=x1063 & new_primPlusNat0(x1063, x1059)=Succ(Zero) ==> new_pr2F34(x1062, Pos(x1059), x1064, x1065, x1066)_>=_new_pr2F31(new_primPlusNat0(Succ(x1062), x1059), x1064, new_primPlusNat0(Succ(x1062), x1059), x1065, x1066)) ==> new_pr2F34(x264, Pos(Succ(x1059)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1059)), x266, new_primPlusNat0(Succ(x264), Succ(x1059)), x267, x268)) (4) (Succ(x1067)=Succ(x269) & Succ(x264)=Succ(x1067) & Succ(x264)=x1058 & new_primPlusNat0(x1058, Zero)=Succ(Zero) ==> new_pr2F34(x264, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Zero), x266, new_primPlusNat0(Succ(x264), Zero), x267, x268)) (5) (Succ(x1068)=Succ(x269) & Succ(x264)=Zero & Succ(x264)=x1058 & new_primPlusNat0(x1058, Succ(x1068))=Succ(Zero) ==> new_pr2F34(x264, Pos(Succ(x1068)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1068)), x266, new_primPlusNat0(Succ(x264), Succ(x1068)), x267, x268)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x264)=x1058 & Succ(x1059)=x1069 & new_primPlusNat0(x1058, x1069)=Succ(Zero) & (\/x1061,x1062,x1063,x1064,x1065,x1066:new_primPlusNat0(x264, x1059)=Succ(x1061) & Succ(x1062)=x264 & Succ(x1062)=x1063 & new_primPlusNat0(x1063, x1059)=Succ(Zero) ==> new_pr2F34(x1062, Pos(x1059), x1064, x1065, x1066)_>=_new_pr2F31(new_primPlusNat0(Succ(x1062), x1059), x1064, new_primPlusNat0(Succ(x1062), x1059), x1065, x1066)) ==> new_pr2F34(x264, Pos(Succ(x1059)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1059)), x266, new_primPlusNat0(Succ(x264), Succ(x1059)), x267, x268)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x264)=x1058 & Zero=x1085 & new_primPlusNat0(x1058, x1085)=Succ(Zero) ==> new_pr2F34(x264, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Zero), x266, new_primPlusNat0(Succ(x264), Zero), x267, x268)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1058, x1069)=Succ(Zero) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1071, x1070)))=Succ(Zero) & Succ(x264)=Succ(x1071) & Succ(x1059)=Succ(x1070) & (\/x1061,x1062,x1063,x1064,x1065,x1066:new_primPlusNat0(x264, x1059)=Succ(x1061) & Succ(x1062)=x264 & Succ(x1062)=x1063 & new_primPlusNat0(x1063, x1059)=Succ(Zero) ==> new_pr2F34(x1062, Pos(x1059), x1064, x1065, x1066)_>=_new_pr2F31(new_primPlusNat0(Succ(x1062), x1059), x1064, new_primPlusNat0(Succ(x1062), x1059), x1065, x1066)) & (\/x1072,x1073,x1074,x1075,x1076,x1077,x1078,x1079,x1080,x1081,x1082:new_primPlusNat0(x1071, x1070)=Succ(Zero) & Succ(x1072)=x1071 & Succ(x1073)=x1070 & (\/x1074,x1075,x1076,x1077,x1078,x1079:new_primPlusNat0(x1072, x1073)=Succ(x1074) & Succ(x1075)=x1072 & Succ(x1075)=x1076 & new_primPlusNat0(x1076, x1073)=Succ(Zero) ==> new_pr2F34(x1075, Pos(x1073), x1077, x1078, x1079)_>=_new_pr2F31(new_primPlusNat0(Succ(x1075), x1073), x1077, new_primPlusNat0(Succ(x1075), x1073), x1078, x1079)) ==> new_pr2F34(x1072, Pos(Succ(x1073)), x1080, x1081, x1082)_>=_new_pr2F31(new_primPlusNat0(Succ(x1072), Succ(x1073)), x1080, new_primPlusNat0(Succ(x1072), Succ(x1073)), x1081, x1082)) ==> new_pr2F34(x264, Pos(Succ(x1059)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1059)), x266, new_primPlusNat0(Succ(x264), Succ(x1059)), x267, x268)) (9) (Succ(x1083)=Succ(Zero) & Succ(x264)=Succ(x1083) & Succ(x1059)=Zero & (\/x1061,x1062,x1063,x1064,x1065,x1066:new_primPlusNat0(x264, x1059)=Succ(x1061) & Succ(x1062)=x264 & Succ(x1062)=x1063 & new_primPlusNat0(x1063, x1059)=Succ(Zero) ==> new_pr2F34(x1062, Pos(x1059), x1064, x1065, x1066)_>=_new_pr2F31(new_primPlusNat0(Succ(x1062), x1059), x1064, new_primPlusNat0(Succ(x1062), x1059), x1065, x1066)) ==> new_pr2F34(x264, Pos(Succ(x1059)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1059)), x266, new_primPlusNat0(Succ(x264), Succ(x1059)), x267, x268)) (10) (Succ(x1084)=Succ(Zero) & Succ(x264)=Zero & Succ(x1059)=Succ(x1084) & (\/x1061,x1062,x1063,x1064,x1065,x1066:new_primPlusNat0(x264, x1059)=Succ(x1061) & Succ(x1062)=x264 & Succ(x1062)=x1063 & new_primPlusNat0(x1063, x1059)=Succ(Zero) ==> new_pr2F34(x1062, Pos(x1059), x1064, x1065, x1066)_>=_new_pr2F31(new_primPlusNat0(Succ(x1062), x1059), x1064, new_primPlusNat0(Succ(x1062), x1059), x1065, x1066)) ==> new_pr2F34(x264, Pos(Succ(x1059)), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Succ(x1059)), x266, new_primPlusNat0(Succ(x264), Succ(x1059)), x267, x268)) We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1058, x1085)=Succ(Zero) which results in the following new constraints: (11) (Succ(Succ(new_primPlusNat0(x1087, x1086)))=Succ(Zero) & Succ(x264)=Succ(x1087) & Zero=Succ(x1086) & (\/x1088,x1089,x1090,x1091:new_primPlusNat0(x1087, x1086)=Succ(Zero) & Succ(x1088)=x1087 & Zero=x1086 ==> new_pr2F34(x1088, Pos(Zero), x1089, x1090, x1091)_>=_new_pr2F31(new_primPlusNat0(Succ(x1088), Zero), x1089, new_primPlusNat0(Succ(x1088), Zero), x1090, x1091)) ==> new_pr2F34(x264, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Zero), x266, new_primPlusNat0(Succ(x264), Zero), x267, x268)) (12) (Succ(x1092)=Succ(Zero) & Succ(x264)=Succ(x1092) & Zero=Zero ==> new_pr2F34(x264, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Zero), x266, new_primPlusNat0(Succ(x264), Zero), x267, x268)) (13) (Succ(x1093)=Succ(Zero) & Succ(x264)=Zero & Zero=Succ(x1093) ==> new_pr2F34(x264, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(x264), Zero), x266, new_primPlusNat0(Succ(x264), Zero), x267, x268)) We solved constraint (11) using rules (I), (II).We simplified constraint (12) using rules (I), (II), (III) which results in the following new constraint: (14) (new_pr2F34(Zero, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x266, new_primPlusNat0(Succ(Zero), Zero), x267, x268)) We solved constraint (13) using rules (I), (II). For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x298), x299, Succ(Succ(x300)), x301, x302) -> new_pr2F0G12(x299, x301, Succ(x300), x300, x302), new_pr2F0G12(x303, x304, x305, Succ(Succ(x306)), x307) -> new_pr2F0G12(x303, x304, x305, x306, x307) which results in the following constraint: (1) (new_pr2F0G12(x299, x301, Succ(x300), x300, x302)=new_pr2F0G12(x303, x304, x305, Succ(Succ(x306)), x307) ==> new_pr2F31(Succ(x298), x299, Succ(Succ(x300)), x301, x302)_>=_new_pr2F0G12(x299, x301, Succ(x300), x300, x302)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x298), x299, Succ(Succ(Succ(Succ(x306)))), x301, x302)_>=_new_pr2F0G12(x299, x301, Succ(Succ(Succ(x306))), Succ(Succ(x306)), x302)) *We consider the chain new_pr2F31(Succ(x308), x309, Succ(Succ(x310)), x311, x312) -> new_pr2F0G12(x309, x311, Succ(x310), x310, x312), new_pr2F0G12(x313, x314, x315, Succ(Zero), x316) -> new_pr2F1(x313, x315, new_fromInt, x314, x316) which results in the following constraint: (1) (new_pr2F0G12(x309, x311, Succ(x310), x310, x312)=new_pr2F0G12(x313, x314, x315, Succ(Zero), x316) ==> new_pr2F31(Succ(x308), x309, Succ(Succ(x310)), x311, x312)_>=_new_pr2F0G12(x309, x311, Succ(x310), x310, x312)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x308), x309, Succ(Succ(Succ(Zero))), x311, x312)_>=_new_pr2F0G12(x309, x311, Succ(Succ(Zero)), Succ(Zero), x312)) *We consider the chain new_pr2F31(Succ(x332), x333, Succ(Succ(x334)), x335, x336) -> new_pr2F0G12(x333, x335, Succ(x334), x334, x336), new_pr2F0G12(x337, x338, x339, Zero, x340) -> new_pr2F0G13(new_sr8(x337, x338, x340), x337, new_primDivNatS1(Succ(x339)), new_primDivNatS1(Succ(x339)), x340) which results in the following constraint: (1) (new_pr2F0G12(x333, x335, Succ(x334), x334, x336)=new_pr2F0G12(x337, x338, x339, Zero, x340) ==> new_pr2F31(Succ(x332), x333, Succ(Succ(x334)), x335, x336)_>=_new_pr2F0G12(x333, x335, Succ(x334), x334, x336)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x332), x333, Succ(Succ(Zero)), x335, x336)_>=_new_pr2F0G12(x333, x335, Succ(Zero), Zero, x336)) For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) the following chains were created: *We consider the chain new_pr2F0G12(x405, x406, x407, Zero, x408) -> new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x407)), new_primDivNatS1(Succ(x407)), x408), new_pr2F0G13(x409, x410, x411, Succ(Zero), x412) -> new_pr2F2(x410, x411, new_fromInt, x409, x412) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x407)), new_primDivNatS1(Succ(x407)), x408)=new_pr2F0G13(x409, x410, x411, Succ(Zero), x412) ==> new_pr2F0G12(x405, x406, x407, Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x407)), new_primDivNatS1(Succ(x407)), x408)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x407)=x1094 & new_primDivNatS1(x1094)=Succ(Zero) ==> new_pr2F0G12(x405, x406, x407, Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x407)), new_primDivNatS1(Succ(x407)), x408)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1094)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1095)=Succ(Zero) & Succ(x407)=Succ(x1095) ==> new_pr2F0G12(x405, x406, x407, Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x407)), new_primDivNatS1(Succ(x407)), x408)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1095)=Succ(Zero) ==> new_pr2F0G12(x405, x406, x1095, Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(x1095)), new_primDivNatS1(Succ(x1095)), x408)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1095)=Succ(Zero) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1096))=Succ(Zero) ==> new_pr2F0G12(x405, x406, Succ(Succ(x1096)), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Succ(x1096)))), new_primDivNatS1(Succ(Succ(Succ(x1096)))), x408)) (6) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G12(x405, x406, Succ(Zero), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x408)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x405, x406, Succ(Succ(x1096)), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Succ(x1096)))), new_primDivNatS1(Succ(Succ(Succ(x1096)))), x408)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x405, x406, Succ(Zero), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x408)) *We consider the chain new_pr2F0G12(x421, x422, x423, Zero, x424) -> new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x423)), new_primDivNatS1(Succ(x423)), x424), new_pr2F0G13(x425, x426, x427, Succ(Succ(x428)), x429) -> new_pr2F0G14(x425, x426, x427, x428, x429) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x423)), new_primDivNatS1(Succ(x423)), x424)=new_pr2F0G13(x425, x426, x427, Succ(Succ(x428)), x429) ==> new_pr2F0G12(x421, x422, x423, Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x423)), new_primDivNatS1(Succ(x423)), x424)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x423)=x1097 & new_primDivNatS1(x1097)=Succ(Succ(x428)) ==> new_pr2F0G12(x421, x422, x423, Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x423)), new_primDivNatS1(Succ(x423)), x424)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1097)=Succ(Succ(x428)) which results in the following new constraint: (3) (new_primDivNatS01(x1098)=Succ(Succ(x428)) & Succ(x423)=Succ(x1098) ==> new_pr2F0G12(x421, x422, x423, Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x423)), new_primDivNatS1(Succ(x423)), x424)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1098)=Succ(Succ(x428)) ==> new_pr2F0G12(x421, x422, x1098, Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(x1098)), new_primDivNatS1(Succ(x1098)), x424)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1098)=Succ(Succ(x428)) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1099))=Succ(Succ(x428)) ==> new_pr2F0G12(x421, x422, Succ(Succ(x1099)), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Succ(x1099)))), new_primDivNatS1(Succ(Succ(Succ(x1099)))), x424)) (6) (Succ(new_primDivNatS2)=Succ(Succ(x428)) ==> new_pr2F0G12(x421, x422, Succ(Zero), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x424)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x421, x422, Succ(Succ(x1099)), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Succ(x1099)))), new_primDivNatS1(Succ(Succ(Succ(x1099)))), x424)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x421, x422, Succ(Zero), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x424)) *We consider the chain new_pr2F0G12(x442, x443, x444, Zero, x445) -> new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445), new_pr2F0G13(x446, x447, x448, Zero, x449) -> new_pr2F0G13(x446, new_sr10(x447, x449), new_primDivNatS1(x448), new_primDivNatS1(x448), x449) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445)=new_pr2F0G13(x446, x447, x448, Zero, x449) ==> new_pr2F0G12(x442, x443, x444, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x444)=x1100 & new_primDivNatS1(x1100)=Zero ==> new_pr2F0G12(x442, x443, x444, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1100)=Zero which results in the following new constraints: (3) (Zero=Zero & Succ(x444)=Zero ==> new_pr2F0G12(x442, x443, x444, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445)) (4) (new_primDivNatS01(x1101)=Zero & Succ(x444)=Succ(x1101) ==> new_pr2F0G12(x442, x443, x444, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x444)), new_primDivNatS1(Succ(x444)), x445)) We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III) which results in the following new constraint: (5) (new_primDivNatS01(x1101)=Zero ==> new_pr2F0G12(x442, x443, x1101, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(x1101)), new_primDivNatS1(Succ(x1101)), x445)) We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1101)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G12(x442, x443, Zero, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x445)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G12(x442, x443, Zero, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x445)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G13(x478, x479, x480, Succ(Zero), x481) -> new_pr2F2(x479, x480, new_fromInt, x478, x481), new_pr2F2(x482, x483, Pos(x484), x485, x486) -> new_pr2F31(new_primPlusNat0(x483, x484), new_sr11(x482, x486), new_primPlusNat0(x483, x484), x485, x486) which results in the following constraint: (1) (new_pr2F2(x479, x480, new_fromInt, x478, x481)=new_pr2F2(x482, x483, Pos(x484), x485, x486) ==> new_pr2F0G13(x478, x479, x480, Succ(Zero), x481)_>=_new_pr2F2(x479, x480, new_fromInt, x478, x481)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x484) ==> new_pr2F0G13(x478, x479, x480, Succ(Zero), x481)_>=_new_pr2F2(x479, x480, new_fromInt, x478, x481)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x484) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x484) ==> new_pr2F0G13(x478, x479, x480, Succ(Zero), x481)_>=_new_pr2F2(x479, x480, new_fromInt, x478, x481)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G13(x478, x479, x480, Succ(Zero), x481)_>=_new_pr2F2(x479, x480, new_fromInt, x478, x481)) For Pair new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) the following chains were created: *We consider the chain new_pr2F2(x531, x532, Pos(x533), x534, x535) -> new_pr2F31(new_primPlusNat0(x532, x533), new_sr11(x531, x535), new_primPlusNat0(x532, x533), x534, x535), new_pr2F31(Succ(x536), x537, Succ(Succ(x538)), x539, x540) -> new_pr2F0G12(x537, x539, Succ(x538), x538, x540) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x532, x533), new_sr11(x531, x535), new_primPlusNat0(x532, x533), x534, x535)=new_pr2F31(Succ(x536), x537, Succ(Succ(x538)), x539, x540) ==> new_pr2F2(x531, x532, Pos(x533), x534, x535)_>=_new_pr2F31(new_primPlusNat0(x532, x533), new_sr11(x531, x535), new_primPlusNat0(x532, x533), x534, x535)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x532, x533)=Succ(x536) & new_primPlusNat0(x532, x533)=Succ(Succ(x538)) ==> new_pr2F2(x531, x532, Pos(x533), x534, x535)_>=_new_pr2F31(new_primPlusNat0(x532, x533), new_sr11(x531, x535), new_primPlusNat0(x532, x533), x534, x535)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x532, x533)=Succ(x536) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1104, x1103)))=Succ(x536) & new_primPlusNat0(Succ(x1104), Succ(x1103))=Succ(Succ(x538)) & (\/x1105,x1106,x1107,x1108,x1109:new_primPlusNat0(x1104, x1103)=Succ(x1105) & new_primPlusNat0(x1104, x1103)=Succ(Succ(x1106)) ==> new_pr2F2(x1107, x1104, Pos(x1103), x1108, x1109)_>=_new_pr2F31(new_primPlusNat0(x1104, x1103), new_sr11(x1107, x1109), new_primPlusNat0(x1104, x1103), x1108, x1109)) ==> new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) (4) (Succ(x1110)=Succ(x536) & new_primPlusNat0(Succ(x1110), Zero)=Succ(Succ(x538)) ==> new_pr2F2(x531, Succ(x1110), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1110), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(x1110), Zero), x534, x535)) (5) (Succ(x1111)=Succ(x536) & new_primPlusNat0(Zero, Succ(x1111))=Succ(Succ(x538)) ==> new_pr2F2(x531, Zero, Pos(Succ(x1111)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1111)), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(x1111)), x534, x535)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1104)=x1112 & Succ(x1103)=x1113 & new_primPlusNat0(x1112, x1113)=Succ(Succ(x538)) & (\/x1105,x1106,x1107,x1108,x1109:new_primPlusNat0(x1104, x1103)=Succ(x1105) & new_primPlusNat0(x1104, x1103)=Succ(Succ(x1106)) ==> new_pr2F2(x1107, x1104, Pos(x1103), x1108, x1109)_>=_new_pr2F31(new_primPlusNat0(x1104, x1103), new_sr11(x1107, x1109), new_primPlusNat0(x1104, x1103), x1108, x1109)) ==> new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1110)=x1129 & Zero=x1130 & new_primPlusNat0(x1129, x1130)=Succ(Succ(x538)) ==> new_pr2F2(x531, Succ(x1110), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1110), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(x1110), Zero), x534, x535)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1140 & Succ(x1111)=x1141 & new_primPlusNat0(x1140, x1141)=Succ(Succ(x538)) ==> new_pr2F2(x531, Zero, Pos(Succ(x1111)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1111)), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(x1111)), x534, x535)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1112, x1113)=Succ(Succ(x538)) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1115, x1114)))=Succ(Succ(x538)) & Succ(x1104)=Succ(x1115) & Succ(x1103)=Succ(x1114) & (\/x1105,x1106,x1107,x1108,x1109:new_primPlusNat0(x1104, x1103)=Succ(x1105) & new_primPlusNat0(x1104, x1103)=Succ(Succ(x1106)) ==> new_pr2F2(x1107, x1104, Pos(x1103), x1108, x1109)_>=_new_pr2F31(new_primPlusNat0(x1104, x1103), new_sr11(x1107, x1109), new_primPlusNat0(x1104, x1103), x1108, x1109)) & (\/x1116,x1117,x1118,x1119,x1120,x1121,x1122,x1123,x1124,x1125,x1126:new_primPlusNat0(x1115, x1114)=Succ(Succ(x1116)) & Succ(x1117)=x1115 & Succ(x1118)=x1114 & (\/x1119,x1120,x1121,x1122,x1123:new_primPlusNat0(x1117, x1118)=Succ(x1119) & new_primPlusNat0(x1117, x1118)=Succ(Succ(x1120)) ==> new_pr2F2(x1121, x1117, Pos(x1118), x1122, x1123)_>=_new_pr2F31(new_primPlusNat0(x1117, x1118), new_sr11(x1121, x1123), new_primPlusNat0(x1117, x1118), x1122, x1123)) ==> new_pr2F2(x1124, Succ(x1117), Pos(Succ(x1118)), x1125, x1126)_>=_new_pr2F31(new_primPlusNat0(Succ(x1117), Succ(x1118)), new_sr11(x1124, x1126), new_primPlusNat0(Succ(x1117), Succ(x1118)), x1125, x1126)) ==> new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) (10) (Succ(x1127)=Succ(Succ(x538)) & Succ(x1104)=Succ(x1127) & Succ(x1103)=Zero & (\/x1105,x1106,x1107,x1108,x1109:new_primPlusNat0(x1104, x1103)=Succ(x1105) & new_primPlusNat0(x1104, x1103)=Succ(Succ(x1106)) ==> new_pr2F2(x1107, x1104, Pos(x1103), x1108, x1109)_>=_new_pr2F31(new_primPlusNat0(x1104, x1103), new_sr11(x1107, x1109), new_primPlusNat0(x1104, x1103), x1108, x1109)) ==> new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) (11) (Succ(x1128)=Succ(Succ(x538)) & Succ(x1104)=Zero & Succ(x1103)=Succ(x1128) & (\/x1105,x1106,x1107,x1108,x1109:new_primPlusNat0(x1104, x1103)=Succ(x1105) & new_primPlusNat0(x1104, x1103)=Succ(Succ(x1106)) ==> new_pr2F2(x1107, x1104, Pos(x1103), x1108, x1109)_>=_new_pr2F31(new_primPlusNat0(x1104, x1103), new_sr11(x1107, x1109), new_primPlusNat0(x1104, x1103), x1108, x1109)) ==> new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: (12) (new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1129, x1130)=Succ(Succ(x538)) which results in the following new constraints: (13) (Succ(Succ(new_primPlusNat0(x1132, x1131)))=Succ(Succ(x538)) & Succ(x1110)=Succ(x1132) & Zero=Succ(x1131) & (\/x1133,x1134,x1135,x1136,x1137:new_primPlusNat0(x1132, x1131)=Succ(Succ(x1133)) & Succ(x1134)=x1132 & Zero=x1131 ==> new_pr2F2(x1135, Succ(x1134), Pos(Zero), x1136, x1137)_>=_new_pr2F31(new_primPlusNat0(Succ(x1134), Zero), new_sr11(x1135, x1137), new_primPlusNat0(Succ(x1134), Zero), x1136, x1137)) ==> new_pr2F2(x531, Succ(x1110), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1110), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(x1110), Zero), x534, x535)) (14) (Succ(x1138)=Succ(Succ(x538)) & Succ(x1110)=Succ(x1138) & Zero=Zero ==> new_pr2F2(x531, Succ(x1110), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1110), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(x1110), Zero), x534, x535)) (15) (Succ(x1139)=Succ(Succ(x538)) & Succ(x1110)=Zero & Zero=Succ(x1139) ==> new_pr2F2(x531, Succ(x1110), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1110), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(x1110), Zero), x534, x535)) We solved constraint (13) using rules (I), (II).We simplified constraint (14) using rules (I), (II), (III) which results in the following new constraint: (16) (new_pr2F2(x531, Succ(Succ(x538)), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x538)), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(Succ(x538)), Zero), x534, x535)) We solved constraint (15) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1140, x1141)=Succ(Succ(x538)) which results in the following new constraints: (17) (Succ(Succ(new_primPlusNat0(x1143, x1142)))=Succ(Succ(x538)) & Zero=Succ(x1143) & Succ(x1111)=Succ(x1142) & (\/x1144,x1145,x1146,x1147,x1148:new_primPlusNat0(x1143, x1142)=Succ(Succ(x1144)) & Zero=x1143 & Succ(x1145)=x1142 ==> new_pr2F2(x1146, Zero, Pos(Succ(x1145)), x1147, x1148)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1145)), new_sr11(x1146, x1148), new_primPlusNat0(Zero, Succ(x1145)), x1147, x1148)) ==> new_pr2F2(x531, Zero, Pos(Succ(x1111)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1111)), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(x1111)), x534, x535)) (18) (Succ(x1149)=Succ(Succ(x538)) & Zero=Succ(x1149) & Succ(x1111)=Zero ==> new_pr2F2(x531, Zero, Pos(Succ(x1111)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1111)), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(x1111)), x534, x535)) (19) (Succ(x1150)=Succ(Succ(x538)) & Zero=Zero & Succ(x1111)=Succ(x1150) ==> new_pr2F2(x531, Zero, Pos(Succ(x1111)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1111)), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(x1111)), x534, x535)) We solved constraint (17) using rules (I), (II).We solved constraint (18) using rules (I), (II).We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint: (20) (new_pr2F2(x531, Zero, Pos(Succ(Succ(x538))), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x538))), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(Succ(x538))), x534, x535)) *We consider the chain new_pr2F2(x556, x557, Pos(x558), x559, x560) -> new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560), new_pr2F31(Succ(x561), x562, Succ(Zero), x563, x564) -> new_pr2F1(x562, Zero, new_fromInt, x563, x564) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)=new_pr2F31(Succ(x561), x562, Succ(Zero), x563, x564) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x557, x558)=Succ(x561) & new_primPlusNat0(x557, x558)=Succ(Zero) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x557, x558)=Succ(x561) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1152, x1151)))=Succ(x561) & new_primPlusNat0(Succ(x1152), Succ(x1151))=Succ(Zero) & (\/x1153,x1154,x1155,x1156:new_primPlusNat0(x1152, x1151)=Succ(x1153) & new_primPlusNat0(x1152, x1151)=Succ(Zero) ==> new_pr2F2(x1154, x1152, Pos(x1151), x1155, x1156)_>=_new_pr2F31(new_primPlusNat0(x1152, x1151), new_sr11(x1154, x1156), new_primPlusNat0(x1152, x1151), x1155, x1156)) ==> new_pr2F2(x556, Succ(x1152), Pos(Succ(x1151)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1152), Succ(x1151)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1152), Succ(x1151)), x559, x560)) (4) (Succ(x1157)=Succ(x561) & new_primPlusNat0(Succ(x1157), Zero)=Succ(Zero) ==> new_pr2F2(x556, Succ(x1157), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1157), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1157), Zero), x559, x560)) (5) (Succ(x1158)=Succ(x561) & new_primPlusNat0(Zero, Succ(x1158))=Succ(Zero) ==> new_pr2F2(x556, Zero, Pos(Succ(x1158)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1158)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1158)), x559, x560)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1152)=x1159 & Succ(x1151)=x1160 & new_primPlusNat0(x1159, x1160)=Succ(Zero) & (\/x1153,x1154,x1155,x1156:new_primPlusNat0(x1152, x1151)=Succ(x1153) & new_primPlusNat0(x1152, x1151)=Succ(Zero) ==> new_pr2F2(x1154, x1152, Pos(x1151), x1155, x1156)_>=_new_pr2F31(new_primPlusNat0(x1152, x1151), new_sr11(x1154, x1156), new_primPlusNat0(x1152, x1151), x1155, x1156)) ==> new_pr2F2(x556, Succ(x1152), Pos(Succ(x1151)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1152), Succ(x1151)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1152), Succ(x1151)), x559, x560)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1157)=x1174 & Zero=x1175 & new_primPlusNat0(x1174, x1175)=Succ(Zero) ==> new_pr2F2(x556, Succ(x1157), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1157), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1157), Zero), x559, x560)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1184 & Succ(x1158)=x1185 & new_primPlusNat0(x1184, x1185)=Succ(Zero) ==> new_pr2F2(x556, Zero, Pos(Succ(x1158)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1158)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1158)), x559, x560)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1159, x1160)=Succ(Zero) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1162, x1161)))=Succ(Zero) & Succ(x1152)=Succ(x1162) & Succ(x1151)=Succ(x1161) & (\/x1153,x1154,x1155,x1156:new_primPlusNat0(x1152, x1151)=Succ(x1153) & new_primPlusNat0(x1152, x1151)=Succ(Zero) ==> new_pr2F2(x1154, x1152, Pos(x1151), x1155, x1156)_>=_new_pr2F31(new_primPlusNat0(x1152, x1151), new_sr11(x1154, x1156), new_primPlusNat0(x1152, x1151), x1155, x1156)) & (\/x1163,x1164,x1165,x1166,x1167,x1168,x1169,x1170,x1171:new_primPlusNat0(x1162, x1161)=Succ(Zero) & Succ(x1163)=x1162 & Succ(x1164)=x1161 & (\/x1165,x1166,x1167,x1168:new_primPlusNat0(x1163, x1164)=Succ(x1165) & new_primPlusNat0(x1163, x1164)=Succ(Zero) ==> new_pr2F2(x1166, x1163, Pos(x1164), x1167, x1168)_>=_new_pr2F31(new_primPlusNat0(x1163, x1164), new_sr11(x1166, x1168), new_primPlusNat0(x1163, x1164), x1167, x1168)) ==> new_pr2F2(x1169, Succ(x1163), Pos(Succ(x1164)), x1170, x1171)_>=_new_pr2F31(new_primPlusNat0(Succ(x1163), Succ(x1164)), new_sr11(x1169, x1171), new_primPlusNat0(Succ(x1163), Succ(x1164)), x1170, x1171)) ==> new_pr2F2(x556, Succ(x1152), Pos(Succ(x1151)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1152), Succ(x1151)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1152), Succ(x1151)), x559, x560)) (10) (Succ(x1172)=Succ(Zero) & Succ(x1152)=Succ(x1172) & Succ(x1151)=Zero & (\/x1153,x1154,x1155,x1156:new_primPlusNat0(x1152, x1151)=Succ(x1153) & new_primPlusNat0(x1152, x1151)=Succ(Zero) ==> new_pr2F2(x1154, x1152, Pos(x1151), x1155, x1156)_>=_new_pr2F31(new_primPlusNat0(x1152, x1151), new_sr11(x1154, x1156), new_primPlusNat0(x1152, x1151), x1155, x1156)) ==> new_pr2F2(x556, Succ(x1152), Pos(Succ(x1151)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1152), Succ(x1151)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1152), Succ(x1151)), x559, x560)) (11) (Succ(x1173)=Succ(Zero) & Succ(x1152)=Zero & Succ(x1151)=Succ(x1173) & (\/x1153,x1154,x1155,x1156:new_primPlusNat0(x1152, x1151)=Succ(x1153) & new_primPlusNat0(x1152, x1151)=Succ(Zero) ==> new_pr2F2(x1154, x1152, Pos(x1151), x1155, x1156)_>=_new_pr2F31(new_primPlusNat0(x1152, x1151), new_sr11(x1154, x1156), new_primPlusNat0(x1152, x1151), x1155, x1156)) ==> new_pr2F2(x556, Succ(x1152), Pos(Succ(x1151)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1152), Succ(x1151)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1152), Succ(x1151)), x559, x560)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1174, x1175)=Succ(Zero) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1177, x1176)))=Succ(Zero) & Succ(x1157)=Succ(x1177) & Zero=Succ(x1176) & (\/x1178,x1179,x1180,x1181:new_primPlusNat0(x1177, x1176)=Succ(Zero) & Succ(x1178)=x1177 & Zero=x1176 ==> new_pr2F2(x1179, Succ(x1178), Pos(Zero), x1180, x1181)_>=_new_pr2F31(new_primPlusNat0(Succ(x1178), Zero), new_sr11(x1179, x1181), new_primPlusNat0(Succ(x1178), Zero), x1180, x1181)) ==> new_pr2F2(x556, Succ(x1157), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1157), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1157), Zero), x559, x560)) (13) (Succ(x1182)=Succ(Zero) & Succ(x1157)=Succ(x1182) & Zero=Zero ==> new_pr2F2(x556, Succ(x1157), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1157), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1157), Zero), x559, x560)) (14) (Succ(x1183)=Succ(Zero) & Succ(x1157)=Zero & Zero=Succ(x1183) ==> new_pr2F2(x556, Succ(x1157), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1157), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1157), Zero), x559, x560)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F2(x556, Succ(Zero), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Zero), Zero), x559, x560)) We solved constraint (14) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1184, x1185)=Succ(Zero) which results in the following new constraints: (16) (Succ(Succ(new_primPlusNat0(x1187, x1186)))=Succ(Zero) & Zero=Succ(x1187) & Succ(x1158)=Succ(x1186) & (\/x1188,x1189,x1190,x1191:new_primPlusNat0(x1187, x1186)=Succ(Zero) & Zero=x1187 & Succ(x1188)=x1186 ==> new_pr2F2(x1189, Zero, Pos(Succ(x1188)), x1190, x1191)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1188)), new_sr11(x1189, x1191), new_primPlusNat0(Zero, Succ(x1188)), x1190, x1191)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1158)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1158)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1158)), x559, x560)) (17) (Succ(x1192)=Succ(Zero) & Zero=Succ(x1192) & Succ(x1158)=Zero ==> new_pr2F2(x556, Zero, Pos(Succ(x1158)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1158)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1158)), x559, x560)) (18) (Succ(x1193)=Succ(Zero) & Zero=Zero & Succ(x1158)=Succ(x1193) ==> new_pr2F2(x556, Zero, Pos(Succ(x1158)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1158)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1158)), x559, x560)) We solved constraint (16) using rules (I), (II).We solved constraint (17) using rules (I), (II).We simplified constraint (18) using rules (I), (II), (III) which results in the following new constraint: (19) (new_pr2F2(x556, Zero, Pos(Succ(Zero)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Zero)), x559, x560)) For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x598), x599, Succ(Zero), x600, x601) -> new_pr2F1(x599, Zero, new_fromInt, x600, x601), new_pr2F1(x602, x603, x604, x605, x606) -> new_pr2F34(x603, x604, x602, new_sr9(x602, x605, x606), x606) which results in the following constraint: (1) (new_pr2F1(x599, Zero, new_fromInt, x600, x601)=new_pr2F1(x602, x603, x604, x605, x606) ==> new_pr2F31(Succ(x598), x599, Succ(Zero), x600, x601)_>=_new_pr2F1(x599, Zero, new_fromInt, x600, x601)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x598), x599, Succ(Zero), x600, x601)_>=_new_pr2F1(x599, Zero, new_fromInt, x600, x601)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) the following chains were created: *We consider the chain new_pr2F0G13(x701, x702, x703, Succ(Succ(x704)), x705) -> new_pr2F0G14(x701, x702, x703, x704, x705), new_pr2F0G14(x706, x707, x708, Succ(Zero), x709) -> new_pr2F2(x707, x708, new_fromInt, x706, x709) which results in the following constraint: (1) (new_pr2F0G14(x701, x702, x703, x704, x705)=new_pr2F0G14(x706, x707, x708, Succ(Zero), x709) ==> new_pr2F0G13(x701, x702, x703, Succ(Succ(x704)), x705)_>=_new_pr2F0G14(x701, x702, x703, x704, x705)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x701, x702, x703, Succ(Succ(Succ(Zero))), x705)_>=_new_pr2F0G14(x701, x702, x703, Succ(Zero), x705)) *We consider the chain new_pr2F0G13(x710, x711, x712, Succ(Succ(x713)), x714) -> new_pr2F0G14(x710, x711, x712, x713, x714), new_pr2F0G14(x715, x716, x717, Succ(Succ(x718)), x719) -> new_pr2F0G14(x715, x716, x717, x718, x719) which results in the following constraint: (1) (new_pr2F0G14(x710, x711, x712, x713, x714)=new_pr2F0G14(x715, x716, x717, Succ(Succ(x718)), x719) ==> new_pr2F0G13(x710, x711, x712, Succ(Succ(x713)), x714)_>=_new_pr2F0G14(x710, x711, x712, x713, x714)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x710, x711, x712, Succ(Succ(Succ(Succ(x718)))), x714)_>=_new_pr2F0G14(x710, x711, x712, Succ(Succ(x718)), x714)) *We consider the chain new_pr2F0G13(x720, x721, x722, Succ(Succ(x723)), x724) -> new_pr2F0G14(x720, x721, x722, x723, x724), new_pr2F0G14(x725, x726, x727, Zero, x728) -> new_pr2F0G13(x725, new_sr10(x726, x728), new_primDivNatS1(x727), new_primDivNatS1(x727), x728) which results in the following constraint: (1) (new_pr2F0G14(x720, x721, x722, x723, x724)=new_pr2F0G14(x725, x726, x727, Zero, x728) ==> new_pr2F0G13(x720, x721, x722, Succ(Succ(x723)), x724)_>=_new_pr2F0G14(x720, x721, x722, x723, x724)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x720, x721, x722, Succ(Succ(Zero)), x724)_>=_new_pr2F0G14(x720, x721, x722, Zero, x724)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G14(x762, x763, x764, Succ(Zero), x765) -> new_pr2F2(x763, x764, new_fromInt, x762, x765), new_pr2F2(x766, x767, Pos(x768), x769, x770) -> new_pr2F31(new_primPlusNat0(x767, x768), new_sr11(x766, x770), new_primPlusNat0(x767, x768), x769, x770) which results in the following constraint: (1) (new_pr2F2(x763, x764, new_fromInt, x762, x765)=new_pr2F2(x766, x767, Pos(x768), x769, x770) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x768) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x768) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x768) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) the following chains were created: *We consider the chain new_pr2F0G14(x845, x846, x847, Succ(Succ(x848)), x849) -> new_pr2F0G14(x845, x846, x847, x848, x849), new_pr2F0G14(x850, x851, x852, Succ(Zero), x853) -> new_pr2F2(x851, x852, new_fromInt, x850, x853) which results in the following constraint: (1) (new_pr2F0G14(x845, x846, x847, x848, x849)=new_pr2F0G14(x850, x851, x852, Succ(Zero), x853) ==> new_pr2F0G14(x845, x846, x847, Succ(Succ(x848)), x849)_>=_new_pr2F0G14(x845, x846, x847, x848, x849)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x845, x846, x847, Succ(Succ(Succ(Zero))), x849)_>=_new_pr2F0G14(x845, x846, x847, Succ(Zero), x849)) *We consider the chain new_pr2F0G14(x854, x855, x856, Succ(Succ(x857)), x858) -> new_pr2F0G14(x854, x855, x856, x857, x858), new_pr2F0G14(x859, x860, x861, Succ(Succ(x862)), x863) -> new_pr2F0G14(x859, x860, x861, x862, x863) which results in the following constraint: (1) (new_pr2F0G14(x854, x855, x856, x857, x858)=new_pr2F0G14(x859, x860, x861, Succ(Succ(x862)), x863) ==> new_pr2F0G14(x854, x855, x856, Succ(Succ(x857)), x858)_>=_new_pr2F0G14(x854, x855, x856, x857, x858)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x854, x855, x856, Succ(Succ(Succ(Succ(x862)))), x858)_>=_new_pr2F0G14(x854, x855, x856, Succ(Succ(x862)), x858)) *We consider the chain new_pr2F0G14(x864, x865, x866, Succ(Succ(x867)), x868) -> new_pr2F0G14(x864, x865, x866, x867, x868), new_pr2F0G14(x869, x870, x871, Zero, x872) -> new_pr2F0G13(x869, new_sr10(x870, x872), new_primDivNatS1(x871), new_primDivNatS1(x871), x872) which results in the following constraint: (1) (new_pr2F0G14(x864, x865, x866, x867, x868)=new_pr2F0G14(x869, x870, x871, Zero, x872) ==> new_pr2F0G14(x864, x865, x866, Succ(Succ(x867)), x868)_>=_new_pr2F0G14(x864, x865, x866, x867, x868)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x864, x865, x866, Succ(Succ(Zero)), x868)_>=_new_pr2F0G14(x864, x865, x866, Zero, x868)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G14(x902, x903, x904, Zero, x905) -> new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(x904), new_primDivNatS1(x904), x905), new_pr2F0G13(x906, x907, x908, Succ(Zero), x909) -> new_pr2F2(x907, x908, new_fromInt, x906, x909) which results in the following constraint: (1) (new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(x904), new_primDivNatS1(x904), x905)=new_pr2F0G13(x906, x907, x908, Succ(Zero), x909) ==> new_pr2F0G14(x902, x903, x904, Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(x904), new_primDivNatS1(x904), x905)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x904)=Succ(Zero) ==> new_pr2F0G14(x902, x903, x904, Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(x904), new_primDivNatS1(x904), x905)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x904)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1194)=Succ(Zero) ==> new_pr2F0G14(x902, x903, Succ(x1194), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(x1194)), new_primDivNatS1(Succ(x1194)), x905)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1194)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1195))=Succ(Zero) ==> new_pr2F0G14(x902, x903, Succ(Succ(Succ(x1195))), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Succ(x1195)))), new_primDivNatS1(Succ(Succ(Succ(x1195)))), x905)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G14(x902, x903, Succ(Succ(Zero)), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x905)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x902, x903, Succ(Succ(Succ(x1195))), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Succ(x1195)))), new_primDivNatS1(Succ(Succ(Succ(x1195)))), x905)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x902, x903, Succ(Succ(Zero)), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x905)) *We consider the chain new_pr2F0G14(x918, x919, x920, Zero, x921) -> new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(x920), new_primDivNatS1(x920), x921), new_pr2F0G13(x922, x923, x924, Succ(Succ(x925)), x926) -> new_pr2F0G14(x922, x923, x924, x925, x926) which results in the following constraint: (1) (new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(x920), new_primDivNatS1(x920), x921)=new_pr2F0G13(x922, x923, x924, Succ(Succ(x925)), x926) ==> new_pr2F0G14(x918, x919, x920, Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(x920), new_primDivNatS1(x920), x921)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x920)=Succ(Succ(x925)) ==> new_pr2F0G14(x918, x919, x920, Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(x920), new_primDivNatS1(x920), x921)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x920)=Succ(Succ(x925)) which results in the following new constraint: (3) (new_primDivNatS01(x1196)=Succ(Succ(x925)) ==> new_pr2F0G14(x918, x919, Succ(x1196), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(x1196)), new_primDivNatS1(Succ(x1196)), x921)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1196)=Succ(Succ(x925)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1197))=Succ(Succ(x925)) ==> new_pr2F0G14(x918, x919, Succ(Succ(Succ(x1197))), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Succ(x1197)))), new_primDivNatS1(Succ(Succ(Succ(x1197)))), x921)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x925)) ==> new_pr2F0G14(x918, x919, Succ(Succ(Zero)), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x921)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x918, x919, Succ(Succ(Succ(x1197))), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Succ(x1197)))), new_primDivNatS1(Succ(Succ(Succ(x1197)))), x921)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x918, x919, Succ(Succ(Zero)), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x921)) *We consider the chain new_pr2F0G14(x939, x940, x941, Zero, x942) -> new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(x941), new_primDivNatS1(x941), x942), new_pr2F0G13(x943, x944, x945, Zero, x946) -> new_pr2F0G13(x943, new_sr10(x944, x946), new_primDivNatS1(x945), new_primDivNatS1(x945), x946) which results in the following constraint: (1) (new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(x941), new_primDivNatS1(x941), x942)=new_pr2F0G13(x943, x944, x945, Zero, x946) ==> new_pr2F0G14(x939, x940, x941, Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(x941), new_primDivNatS1(x941), x942)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x941)=Zero ==> new_pr2F0G14(x939, x940, x941, Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(x941), new_primDivNatS1(x941), x942)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x941)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G14(x939, x940, Zero, Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x942)) (4) (new_primDivNatS01(x1198)=Zero ==> new_pr2F0G14(x939, x940, Succ(x1198), Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Succ(x1198)), new_primDivNatS1(Succ(x1198)), x942)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G14(x939, x940, Zero, Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x942)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1198)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G14(x939, x940, Succ(Zero), Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x942)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G14(x939, x940, Succ(Zero), Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x942)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G13(x971, x972, x973, Zero, x974) -> new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(x973), new_primDivNatS1(x973), x974), new_pr2F0G13(x975, x976, x977, Succ(Zero), x978) -> new_pr2F2(x976, x977, new_fromInt, x975, x978) which results in the following constraint: (1) (new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(x973), new_primDivNatS1(x973), x974)=new_pr2F0G13(x975, x976, x977, Succ(Zero), x978) ==> new_pr2F0G13(x971, x972, x973, Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(x973), new_primDivNatS1(x973), x974)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x973)=Succ(Zero) ==> new_pr2F0G13(x971, x972, x973, Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(x973), new_primDivNatS1(x973), x974)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x973)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1200)=Succ(Zero) ==> new_pr2F0G13(x971, x972, Succ(x1200), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(x1200)), new_primDivNatS1(Succ(x1200)), x974)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1200)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1201))=Succ(Zero) ==> new_pr2F0G13(x971, x972, Succ(Succ(Succ(x1201))), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Succ(x1201)))), new_primDivNatS1(Succ(Succ(Succ(x1201)))), x974)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G13(x971, x972, Succ(Succ(Zero)), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x974)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x971, x972, Succ(Succ(Succ(x1201))), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Succ(x1201)))), new_primDivNatS1(Succ(Succ(Succ(x1201)))), x974)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x971, x972, Succ(Succ(Zero)), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x974)) *We consider the chain new_pr2F0G13(x987, x988, x989, Zero, x990) -> new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(x989), new_primDivNatS1(x989), x990), new_pr2F0G13(x991, x992, x993, Succ(Succ(x994)), x995) -> new_pr2F0G14(x991, x992, x993, x994, x995) which results in the following constraint: (1) (new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(x989), new_primDivNatS1(x989), x990)=new_pr2F0G13(x991, x992, x993, Succ(Succ(x994)), x995) ==> new_pr2F0G13(x987, x988, x989, Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(x989), new_primDivNatS1(x989), x990)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x989)=Succ(Succ(x994)) ==> new_pr2F0G13(x987, x988, x989, Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(x989), new_primDivNatS1(x989), x990)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x989)=Succ(Succ(x994)) which results in the following new constraint: (3) (new_primDivNatS01(x1202)=Succ(Succ(x994)) ==> new_pr2F0G13(x987, x988, Succ(x1202), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(x1202)), new_primDivNatS1(Succ(x1202)), x990)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1202)=Succ(Succ(x994)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1203))=Succ(Succ(x994)) ==> new_pr2F0G13(x987, x988, Succ(Succ(Succ(x1203))), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Succ(x1203)))), new_primDivNatS1(Succ(Succ(Succ(x1203)))), x990)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x994)) ==> new_pr2F0G13(x987, x988, Succ(Succ(Zero)), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x990)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x987, x988, Succ(Succ(Succ(x1203))), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Succ(x1203)))), new_primDivNatS1(Succ(Succ(Succ(x1203)))), x990)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x987, x988, Succ(Succ(Zero)), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x990)) *We consider the chain new_pr2F0G13(x1008, x1009, x1010, Zero, x1011) -> new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(x1010), new_primDivNatS1(x1010), x1011), new_pr2F0G13(x1012, x1013, x1014, Zero, x1015) -> new_pr2F0G13(x1012, new_sr10(x1013, x1015), new_primDivNatS1(x1014), new_primDivNatS1(x1014), x1015) which results in the following constraint: (1) (new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(x1010), new_primDivNatS1(x1010), x1011)=new_pr2F0G13(x1012, x1013, x1014, Zero, x1015) ==> new_pr2F0G13(x1008, x1009, x1010, Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(x1010), new_primDivNatS1(x1010), x1011)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1010)=Zero ==> new_pr2F0G13(x1008, x1009, x1010, Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(x1010), new_primDivNatS1(x1010), x1011)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1010)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G13(x1008, x1009, Zero, Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1011)) (4) (new_primDivNatS01(x1204)=Zero ==> new_pr2F0G13(x1008, x1009, Succ(x1204), Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Succ(x1204)), new_primDivNatS1(Succ(x1204)), x1011)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G13(x1008, x1009, Zero, Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1011)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1204)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G13(x1008, x1009, Succ(Zero), Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1011)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G13(x1008, x1009, Succ(Zero), Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1011)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) *(new_pr2F0G12(x0, x1, x2, Succ(Succ(Succ(Succ(x8)))), x4)_>=_new_pr2F0G12(x0, x1, x2, Succ(Succ(x8)), x4)) *(new_pr2F0G12(x10, x11, x12, Succ(Succ(Succ(Zero))), x14)_>=_new_pr2F0G12(x10, x11, x12, Succ(Zero), x14)) *(new_pr2F0G12(x34, x35, x36, Succ(Succ(Zero)), x38)_>=_new_pr2F0G12(x34, x35, x36, Zero, x38)) *new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) *(new_pr2F0G12(x91, x92, x93, Succ(Zero), x94)_>=_new_pr2F1(x91, x93, new_fromInt, x92, x94)) *new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) *(new_pr2F1(x159, x160, Pos(x165), x162, x163)_>=_new_pr2F34(x160, Pos(x165), x159, new_sr9(x159, x162, x163), x163)) *new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) *(new_pr2F34(Succ(x246), Pos(Zero), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x246)), Zero), x241, new_primPlusNat0(Succ(Succ(x246)), Zero), x242, x243)) *(new_pr2F34(x239, Pos(Succ(x1018)), x241, x242, x243)_>=_new_pr2F31(new_primPlusNat0(Succ(x239), Succ(x1018)), x241, new_primPlusNat0(Succ(x239), Succ(x1018)), x242, x243)) *(new_pr2F34(Zero, Pos(Zero), x266, x267, x268)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x266, new_primPlusNat0(Succ(Zero), Zero), x267, x268)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) *(new_pr2F31(Succ(x298), x299, Succ(Succ(Succ(Succ(x306)))), x301, x302)_>=_new_pr2F0G12(x299, x301, Succ(Succ(Succ(x306))), Succ(Succ(x306)), x302)) *(new_pr2F31(Succ(x308), x309, Succ(Succ(Succ(Zero))), x311, x312)_>=_new_pr2F0G12(x309, x311, Succ(Succ(Zero)), Succ(Zero), x312)) *(new_pr2F31(Succ(x332), x333, Succ(Succ(Zero)), x335, x336)_>=_new_pr2F0G12(x333, x335, Succ(Zero), Zero, x336)) *new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) *(new_pr2F0G12(x405, x406, Succ(Succ(x1096)), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Succ(x1096)))), new_primDivNatS1(Succ(Succ(Succ(x1096)))), x408)) *(new_pr2F0G12(x405, x406, Succ(Zero), Zero, x408)_>=_new_pr2F0G13(new_sr8(x405, x406, x408), x405, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x408)) *(new_pr2F0G12(x421, x422, Succ(Succ(x1099)), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Succ(x1099)))), new_primDivNatS1(Succ(Succ(Succ(x1099)))), x424)) *(new_pr2F0G12(x421, x422, Succ(Zero), Zero, x424)_>=_new_pr2F0G13(new_sr8(x421, x422, x424), x421, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x424)) *(new_pr2F0G12(x442, x443, Zero, Zero, x445)_>=_new_pr2F0G13(new_sr8(x442, x443, x445), x442, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x445)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G13(x478, x479, x480, Succ(Zero), x481)_>=_new_pr2F2(x479, x480, new_fromInt, x478, x481)) *new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) *(new_pr2F2(x531, Succ(Succ(x538)), Pos(Zero), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x538)), Zero), new_sr11(x531, x535), new_primPlusNat0(Succ(Succ(x538)), Zero), x534, x535)) *(new_pr2F2(x531, Zero, Pos(Succ(Succ(x538))), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x538))), new_sr11(x531, x535), new_primPlusNat0(Zero, Succ(Succ(x538))), x534, x535)) *(new_pr2F2(x531, Succ(x1104), Pos(Succ(x1103)), x534, x535)_>=_new_pr2F31(new_primPlusNat0(Succ(x1104), Succ(x1103)), new_sr11(x531, x535), new_primPlusNat0(Succ(x1104), Succ(x1103)), x534, x535)) *(new_pr2F2(x556, Succ(Zero), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Zero), Zero), x559, x560)) *(new_pr2F2(x556, Zero, Pos(Succ(Zero)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Zero)), x559, x560)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) *(new_pr2F31(Succ(x598), x599, Succ(Zero), x600, x601)_>=_new_pr2F1(x599, Zero, new_fromInt, x600, x601)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) *(new_pr2F0G13(x701, x702, x703, Succ(Succ(Succ(Zero))), x705)_>=_new_pr2F0G14(x701, x702, x703, Succ(Zero), x705)) *(new_pr2F0G13(x710, x711, x712, Succ(Succ(Succ(Succ(x718)))), x714)_>=_new_pr2F0G14(x710, x711, x712, Succ(Succ(x718)), x714)) *(new_pr2F0G13(x720, x721, x722, Succ(Succ(Zero)), x724)_>=_new_pr2F0G14(x720, x721, x722, Zero, x724)) *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) *(new_pr2F0G14(x845, x846, x847, Succ(Succ(Succ(Zero))), x849)_>=_new_pr2F0G14(x845, x846, x847, Succ(Zero), x849)) *(new_pr2F0G14(x854, x855, x856, Succ(Succ(Succ(Succ(x862)))), x858)_>=_new_pr2F0G14(x854, x855, x856, Succ(Succ(x862)), x858)) *(new_pr2F0G14(x864, x865, x866, Succ(Succ(Zero)), x868)_>=_new_pr2F0G14(x864, x865, x866, Zero, x868)) *new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G14(x902, x903, Succ(Succ(Succ(x1195))), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Succ(x1195)))), new_primDivNatS1(Succ(Succ(Succ(x1195)))), x905)) *(new_pr2F0G14(x902, x903, Succ(Succ(Zero)), Zero, x905)_>=_new_pr2F0G13(x902, new_sr10(x903, x905), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x905)) *(new_pr2F0G14(x918, x919, Succ(Succ(Succ(x1197))), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Succ(x1197)))), new_primDivNatS1(Succ(Succ(Succ(x1197)))), x921)) *(new_pr2F0G14(x918, x919, Succ(Succ(Zero)), Zero, x921)_>=_new_pr2F0G13(x918, new_sr10(x919, x921), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x921)) *(new_pr2F0G14(x939, x940, Zero, Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x942)) *(new_pr2F0G14(x939, x940, Succ(Zero), Zero, x942)_>=_new_pr2F0G13(x939, new_sr10(x940, x942), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x942)) *new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G13(x971, x972, Succ(Succ(Succ(x1201))), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Succ(x1201)))), new_primDivNatS1(Succ(Succ(Succ(x1201)))), x974)) *(new_pr2F0G13(x971, x972, Succ(Succ(Zero)), Zero, x974)_>=_new_pr2F0G13(x971, new_sr10(x972, x974), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x974)) *(new_pr2F0G13(x987, x988, Succ(Succ(Succ(x1203))), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Succ(x1203)))), new_primDivNatS1(Succ(Succ(Succ(x1203)))), x990)) *(new_pr2F0G13(x987, x988, Succ(Succ(Zero)), Zero, x990)_>=_new_pr2F0G13(x987, new_sr10(x988, x990), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x990)) *(new_pr2F0G13(x1008, x1009, Zero, Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1011)) *(new_pr2F0G13(x1008, x1009, Succ(Zero), Zero, x1011)_>=_new_pr2F0G13(x1008, new_sr10(x1009, x1011), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1011)) 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. ---------------------------------------- (37) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (38) QDPPairToRuleProof (EQUIVALENT) The dependency pair new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) was transformed to the following new rules: anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) the following new pairs maintain the fan-in: new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) the following new pairs maintain the fan-out: H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) ---------------------------------------- (39) Complex Obligation (AND) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) new_new_pr2F0G12(Succ(Succ(x0))) anew_new_pr2F0G12(Succ(Succ(x0))) new_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (43) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) the following chains were created: *We consider the chain new_pr2F0G12(x4, x5, x6, Succ(Zero), x7) -> new_pr2F1(x4, x6, new_fromInt, x5, x7), new_pr2F1(x8, x9, x10, x11, x12) -> new_pr2F34(x9, x10, x8, new_sr9(x8, x11, x12), x12) which results in the following constraint: (1) (new_pr2F1(x4, x6, new_fromInt, x5, x7)=new_pr2F1(x8, x9, x10, x11, x12) ==> new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) For Pair new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) the following chains were created: *We consider the chain new_pr2F1(x79, x80, x81, x82, x83) -> new_pr2F34(x80, x81, x79, new_sr9(x79, x82, x83), x83), new_pr2F34(x84, Pos(x85), x86, x87, x88) -> new_pr2F31(new_primPlusNat0(Succ(x84), x85), x86, new_primPlusNat0(Succ(x84), x85), x87, x88) which results in the following constraint: (1) (new_pr2F34(x80, x81, x79, new_sr9(x79, x82, x83), x83)=new_pr2F34(x84, Pos(x85), x86, x87, x88) ==> new_pr2F1(x79, x80, x81, x82, x83)_>=_new_pr2F34(x80, x81, x79, new_sr9(x79, x82, x83), x83)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F1(x79, x80, Pos(x85), x82, x83)_>=_new_pr2F34(x80, Pos(x85), x79, new_sr9(x79, x82, x83), x83)) For Pair new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) the following chains were created: *We consider the chain new_pr2F34(x169, Pos(x170), x171, x172, x173) -> new_pr2F31(new_primPlusNat0(Succ(x169), x170), x171, new_primPlusNat0(Succ(x169), x170), x172, x173), new_pr2F31(Succ(x174), x175, Succ(Succ(x176)), x177, x178) -> new_pr2F0G12(x175, x177, Succ(x176), x176, x178) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x169), x170), x171, new_primPlusNat0(Succ(x169), x170), x172, x173)=new_pr2F31(Succ(x174), x175, Succ(Succ(x176)), x177, x178) ==> new_pr2F34(x169, Pos(x170), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), x170), x171, new_primPlusNat0(Succ(x169), x170), x172, x173)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x169)=x1278 & new_primPlusNat0(x1278, x170)=Succ(x174) & Succ(x169)=x1279 & new_primPlusNat0(x1279, x170)=Succ(Succ(x176)) ==> new_pr2F34(x169, Pos(x170), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), x170), x171, new_primPlusNat0(Succ(x169), x170), x172, x173)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1278, x170)=Succ(x174) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1281, x1280)))=Succ(x174) & Succ(x169)=Succ(x1281) & Succ(x169)=x1279 & new_primPlusNat0(x1279, Succ(x1280))=Succ(Succ(x176)) & (\/x1282,x1283,x1284,x1285,x1286,x1287,x1288:new_primPlusNat0(x1281, x1280)=Succ(x1282) & Succ(x1283)=x1281 & Succ(x1283)=x1284 & new_primPlusNat0(x1284, x1280)=Succ(Succ(x1285)) ==> new_pr2F34(x1283, Pos(x1280), x1286, x1287, x1288)_>=_new_pr2F31(new_primPlusNat0(Succ(x1283), x1280), x1286, new_primPlusNat0(Succ(x1283), x1280), x1287, x1288)) ==> new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) (4) (Succ(x1289)=Succ(x174) & Succ(x169)=Succ(x1289) & Succ(x169)=x1279 & new_primPlusNat0(x1279, Zero)=Succ(Succ(x176)) ==> new_pr2F34(x169, Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Zero), x171, new_primPlusNat0(Succ(x169), Zero), x172, x173)) (5) (Succ(x1290)=Succ(x174) & Succ(x169)=Zero & Succ(x169)=x1279 & new_primPlusNat0(x1279, Succ(x1290))=Succ(Succ(x176)) ==> new_pr2F34(x169, Pos(Succ(x1290)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1290)), x171, new_primPlusNat0(Succ(x169), Succ(x1290)), x172, x173)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x169)=x1279 & Succ(x1280)=x1291 & new_primPlusNat0(x1279, x1291)=Succ(Succ(x176)) & (\/x1282,x1283,x1284,x1285,x1286,x1287,x1288:new_primPlusNat0(x169, x1280)=Succ(x1282) & Succ(x1283)=x169 & Succ(x1283)=x1284 & new_primPlusNat0(x1284, x1280)=Succ(Succ(x1285)) ==> new_pr2F34(x1283, Pos(x1280), x1286, x1287, x1288)_>=_new_pr2F31(new_primPlusNat0(Succ(x1283), x1280), x1286, new_primPlusNat0(Succ(x1283), x1280), x1287, x1288)) ==> new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x169)=x1279 & Zero=x1309 & new_primPlusNat0(x1279, x1309)=Succ(Succ(x176)) ==> new_pr2F34(x169, Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Zero), x171, new_primPlusNat0(Succ(x169), Zero), x172, x173)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1279, x1291)=Succ(Succ(x176)) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1293, x1292)))=Succ(Succ(x176)) & Succ(x169)=Succ(x1293) & Succ(x1280)=Succ(x1292) & (\/x1282,x1283,x1284,x1285,x1286,x1287,x1288:new_primPlusNat0(x169, x1280)=Succ(x1282) & Succ(x1283)=x169 & Succ(x1283)=x1284 & new_primPlusNat0(x1284, x1280)=Succ(Succ(x1285)) ==> new_pr2F34(x1283, Pos(x1280), x1286, x1287, x1288)_>=_new_pr2F31(new_primPlusNat0(Succ(x1283), x1280), x1286, new_primPlusNat0(Succ(x1283), x1280), x1287, x1288)) & (\/x1294,x1295,x1296,x1297,x1298,x1299,x1300,x1301,x1302,x1303,x1304,x1305,x1306:new_primPlusNat0(x1293, x1292)=Succ(Succ(x1294)) & Succ(x1295)=x1293 & Succ(x1296)=x1292 & (\/x1297,x1298,x1299,x1300,x1301,x1302,x1303:new_primPlusNat0(x1295, x1296)=Succ(x1297) & Succ(x1298)=x1295 & Succ(x1298)=x1299 & new_primPlusNat0(x1299, x1296)=Succ(Succ(x1300)) ==> new_pr2F34(x1298, Pos(x1296), x1301, x1302, x1303)_>=_new_pr2F31(new_primPlusNat0(Succ(x1298), x1296), x1301, new_primPlusNat0(Succ(x1298), x1296), x1302, x1303)) ==> new_pr2F34(x1295, Pos(Succ(x1296)), x1304, x1305, x1306)_>=_new_pr2F31(new_primPlusNat0(Succ(x1295), Succ(x1296)), x1304, new_primPlusNat0(Succ(x1295), Succ(x1296)), x1305, x1306)) ==> new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) (9) (Succ(x1307)=Succ(Succ(x176)) & Succ(x169)=Succ(x1307) & Succ(x1280)=Zero & (\/x1282,x1283,x1284,x1285,x1286,x1287,x1288:new_primPlusNat0(x169, x1280)=Succ(x1282) & Succ(x1283)=x169 & Succ(x1283)=x1284 & new_primPlusNat0(x1284, x1280)=Succ(Succ(x1285)) ==> new_pr2F34(x1283, Pos(x1280), x1286, x1287, x1288)_>=_new_pr2F31(new_primPlusNat0(Succ(x1283), x1280), x1286, new_primPlusNat0(Succ(x1283), x1280), x1287, x1288)) ==> new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) (10) (Succ(x1308)=Succ(Succ(x176)) & Succ(x169)=Zero & Succ(x1280)=Succ(x1308) & (\/x1282,x1283,x1284,x1285,x1286,x1287,x1288:new_primPlusNat0(x169, x1280)=Succ(x1282) & Succ(x1283)=x169 & Succ(x1283)=x1284 & new_primPlusNat0(x1284, x1280)=Succ(Succ(x1285)) ==> new_pr2F34(x1283, Pos(x1280), x1286, x1287, x1288)_>=_new_pr2F31(new_primPlusNat0(Succ(x1283), x1280), x1286, new_primPlusNat0(Succ(x1283), x1280), x1287, x1288)) ==> new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint: (11) (new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1279, x1309)=Succ(Succ(x176)) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1311, x1310)))=Succ(Succ(x176)) & Succ(x169)=Succ(x1311) & Zero=Succ(x1310) & (\/x1312,x1313,x1314,x1315,x1316:new_primPlusNat0(x1311, x1310)=Succ(Succ(x1312)) & Succ(x1313)=x1311 & Zero=x1310 ==> new_pr2F34(x1313, Pos(Zero), x1314, x1315, x1316)_>=_new_pr2F31(new_primPlusNat0(Succ(x1313), Zero), x1314, new_primPlusNat0(Succ(x1313), Zero), x1315, x1316)) ==> new_pr2F34(x169, Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Zero), x171, new_primPlusNat0(Succ(x169), Zero), x172, x173)) (13) (Succ(x1317)=Succ(Succ(x176)) & Succ(x169)=Succ(x1317) & Zero=Zero ==> new_pr2F34(x169, Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Zero), x171, new_primPlusNat0(Succ(x169), Zero), x172, x173)) (14) (Succ(x1318)=Succ(Succ(x176)) & Succ(x169)=Zero & Zero=Succ(x1318) ==> new_pr2F34(x169, Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Zero), x171, new_primPlusNat0(Succ(x169), Zero), x172, x173)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F34(Succ(x176), Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x176)), Zero), x171, new_primPlusNat0(Succ(Succ(x176)), Zero), x172, x173)) We solved constraint (14) using rules (I), (II). *We consider the chain new_pr2F34(x194, Pos(x195), x196, x197, x198) -> new_pr2F31(new_primPlusNat0(Succ(x194), x195), x196, new_primPlusNat0(Succ(x194), x195), x197, x198), new_pr2F31(Succ(x199), x200, Succ(Zero), x201, x202) -> new_pr2F1(x200, Zero, new_fromInt, x201, x202) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x194), x195), x196, new_primPlusNat0(Succ(x194), x195), x197, x198)=new_pr2F31(Succ(x199), x200, Succ(Zero), x201, x202) ==> new_pr2F34(x194, Pos(x195), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), x195), x196, new_primPlusNat0(Succ(x194), x195), x197, x198)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x194)=x1319 & new_primPlusNat0(x1319, x195)=Succ(x199) & Succ(x194)=x1320 & new_primPlusNat0(x1320, x195)=Succ(Zero) ==> new_pr2F34(x194, Pos(x195), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), x195), x196, new_primPlusNat0(Succ(x194), x195), x197, x198)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1319, x195)=Succ(x199) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1322, x1321)))=Succ(x199) & Succ(x194)=Succ(x1322) & Succ(x194)=x1320 & new_primPlusNat0(x1320, Succ(x1321))=Succ(Zero) & (\/x1323,x1324,x1325,x1326,x1327,x1328:new_primPlusNat0(x1322, x1321)=Succ(x1323) & Succ(x1324)=x1322 & Succ(x1324)=x1325 & new_primPlusNat0(x1325, x1321)=Succ(Zero) ==> new_pr2F34(x1324, Pos(x1321), x1326, x1327, x1328)_>=_new_pr2F31(new_primPlusNat0(Succ(x1324), x1321), x1326, new_primPlusNat0(Succ(x1324), x1321), x1327, x1328)) ==> new_pr2F34(x194, Pos(Succ(x1321)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1321)), x196, new_primPlusNat0(Succ(x194), Succ(x1321)), x197, x198)) (4) (Succ(x1329)=Succ(x199) & Succ(x194)=Succ(x1329) & Succ(x194)=x1320 & new_primPlusNat0(x1320, Zero)=Succ(Zero) ==> new_pr2F34(x194, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Zero), x196, new_primPlusNat0(Succ(x194), Zero), x197, x198)) (5) (Succ(x1330)=Succ(x199) & Succ(x194)=Zero & Succ(x194)=x1320 & new_primPlusNat0(x1320, Succ(x1330))=Succ(Zero) ==> new_pr2F34(x194, Pos(Succ(x1330)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1330)), x196, new_primPlusNat0(Succ(x194), Succ(x1330)), x197, x198)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x194)=x1320 & Succ(x1321)=x1331 & new_primPlusNat0(x1320, x1331)=Succ(Zero) & (\/x1323,x1324,x1325,x1326,x1327,x1328:new_primPlusNat0(x194, x1321)=Succ(x1323) & Succ(x1324)=x194 & Succ(x1324)=x1325 & new_primPlusNat0(x1325, x1321)=Succ(Zero) ==> new_pr2F34(x1324, Pos(x1321), x1326, x1327, x1328)_>=_new_pr2F31(new_primPlusNat0(Succ(x1324), x1321), x1326, new_primPlusNat0(Succ(x1324), x1321), x1327, x1328)) ==> new_pr2F34(x194, Pos(Succ(x1321)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1321)), x196, new_primPlusNat0(Succ(x194), Succ(x1321)), x197, x198)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x194)=x1320 & Zero=x1347 & new_primPlusNat0(x1320, x1347)=Succ(Zero) ==> new_pr2F34(x194, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Zero), x196, new_primPlusNat0(Succ(x194), Zero), x197, x198)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1320, x1331)=Succ(Zero) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1333, x1332)))=Succ(Zero) & Succ(x194)=Succ(x1333) & Succ(x1321)=Succ(x1332) & (\/x1323,x1324,x1325,x1326,x1327,x1328:new_primPlusNat0(x194, x1321)=Succ(x1323) & Succ(x1324)=x194 & Succ(x1324)=x1325 & new_primPlusNat0(x1325, x1321)=Succ(Zero) ==> new_pr2F34(x1324, Pos(x1321), x1326, x1327, x1328)_>=_new_pr2F31(new_primPlusNat0(Succ(x1324), x1321), x1326, new_primPlusNat0(Succ(x1324), x1321), x1327, x1328)) & (\/x1334,x1335,x1336,x1337,x1338,x1339,x1340,x1341,x1342,x1343,x1344:new_primPlusNat0(x1333, x1332)=Succ(Zero) & Succ(x1334)=x1333 & Succ(x1335)=x1332 & (\/x1336,x1337,x1338,x1339,x1340,x1341:new_primPlusNat0(x1334, x1335)=Succ(x1336) & Succ(x1337)=x1334 & Succ(x1337)=x1338 & new_primPlusNat0(x1338, x1335)=Succ(Zero) ==> new_pr2F34(x1337, Pos(x1335), x1339, x1340, x1341)_>=_new_pr2F31(new_primPlusNat0(Succ(x1337), x1335), x1339, new_primPlusNat0(Succ(x1337), x1335), x1340, x1341)) ==> new_pr2F34(x1334, Pos(Succ(x1335)), x1342, x1343, x1344)_>=_new_pr2F31(new_primPlusNat0(Succ(x1334), Succ(x1335)), x1342, new_primPlusNat0(Succ(x1334), Succ(x1335)), x1343, x1344)) ==> new_pr2F34(x194, Pos(Succ(x1321)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1321)), x196, new_primPlusNat0(Succ(x194), Succ(x1321)), x197, x198)) (9) (Succ(x1345)=Succ(Zero) & Succ(x194)=Succ(x1345) & Succ(x1321)=Zero & (\/x1323,x1324,x1325,x1326,x1327,x1328:new_primPlusNat0(x194, x1321)=Succ(x1323) & Succ(x1324)=x194 & Succ(x1324)=x1325 & new_primPlusNat0(x1325, x1321)=Succ(Zero) ==> new_pr2F34(x1324, Pos(x1321), x1326, x1327, x1328)_>=_new_pr2F31(new_primPlusNat0(Succ(x1324), x1321), x1326, new_primPlusNat0(Succ(x1324), x1321), x1327, x1328)) ==> new_pr2F34(x194, Pos(Succ(x1321)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1321)), x196, new_primPlusNat0(Succ(x194), Succ(x1321)), x197, x198)) (10) (Succ(x1346)=Succ(Zero) & Succ(x194)=Zero & Succ(x1321)=Succ(x1346) & (\/x1323,x1324,x1325,x1326,x1327,x1328:new_primPlusNat0(x194, x1321)=Succ(x1323) & Succ(x1324)=x194 & Succ(x1324)=x1325 & new_primPlusNat0(x1325, x1321)=Succ(Zero) ==> new_pr2F34(x1324, Pos(x1321), x1326, x1327, x1328)_>=_new_pr2F31(new_primPlusNat0(Succ(x1324), x1321), x1326, new_primPlusNat0(Succ(x1324), x1321), x1327, x1328)) ==> new_pr2F34(x194, Pos(Succ(x1321)), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Succ(x1321)), x196, new_primPlusNat0(Succ(x194), Succ(x1321)), x197, x198)) We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1320, x1347)=Succ(Zero) which results in the following new constraints: (11) (Succ(Succ(new_primPlusNat0(x1349, x1348)))=Succ(Zero) & Succ(x194)=Succ(x1349) & Zero=Succ(x1348) & (\/x1350,x1351,x1352,x1353:new_primPlusNat0(x1349, x1348)=Succ(Zero) & Succ(x1350)=x1349 & Zero=x1348 ==> new_pr2F34(x1350, Pos(Zero), x1351, x1352, x1353)_>=_new_pr2F31(new_primPlusNat0(Succ(x1350), Zero), x1351, new_primPlusNat0(Succ(x1350), Zero), x1352, x1353)) ==> new_pr2F34(x194, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Zero), x196, new_primPlusNat0(Succ(x194), Zero), x197, x198)) (12) (Succ(x1354)=Succ(Zero) & Succ(x194)=Succ(x1354) & Zero=Zero ==> new_pr2F34(x194, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Zero), x196, new_primPlusNat0(Succ(x194), Zero), x197, x198)) (13) (Succ(x1355)=Succ(Zero) & Succ(x194)=Zero & Zero=Succ(x1355) ==> new_pr2F34(x194, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(x194), Zero), x196, new_primPlusNat0(Succ(x194), Zero), x197, x198)) We solved constraint (11) using rules (I), (II).We simplified constraint (12) using rules (I), (II), (III) which results in the following new constraint: (14) (new_pr2F34(Zero, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x196, new_primPlusNat0(Succ(Zero), Zero), x197, x198)) We solved constraint (13) using rules (I), (II). *We consider the chain new_pr2F34(x228, Pos(x229), x230, x231, x232) -> new_pr2F31(new_primPlusNat0(Succ(x228), x229), x230, new_primPlusNat0(Succ(x228), x229), x231, x232), new_pr2F31(Succ(x233), x234, Succ(Succ(x235)), x236, x237) -> H(x234, x236, Succ(x235), x237, anew_new_pr2F0G12(x235)) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x228), x229), x230, new_primPlusNat0(Succ(x228), x229), x231, x232)=new_pr2F31(Succ(x233), x234, Succ(Succ(x235)), x236, x237) ==> new_pr2F34(x228, Pos(x229), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), x229), x230, new_primPlusNat0(Succ(x228), x229), x231, x232)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x228)=x1356 & new_primPlusNat0(x1356, x229)=Succ(x233) & Succ(x228)=x1357 & new_primPlusNat0(x1357, x229)=Succ(Succ(x235)) ==> new_pr2F34(x228, Pos(x229), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), x229), x230, new_primPlusNat0(Succ(x228), x229), x231, x232)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1356, x229)=Succ(x233) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1359, x1358)))=Succ(x233) & Succ(x228)=Succ(x1359) & Succ(x228)=x1357 & new_primPlusNat0(x1357, Succ(x1358))=Succ(Succ(x235)) & (\/x1360,x1361,x1362,x1363,x1364,x1365,x1366:new_primPlusNat0(x1359, x1358)=Succ(x1360) & Succ(x1361)=x1359 & Succ(x1361)=x1362 & new_primPlusNat0(x1362, x1358)=Succ(Succ(x1363)) ==> new_pr2F34(x1361, Pos(x1358), x1364, x1365, x1366)_>=_new_pr2F31(new_primPlusNat0(Succ(x1361), x1358), x1364, new_primPlusNat0(Succ(x1361), x1358), x1365, x1366)) ==> new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) (4) (Succ(x1367)=Succ(x233) & Succ(x228)=Succ(x1367) & Succ(x228)=x1357 & new_primPlusNat0(x1357, Zero)=Succ(Succ(x235)) ==> new_pr2F34(x228, Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Zero), x230, new_primPlusNat0(Succ(x228), Zero), x231, x232)) (5) (Succ(x1368)=Succ(x233) & Succ(x228)=Zero & Succ(x228)=x1357 & new_primPlusNat0(x1357, Succ(x1368))=Succ(Succ(x235)) ==> new_pr2F34(x228, Pos(Succ(x1368)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1368)), x230, new_primPlusNat0(Succ(x228), Succ(x1368)), x231, x232)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x228)=x1357 & Succ(x1358)=x1369 & new_primPlusNat0(x1357, x1369)=Succ(Succ(x235)) & (\/x1360,x1361,x1362,x1363,x1364,x1365,x1366:new_primPlusNat0(x228, x1358)=Succ(x1360) & Succ(x1361)=x228 & Succ(x1361)=x1362 & new_primPlusNat0(x1362, x1358)=Succ(Succ(x1363)) ==> new_pr2F34(x1361, Pos(x1358), x1364, x1365, x1366)_>=_new_pr2F31(new_primPlusNat0(Succ(x1361), x1358), x1364, new_primPlusNat0(Succ(x1361), x1358), x1365, x1366)) ==> new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x228)=x1357 & Zero=x1387 & new_primPlusNat0(x1357, x1387)=Succ(Succ(x235)) ==> new_pr2F34(x228, Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Zero), x230, new_primPlusNat0(Succ(x228), Zero), x231, x232)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1357, x1369)=Succ(Succ(x235)) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1371, x1370)))=Succ(Succ(x235)) & Succ(x228)=Succ(x1371) & Succ(x1358)=Succ(x1370) & (\/x1360,x1361,x1362,x1363,x1364,x1365,x1366:new_primPlusNat0(x228, x1358)=Succ(x1360) & Succ(x1361)=x228 & Succ(x1361)=x1362 & new_primPlusNat0(x1362, x1358)=Succ(Succ(x1363)) ==> new_pr2F34(x1361, Pos(x1358), x1364, x1365, x1366)_>=_new_pr2F31(new_primPlusNat0(Succ(x1361), x1358), x1364, new_primPlusNat0(Succ(x1361), x1358), x1365, x1366)) & (\/x1372,x1373,x1374,x1375,x1376,x1377,x1378,x1379,x1380,x1381,x1382,x1383,x1384:new_primPlusNat0(x1371, x1370)=Succ(Succ(x1372)) & Succ(x1373)=x1371 & Succ(x1374)=x1370 & (\/x1375,x1376,x1377,x1378,x1379,x1380,x1381:new_primPlusNat0(x1373, x1374)=Succ(x1375) & Succ(x1376)=x1373 & Succ(x1376)=x1377 & new_primPlusNat0(x1377, x1374)=Succ(Succ(x1378)) ==> new_pr2F34(x1376, Pos(x1374), x1379, x1380, x1381)_>=_new_pr2F31(new_primPlusNat0(Succ(x1376), x1374), x1379, new_primPlusNat0(Succ(x1376), x1374), x1380, x1381)) ==> new_pr2F34(x1373, Pos(Succ(x1374)), x1382, x1383, x1384)_>=_new_pr2F31(new_primPlusNat0(Succ(x1373), Succ(x1374)), x1382, new_primPlusNat0(Succ(x1373), Succ(x1374)), x1383, x1384)) ==> new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) (9) (Succ(x1385)=Succ(Succ(x235)) & Succ(x228)=Succ(x1385) & Succ(x1358)=Zero & (\/x1360,x1361,x1362,x1363,x1364,x1365,x1366:new_primPlusNat0(x228, x1358)=Succ(x1360) & Succ(x1361)=x228 & Succ(x1361)=x1362 & new_primPlusNat0(x1362, x1358)=Succ(Succ(x1363)) ==> new_pr2F34(x1361, Pos(x1358), x1364, x1365, x1366)_>=_new_pr2F31(new_primPlusNat0(Succ(x1361), x1358), x1364, new_primPlusNat0(Succ(x1361), x1358), x1365, x1366)) ==> new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) (10) (Succ(x1386)=Succ(Succ(x235)) & Succ(x228)=Zero & Succ(x1358)=Succ(x1386) & (\/x1360,x1361,x1362,x1363,x1364,x1365,x1366:new_primPlusNat0(x228, x1358)=Succ(x1360) & Succ(x1361)=x228 & Succ(x1361)=x1362 & new_primPlusNat0(x1362, x1358)=Succ(Succ(x1363)) ==> new_pr2F34(x1361, Pos(x1358), x1364, x1365, x1366)_>=_new_pr2F31(new_primPlusNat0(Succ(x1361), x1358), x1364, new_primPlusNat0(Succ(x1361), x1358), x1365, x1366)) ==> new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint: (11) (new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1357, x1387)=Succ(Succ(x235)) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1389, x1388)))=Succ(Succ(x235)) & Succ(x228)=Succ(x1389) & Zero=Succ(x1388) & (\/x1390,x1391,x1392,x1393,x1394:new_primPlusNat0(x1389, x1388)=Succ(Succ(x1390)) & Succ(x1391)=x1389 & Zero=x1388 ==> new_pr2F34(x1391, Pos(Zero), x1392, x1393, x1394)_>=_new_pr2F31(new_primPlusNat0(Succ(x1391), Zero), x1392, new_primPlusNat0(Succ(x1391), Zero), x1393, x1394)) ==> new_pr2F34(x228, Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Zero), x230, new_primPlusNat0(Succ(x228), Zero), x231, x232)) (13) (Succ(x1395)=Succ(Succ(x235)) & Succ(x228)=Succ(x1395) & Zero=Zero ==> new_pr2F34(x228, Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Zero), x230, new_primPlusNat0(Succ(x228), Zero), x231, x232)) (14) (Succ(x1396)=Succ(Succ(x235)) & Succ(x228)=Zero & Zero=Succ(x1396) ==> new_pr2F34(x228, Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Zero), x230, new_primPlusNat0(Succ(x228), Zero), x231, x232)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F34(Succ(x235), Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x235)), Zero), x230, new_primPlusNat0(Succ(Succ(x235)), Zero), x231, x232)) We solved constraint (14) using rules (I), (II). For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x248), x249, Succ(Succ(x250)), x251, x252) -> new_pr2F0G12(x249, x251, Succ(x250), x250, x252), new_pr2F0G12(x253, x254, x255, Succ(Zero), x256) -> new_pr2F1(x253, x255, new_fromInt, x254, x256) which results in the following constraint: (1) (new_pr2F0G12(x249, x251, Succ(x250), x250, x252)=new_pr2F0G12(x253, x254, x255, Succ(Zero), x256) ==> new_pr2F31(Succ(x248), x249, Succ(Succ(x250)), x251, x252)_>=_new_pr2F0G12(x249, x251, Succ(x250), x250, x252)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x248), x249, Succ(Succ(Succ(Zero))), x251, x252)_>=_new_pr2F0G12(x249, x251, Succ(Succ(Zero)), Succ(Zero), x252)) *We consider the chain new_pr2F31(Succ(x272), x273, Succ(Succ(x274)), x275, x276) -> new_pr2F0G12(x273, x275, Succ(x274), x274, x276), new_pr2F0G12(x277, x278, x279, Zero, x280) -> new_pr2F0G13(new_sr8(x277, x278, x280), x277, new_primDivNatS1(Succ(x279)), new_primDivNatS1(Succ(x279)), x280) which results in the following constraint: (1) (new_pr2F0G12(x273, x275, Succ(x274), x274, x276)=new_pr2F0G12(x277, x278, x279, Zero, x280) ==> new_pr2F31(Succ(x272), x273, Succ(Succ(x274)), x275, x276)_>=_new_pr2F0G12(x273, x275, Succ(x274), x274, x276)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x272), x273, Succ(Succ(Zero)), x275, x276)_>=_new_pr2F0G12(x273, x275, Succ(Zero), Zero, x276)) For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) the following chains were created: *We consider the chain new_pr2F0G12(x356, x357, x358, Zero, x359) -> new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x358)), new_primDivNatS1(Succ(x358)), x359), new_pr2F0G13(x360, x361, x362, Succ(Zero), x363) -> new_pr2F2(x361, x362, new_fromInt, x360, x363) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x358)), new_primDivNatS1(Succ(x358)), x359)=new_pr2F0G13(x360, x361, x362, Succ(Zero), x363) ==> new_pr2F0G12(x356, x357, x358, Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x358)), new_primDivNatS1(Succ(x358)), x359)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x358)=x1397 & new_primDivNatS1(x1397)=Succ(Zero) ==> new_pr2F0G12(x356, x357, x358, Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x358)), new_primDivNatS1(Succ(x358)), x359)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1397)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1398)=Succ(Zero) & Succ(x358)=Succ(x1398) ==> new_pr2F0G12(x356, x357, x358, Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x358)), new_primDivNatS1(Succ(x358)), x359)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1398)=Succ(Zero) ==> new_pr2F0G12(x356, x357, x1398, Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(x1398)), new_primDivNatS1(Succ(x1398)), x359)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1398)=Succ(Zero) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1399))=Succ(Zero) ==> new_pr2F0G12(x356, x357, Succ(Succ(x1399)), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Succ(x1399)))), new_primDivNatS1(Succ(Succ(Succ(x1399)))), x359)) (6) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G12(x356, x357, Succ(Zero), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x359)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x356, x357, Succ(Succ(x1399)), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Succ(x1399)))), new_primDivNatS1(Succ(Succ(Succ(x1399)))), x359)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x356, x357, Succ(Zero), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x359)) *We consider the chain new_pr2F0G12(x372, x373, x374, Zero, x375) -> new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x374)), new_primDivNatS1(Succ(x374)), x375), new_pr2F0G13(x376, x377, x378, Succ(Succ(x379)), x380) -> new_pr2F0G14(x376, x377, x378, x379, x380) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x374)), new_primDivNatS1(Succ(x374)), x375)=new_pr2F0G13(x376, x377, x378, Succ(Succ(x379)), x380) ==> new_pr2F0G12(x372, x373, x374, Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x374)), new_primDivNatS1(Succ(x374)), x375)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x374)=x1400 & new_primDivNatS1(x1400)=Succ(Succ(x379)) ==> new_pr2F0G12(x372, x373, x374, Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x374)), new_primDivNatS1(Succ(x374)), x375)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1400)=Succ(Succ(x379)) which results in the following new constraint: (3) (new_primDivNatS01(x1401)=Succ(Succ(x379)) & Succ(x374)=Succ(x1401) ==> new_pr2F0G12(x372, x373, x374, Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x374)), new_primDivNatS1(Succ(x374)), x375)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1401)=Succ(Succ(x379)) ==> new_pr2F0G12(x372, x373, x1401, Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(x1401)), new_primDivNatS1(Succ(x1401)), x375)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1401)=Succ(Succ(x379)) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1402))=Succ(Succ(x379)) ==> new_pr2F0G12(x372, x373, Succ(Succ(x1402)), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Succ(x1402)))), new_primDivNatS1(Succ(Succ(Succ(x1402)))), x375)) (6) (Succ(new_primDivNatS2)=Succ(Succ(x379)) ==> new_pr2F0G12(x372, x373, Succ(Zero), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x375)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x372, x373, Succ(Succ(x1402)), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Succ(x1402)))), new_primDivNatS1(Succ(Succ(Succ(x1402)))), x375)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x372, x373, Succ(Zero), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x375)) *We consider the chain new_pr2F0G12(x393, x394, x395, Zero, x396) -> new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396), new_pr2F0G13(x397, x398, x399, Zero, x400) -> new_pr2F0G13(x397, new_sr10(x398, x400), new_primDivNatS1(x399), new_primDivNatS1(x399), x400) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396)=new_pr2F0G13(x397, x398, x399, Zero, x400) ==> new_pr2F0G12(x393, x394, x395, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x395)=x1403 & new_primDivNatS1(x1403)=Zero ==> new_pr2F0G12(x393, x394, x395, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1403)=Zero which results in the following new constraints: (3) (Zero=Zero & Succ(x395)=Zero ==> new_pr2F0G12(x393, x394, x395, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396)) (4) (new_primDivNatS01(x1404)=Zero & Succ(x395)=Succ(x1404) ==> new_pr2F0G12(x393, x394, x395, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x395)), new_primDivNatS1(Succ(x395)), x396)) We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III) which results in the following new constraint: (5) (new_primDivNatS01(x1404)=Zero ==> new_pr2F0G12(x393, x394, x1404, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(x1404)), new_primDivNatS1(Succ(x1404)), x396)) We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1404)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G12(x393, x394, Zero, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x396)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G12(x393, x394, Zero, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x396)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G13(x437, x438, x439, Succ(Zero), x440) -> new_pr2F2(x438, x439, new_fromInt, x437, x440), new_pr2F2(x441, x442, Pos(x443), x444, x445) -> new_pr2F31(new_primPlusNat0(x442, x443), new_sr11(x441, x445), new_primPlusNat0(x442, x443), x444, x445) which results in the following constraint: (1) (new_pr2F2(x438, x439, new_fromInt, x437, x440)=new_pr2F2(x441, x442, Pos(x443), x444, x445) ==> new_pr2F0G13(x437, x438, x439, Succ(Zero), x440)_>=_new_pr2F2(x438, x439, new_fromInt, x437, x440)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x443) ==> new_pr2F0G13(x437, x438, x439, Succ(Zero), x440)_>=_new_pr2F2(x438, x439, new_fromInt, x437, x440)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x443) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x443) ==> new_pr2F0G13(x437, x438, x439, Succ(Zero), x440)_>=_new_pr2F2(x438, x439, new_fromInt, x437, x440)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G13(x437, x438, x439, Succ(Zero), x440)_>=_new_pr2F2(x438, x439, new_fromInt, x437, x440)) For Pair new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) the following chains were created: *We consider the chain new_pr2F2(x497, x498, Pos(x499), x500, x501) -> new_pr2F31(new_primPlusNat0(x498, x499), new_sr11(x497, x501), new_primPlusNat0(x498, x499), x500, x501), new_pr2F31(Succ(x502), x503, Succ(Succ(x504)), x505, x506) -> new_pr2F0G12(x503, x505, Succ(x504), x504, x506) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x498, x499), new_sr11(x497, x501), new_primPlusNat0(x498, x499), x500, x501)=new_pr2F31(Succ(x502), x503, Succ(Succ(x504)), x505, x506) ==> new_pr2F2(x497, x498, Pos(x499), x500, x501)_>=_new_pr2F31(new_primPlusNat0(x498, x499), new_sr11(x497, x501), new_primPlusNat0(x498, x499), x500, x501)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x498, x499)=Succ(x502) & new_primPlusNat0(x498, x499)=Succ(Succ(x504)) ==> new_pr2F2(x497, x498, Pos(x499), x500, x501)_>=_new_pr2F31(new_primPlusNat0(x498, x499), new_sr11(x497, x501), new_primPlusNat0(x498, x499), x500, x501)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x498, x499)=Succ(x502) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1407, x1406)))=Succ(x502) & new_primPlusNat0(Succ(x1407), Succ(x1406))=Succ(Succ(x504)) & (\/x1408,x1409,x1410,x1411,x1412:new_primPlusNat0(x1407, x1406)=Succ(x1408) & new_primPlusNat0(x1407, x1406)=Succ(Succ(x1409)) ==> new_pr2F2(x1410, x1407, Pos(x1406), x1411, x1412)_>=_new_pr2F31(new_primPlusNat0(x1407, x1406), new_sr11(x1410, x1412), new_primPlusNat0(x1407, x1406), x1411, x1412)) ==> new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) (4) (Succ(x1413)=Succ(x502) & new_primPlusNat0(Succ(x1413), Zero)=Succ(Succ(x504)) ==> new_pr2F2(x497, Succ(x1413), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1413), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(x1413), Zero), x500, x501)) (5) (Succ(x1414)=Succ(x502) & new_primPlusNat0(Zero, Succ(x1414))=Succ(Succ(x504)) ==> new_pr2F2(x497, Zero, Pos(Succ(x1414)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1414)), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(x1414)), x500, x501)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1407)=x1415 & Succ(x1406)=x1416 & new_primPlusNat0(x1415, x1416)=Succ(Succ(x504)) & (\/x1408,x1409,x1410,x1411,x1412:new_primPlusNat0(x1407, x1406)=Succ(x1408) & new_primPlusNat0(x1407, x1406)=Succ(Succ(x1409)) ==> new_pr2F2(x1410, x1407, Pos(x1406), x1411, x1412)_>=_new_pr2F31(new_primPlusNat0(x1407, x1406), new_sr11(x1410, x1412), new_primPlusNat0(x1407, x1406), x1411, x1412)) ==> new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1413)=x1432 & Zero=x1433 & new_primPlusNat0(x1432, x1433)=Succ(Succ(x504)) ==> new_pr2F2(x497, Succ(x1413), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1413), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(x1413), Zero), x500, x501)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1443 & Succ(x1414)=x1444 & new_primPlusNat0(x1443, x1444)=Succ(Succ(x504)) ==> new_pr2F2(x497, Zero, Pos(Succ(x1414)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1414)), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(x1414)), x500, x501)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1415, x1416)=Succ(Succ(x504)) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1418, x1417)))=Succ(Succ(x504)) & Succ(x1407)=Succ(x1418) & Succ(x1406)=Succ(x1417) & (\/x1408,x1409,x1410,x1411,x1412:new_primPlusNat0(x1407, x1406)=Succ(x1408) & new_primPlusNat0(x1407, x1406)=Succ(Succ(x1409)) ==> new_pr2F2(x1410, x1407, Pos(x1406), x1411, x1412)_>=_new_pr2F31(new_primPlusNat0(x1407, x1406), new_sr11(x1410, x1412), new_primPlusNat0(x1407, x1406), x1411, x1412)) & (\/x1419,x1420,x1421,x1422,x1423,x1424,x1425,x1426,x1427,x1428,x1429:new_primPlusNat0(x1418, x1417)=Succ(Succ(x1419)) & Succ(x1420)=x1418 & Succ(x1421)=x1417 & (\/x1422,x1423,x1424,x1425,x1426:new_primPlusNat0(x1420, x1421)=Succ(x1422) & new_primPlusNat0(x1420, x1421)=Succ(Succ(x1423)) ==> new_pr2F2(x1424, x1420, Pos(x1421), x1425, x1426)_>=_new_pr2F31(new_primPlusNat0(x1420, x1421), new_sr11(x1424, x1426), new_primPlusNat0(x1420, x1421), x1425, x1426)) ==> new_pr2F2(x1427, Succ(x1420), Pos(Succ(x1421)), x1428, x1429)_>=_new_pr2F31(new_primPlusNat0(Succ(x1420), Succ(x1421)), new_sr11(x1427, x1429), new_primPlusNat0(Succ(x1420), Succ(x1421)), x1428, x1429)) ==> new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) (10) (Succ(x1430)=Succ(Succ(x504)) & Succ(x1407)=Succ(x1430) & Succ(x1406)=Zero & (\/x1408,x1409,x1410,x1411,x1412:new_primPlusNat0(x1407, x1406)=Succ(x1408) & new_primPlusNat0(x1407, x1406)=Succ(Succ(x1409)) ==> new_pr2F2(x1410, x1407, Pos(x1406), x1411, x1412)_>=_new_pr2F31(new_primPlusNat0(x1407, x1406), new_sr11(x1410, x1412), new_primPlusNat0(x1407, x1406), x1411, x1412)) ==> new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) (11) (Succ(x1431)=Succ(Succ(x504)) & Succ(x1407)=Zero & Succ(x1406)=Succ(x1431) & (\/x1408,x1409,x1410,x1411,x1412:new_primPlusNat0(x1407, x1406)=Succ(x1408) & new_primPlusNat0(x1407, x1406)=Succ(Succ(x1409)) ==> new_pr2F2(x1410, x1407, Pos(x1406), x1411, x1412)_>=_new_pr2F31(new_primPlusNat0(x1407, x1406), new_sr11(x1410, x1412), new_primPlusNat0(x1407, x1406), x1411, x1412)) ==> new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: (12) (new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1432, x1433)=Succ(Succ(x504)) which results in the following new constraints: (13) (Succ(Succ(new_primPlusNat0(x1435, x1434)))=Succ(Succ(x504)) & Succ(x1413)=Succ(x1435) & Zero=Succ(x1434) & (\/x1436,x1437,x1438,x1439,x1440:new_primPlusNat0(x1435, x1434)=Succ(Succ(x1436)) & Succ(x1437)=x1435 & Zero=x1434 ==> new_pr2F2(x1438, Succ(x1437), Pos(Zero), x1439, x1440)_>=_new_pr2F31(new_primPlusNat0(Succ(x1437), Zero), new_sr11(x1438, x1440), new_primPlusNat0(Succ(x1437), Zero), x1439, x1440)) ==> new_pr2F2(x497, Succ(x1413), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1413), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(x1413), Zero), x500, x501)) (14) (Succ(x1441)=Succ(Succ(x504)) & Succ(x1413)=Succ(x1441) & Zero=Zero ==> new_pr2F2(x497, Succ(x1413), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1413), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(x1413), Zero), x500, x501)) (15) (Succ(x1442)=Succ(Succ(x504)) & Succ(x1413)=Zero & Zero=Succ(x1442) ==> new_pr2F2(x497, Succ(x1413), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1413), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(x1413), Zero), x500, x501)) We solved constraint (13) using rules (I), (II).We simplified constraint (14) using rules (I), (II), (III) which results in the following new constraint: (16) (new_pr2F2(x497, Succ(Succ(x504)), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x504)), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(Succ(x504)), Zero), x500, x501)) We solved constraint (15) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1443, x1444)=Succ(Succ(x504)) which results in the following new constraints: (17) (Succ(Succ(new_primPlusNat0(x1446, x1445)))=Succ(Succ(x504)) & Zero=Succ(x1446) & Succ(x1414)=Succ(x1445) & (\/x1447,x1448,x1449,x1450,x1451:new_primPlusNat0(x1446, x1445)=Succ(Succ(x1447)) & Zero=x1446 & Succ(x1448)=x1445 ==> new_pr2F2(x1449, Zero, Pos(Succ(x1448)), x1450, x1451)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1448)), new_sr11(x1449, x1451), new_primPlusNat0(Zero, Succ(x1448)), x1450, x1451)) ==> new_pr2F2(x497, Zero, Pos(Succ(x1414)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1414)), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(x1414)), x500, x501)) (18) (Succ(x1452)=Succ(Succ(x504)) & Zero=Succ(x1452) & Succ(x1414)=Zero ==> new_pr2F2(x497, Zero, Pos(Succ(x1414)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1414)), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(x1414)), x500, x501)) (19) (Succ(x1453)=Succ(Succ(x504)) & Zero=Zero & Succ(x1414)=Succ(x1453) ==> new_pr2F2(x497, Zero, Pos(Succ(x1414)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1414)), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(x1414)), x500, x501)) We solved constraint (17) using rules (I), (II).We solved constraint (18) using rules (I), (II).We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint: (20) (new_pr2F2(x497, Zero, Pos(Succ(Succ(x504))), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x504))), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(Succ(x504))), x500, x501)) *We consider the chain new_pr2F2(x522, x523, Pos(x524), x525, x526) -> new_pr2F31(new_primPlusNat0(x523, x524), new_sr11(x522, x526), new_primPlusNat0(x523, x524), x525, x526), new_pr2F31(Succ(x527), x528, Succ(Zero), x529, x530) -> new_pr2F1(x528, Zero, new_fromInt, x529, x530) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x523, x524), new_sr11(x522, x526), new_primPlusNat0(x523, x524), x525, x526)=new_pr2F31(Succ(x527), x528, Succ(Zero), x529, x530) ==> new_pr2F2(x522, x523, Pos(x524), x525, x526)_>=_new_pr2F31(new_primPlusNat0(x523, x524), new_sr11(x522, x526), new_primPlusNat0(x523, x524), x525, x526)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x523, x524)=Succ(x527) & new_primPlusNat0(x523, x524)=Succ(Zero) ==> new_pr2F2(x522, x523, Pos(x524), x525, x526)_>=_new_pr2F31(new_primPlusNat0(x523, x524), new_sr11(x522, x526), new_primPlusNat0(x523, x524), x525, x526)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x523, x524)=Succ(x527) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1455, x1454)))=Succ(x527) & new_primPlusNat0(Succ(x1455), Succ(x1454))=Succ(Zero) & (\/x1456,x1457,x1458,x1459:new_primPlusNat0(x1455, x1454)=Succ(x1456) & new_primPlusNat0(x1455, x1454)=Succ(Zero) ==> new_pr2F2(x1457, x1455, Pos(x1454), x1458, x1459)_>=_new_pr2F31(new_primPlusNat0(x1455, x1454), new_sr11(x1457, x1459), new_primPlusNat0(x1455, x1454), x1458, x1459)) ==> new_pr2F2(x522, Succ(x1455), Pos(Succ(x1454)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1455), Succ(x1454)), new_sr11(x522, x526), new_primPlusNat0(Succ(x1455), Succ(x1454)), x525, x526)) (4) (Succ(x1460)=Succ(x527) & new_primPlusNat0(Succ(x1460), Zero)=Succ(Zero) ==> new_pr2F2(x522, Succ(x1460), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1460), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(x1460), Zero), x525, x526)) (5) (Succ(x1461)=Succ(x527) & new_primPlusNat0(Zero, Succ(x1461))=Succ(Zero) ==> new_pr2F2(x522, Zero, Pos(Succ(x1461)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1461)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(x1461)), x525, x526)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1455)=x1462 & Succ(x1454)=x1463 & new_primPlusNat0(x1462, x1463)=Succ(Zero) & (\/x1456,x1457,x1458,x1459:new_primPlusNat0(x1455, x1454)=Succ(x1456) & new_primPlusNat0(x1455, x1454)=Succ(Zero) ==> new_pr2F2(x1457, x1455, Pos(x1454), x1458, x1459)_>=_new_pr2F31(new_primPlusNat0(x1455, x1454), new_sr11(x1457, x1459), new_primPlusNat0(x1455, x1454), x1458, x1459)) ==> new_pr2F2(x522, Succ(x1455), Pos(Succ(x1454)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1455), Succ(x1454)), new_sr11(x522, x526), new_primPlusNat0(Succ(x1455), Succ(x1454)), x525, x526)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1460)=x1477 & Zero=x1478 & new_primPlusNat0(x1477, x1478)=Succ(Zero) ==> new_pr2F2(x522, Succ(x1460), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1460), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(x1460), Zero), x525, x526)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1487 & Succ(x1461)=x1488 & new_primPlusNat0(x1487, x1488)=Succ(Zero) ==> new_pr2F2(x522, Zero, Pos(Succ(x1461)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1461)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(x1461)), x525, x526)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1462, x1463)=Succ(Zero) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1465, x1464)))=Succ(Zero) & Succ(x1455)=Succ(x1465) & Succ(x1454)=Succ(x1464) & (\/x1456,x1457,x1458,x1459:new_primPlusNat0(x1455, x1454)=Succ(x1456) & new_primPlusNat0(x1455, x1454)=Succ(Zero) ==> new_pr2F2(x1457, x1455, Pos(x1454), x1458, x1459)_>=_new_pr2F31(new_primPlusNat0(x1455, x1454), new_sr11(x1457, x1459), new_primPlusNat0(x1455, x1454), x1458, x1459)) & (\/x1466,x1467,x1468,x1469,x1470,x1471,x1472,x1473,x1474:new_primPlusNat0(x1465, x1464)=Succ(Zero) & Succ(x1466)=x1465 & Succ(x1467)=x1464 & (\/x1468,x1469,x1470,x1471:new_primPlusNat0(x1466, x1467)=Succ(x1468) & new_primPlusNat0(x1466, x1467)=Succ(Zero) ==> new_pr2F2(x1469, x1466, Pos(x1467), x1470, x1471)_>=_new_pr2F31(new_primPlusNat0(x1466, x1467), new_sr11(x1469, x1471), new_primPlusNat0(x1466, x1467), x1470, x1471)) ==> new_pr2F2(x1472, Succ(x1466), Pos(Succ(x1467)), x1473, x1474)_>=_new_pr2F31(new_primPlusNat0(Succ(x1466), Succ(x1467)), new_sr11(x1472, x1474), new_primPlusNat0(Succ(x1466), Succ(x1467)), x1473, x1474)) ==> new_pr2F2(x522, Succ(x1455), Pos(Succ(x1454)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1455), Succ(x1454)), new_sr11(x522, x526), new_primPlusNat0(Succ(x1455), Succ(x1454)), x525, x526)) (10) (Succ(x1475)=Succ(Zero) & Succ(x1455)=Succ(x1475) & Succ(x1454)=Zero & (\/x1456,x1457,x1458,x1459:new_primPlusNat0(x1455, x1454)=Succ(x1456) & new_primPlusNat0(x1455, x1454)=Succ(Zero) ==> new_pr2F2(x1457, x1455, Pos(x1454), x1458, x1459)_>=_new_pr2F31(new_primPlusNat0(x1455, x1454), new_sr11(x1457, x1459), new_primPlusNat0(x1455, x1454), x1458, x1459)) ==> new_pr2F2(x522, Succ(x1455), Pos(Succ(x1454)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1455), Succ(x1454)), new_sr11(x522, x526), new_primPlusNat0(Succ(x1455), Succ(x1454)), x525, x526)) (11) (Succ(x1476)=Succ(Zero) & Succ(x1455)=Zero & Succ(x1454)=Succ(x1476) & (\/x1456,x1457,x1458,x1459:new_primPlusNat0(x1455, x1454)=Succ(x1456) & new_primPlusNat0(x1455, x1454)=Succ(Zero) ==> new_pr2F2(x1457, x1455, Pos(x1454), x1458, x1459)_>=_new_pr2F31(new_primPlusNat0(x1455, x1454), new_sr11(x1457, x1459), new_primPlusNat0(x1455, x1454), x1458, x1459)) ==> new_pr2F2(x522, Succ(x1455), Pos(Succ(x1454)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1455), Succ(x1454)), new_sr11(x522, x526), new_primPlusNat0(Succ(x1455), Succ(x1454)), x525, x526)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1477, x1478)=Succ(Zero) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1480, x1479)))=Succ(Zero) & Succ(x1460)=Succ(x1480) & Zero=Succ(x1479) & (\/x1481,x1482,x1483,x1484:new_primPlusNat0(x1480, x1479)=Succ(Zero) & Succ(x1481)=x1480 & Zero=x1479 ==> new_pr2F2(x1482, Succ(x1481), Pos(Zero), x1483, x1484)_>=_new_pr2F31(new_primPlusNat0(Succ(x1481), Zero), new_sr11(x1482, x1484), new_primPlusNat0(Succ(x1481), Zero), x1483, x1484)) ==> new_pr2F2(x522, Succ(x1460), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1460), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(x1460), Zero), x525, x526)) (13) (Succ(x1485)=Succ(Zero) & Succ(x1460)=Succ(x1485) & Zero=Zero ==> new_pr2F2(x522, Succ(x1460), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1460), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(x1460), Zero), x525, x526)) (14) (Succ(x1486)=Succ(Zero) & Succ(x1460)=Zero & Zero=Succ(x1486) ==> new_pr2F2(x522, Succ(x1460), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(x1460), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(x1460), Zero), x525, x526)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F2(x522, Succ(Zero), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(Zero), Zero), x525, x526)) We solved constraint (14) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1487, x1488)=Succ(Zero) which results in the following new constraints: (16) (Succ(Succ(new_primPlusNat0(x1490, x1489)))=Succ(Zero) & Zero=Succ(x1490) & Succ(x1461)=Succ(x1489) & (\/x1491,x1492,x1493,x1494:new_primPlusNat0(x1490, x1489)=Succ(Zero) & Zero=x1490 & Succ(x1491)=x1489 ==> new_pr2F2(x1492, Zero, Pos(Succ(x1491)), x1493, x1494)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1491)), new_sr11(x1492, x1494), new_primPlusNat0(Zero, Succ(x1491)), x1493, x1494)) ==> new_pr2F2(x522, Zero, Pos(Succ(x1461)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1461)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(x1461)), x525, x526)) (17) (Succ(x1495)=Succ(Zero) & Zero=Succ(x1495) & Succ(x1461)=Zero ==> new_pr2F2(x522, Zero, Pos(Succ(x1461)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1461)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(x1461)), x525, x526)) (18) (Succ(x1496)=Succ(Zero) & Zero=Zero & Succ(x1461)=Succ(x1496) ==> new_pr2F2(x522, Zero, Pos(Succ(x1461)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1461)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(x1461)), x525, x526)) We solved constraint (16) using rules (I), (II).We solved constraint (17) using rules (I), (II).We simplified constraint (18) using rules (I), (II), (III) which results in the following new constraint: (19) (new_pr2F2(x522, Zero, Pos(Succ(Zero)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(Zero)), x525, x526)) *We consider the chain new_pr2F2(x556, x557, Pos(x558), x559, x560) -> new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560), new_pr2F31(Succ(x561), x562, Succ(Succ(x563)), x564, x565) -> H(x562, x564, Succ(x563), x565, anew_new_pr2F0G12(x563)) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)=new_pr2F31(Succ(x561), x562, Succ(Succ(x563)), x564, x565) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x557, x558)=Succ(x561) & new_primPlusNat0(x557, x558)=Succ(Succ(x563)) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x557, x558)=Succ(x561) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1498, x1497)))=Succ(x561) & new_primPlusNat0(Succ(x1498), Succ(x1497))=Succ(Succ(x563)) & (\/x1499,x1500,x1501,x1502,x1503:new_primPlusNat0(x1498, x1497)=Succ(x1499) & new_primPlusNat0(x1498, x1497)=Succ(Succ(x1500)) ==> new_pr2F2(x1501, x1498, Pos(x1497), x1502, x1503)_>=_new_pr2F31(new_primPlusNat0(x1498, x1497), new_sr11(x1501, x1503), new_primPlusNat0(x1498, x1497), x1502, x1503)) ==> new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) (4) (Succ(x1504)=Succ(x561) & new_primPlusNat0(Succ(x1504), Zero)=Succ(Succ(x563)) ==> new_pr2F2(x556, Succ(x1504), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1504), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1504), Zero), x559, x560)) (5) (Succ(x1505)=Succ(x561) & new_primPlusNat0(Zero, Succ(x1505))=Succ(Succ(x563)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1505)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1505)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1505)), x559, x560)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1498)=x1506 & Succ(x1497)=x1507 & new_primPlusNat0(x1506, x1507)=Succ(Succ(x563)) & (\/x1499,x1500,x1501,x1502,x1503:new_primPlusNat0(x1498, x1497)=Succ(x1499) & new_primPlusNat0(x1498, x1497)=Succ(Succ(x1500)) ==> new_pr2F2(x1501, x1498, Pos(x1497), x1502, x1503)_>=_new_pr2F31(new_primPlusNat0(x1498, x1497), new_sr11(x1501, x1503), new_primPlusNat0(x1498, x1497), x1502, x1503)) ==> new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1504)=x1523 & Zero=x1524 & new_primPlusNat0(x1523, x1524)=Succ(Succ(x563)) ==> new_pr2F2(x556, Succ(x1504), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1504), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1504), Zero), x559, x560)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1534 & Succ(x1505)=x1535 & new_primPlusNat0(x1534, x1535)=Succ(Succ(x563)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1505)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1505)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1505)), x559, x560)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1506, x1507)=Succ(Succ(x563)) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1509, x1508)))=Succ(Succ(x563)) & Succ(x1498)=Succ(x1509) & Succ(x1497)=Succ(x1508) & (\/x1499,x1500,x1501,x1502,x1503:new_primPlusNat0(x1498, x1497)=Succ(x1499) & new_primPlusNat0(x1498, x1497)=Succ(Succ(x1500)) ==> new_pr2F2(x1501, x1498, Pos(x1497), x1502, x1503)_>=_new_pr2F31(new_primPlusNat0(x1498, x1497), new_sr11(x1501, x1503), new_primPlusNat0(x1498, x1497), x1502, x1503)) & (\/x1510,x1511,x1512,x1513,x1514,x1515,x1516,x1517,x1518,x1519,x1520:new_primPlusNat0(x1509, x1508)=Succ(Succ(x1510)) & Succ(x1511)=x1509 & Succ(x1512)=x1508 & (\/x1513,x1514,x1515,x1516,x1517:new_primPlusNat0(x1511, x1512)=Succ(x1513) & new_primPlusNat0(x1511, x1512)=Succ(Succ(x1514)) ==> new_pr2F2(x1515, x1511, Pos(x1512), x1516, x1517)_>=_new_pr2F31(new_primPlusNat0(x1511, x1512), new_sr11(x1515, x1517), new_primPlusNat0(x1511, x1512), x1516, x1517)) ==> new_pr2F2(x1518, Succ(x1511), Pos(Succ(x1512)), x1519, x1520)_>=_new_pr2F31(new_primPlusNat0(Succ(x1511), Succ(x1512)), new_sr11(x1518, x1520), new_primPlusNat0(Succ(x1511), Succ(x1512)), x1519, x1520)) ==> new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) (10) (Succ(x1521)=Succ(Succ(x563)) & Succ(x1498)=Succ(x1521) & Succ(x1497)=Zero & (\/x1499,x1500,x1501,x1502,x1503:new_primPlusNat0(x1498, x1497)=Succ(x1499) & new_primPlusNat0(x1498, x1497)=Succ(Succ(x1500)) ==> new_pr2F2(x1501, x1498, Pos(x1497), x1502, x1503)_>=_new_pr2F31(new_primPlusNat0(x1498, x1497), new_sr11(x1501, x1503), new_primPlusNat0(x1498, x1497), x1502, x1503)) ==> new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) (11) (Succ(x1522)=Succ(Succ(x563)) & Succ(x1498)=Zero & Succ(x1497)=Succ(x1522) & (\/x1499,x1500,x1501,x1502,x1503:new_primPlusNat0(x1498, x1497)=Succ(x1499) & new_primPlusNat0(x1498, x1497)=Succ(Succ(x1500)) ==> new_pr2F2(x1501, x1498, Pos(x1497), x1502, x1503)_>=_new_pr2F31(new_primPlusNat0(x1498, x1497), new_sr11(x1501, x1503), new_primPlusNat0(x1498, x1497), x1502, x1503)) ==> new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: (12) (new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1523, x1524)=Succ(Succ(x563)) which results in the following new constraints: (13) (Succ(Succ(new_primPlusNat0(x1526, x1525)))=Succ(Succ(x563)) & Succ(x1504)=Succ(x1526) & Zero=Succ(x1525) & (\/x1527,x1528,x1529,x1530,x1531:new_primPlusNat0(x1526, x1525)=Succ(Succ(x1527)) & Succ(x1528)=x1526 & Zero=x1525 ==> new_pr2F2(x1529, Succ(x1528), Pos(Zero), x1530, x1531)_>=_new_pr2F31(new_primPlusNat0(Succ(x1528), Zero), new_sr11(x1529, x1531), new_primPlusNat0(Succ(x1528), Zero), x1530, x1531)) ==> new_pr2F2(x556, Succ(x1504), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1504), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1504), Zero), x559, x560)) (14) (Succ(x1532)=Succ(Succ(x563)) & Succ(x1504)=Succ(x1532) & Zero=Zero ==> new_pr2F2(x556, Succ(x1504), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1504), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1504), Zero), x559, x560)) (15) (Succ(x1533)=Succ(Succ(x563)) & Succ(x1504)=Zero & Zero=Succ(x1533) ==> new_pr2F2(x556, Succ(x1504), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1504), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1504), Zero), x559, x560)) We solved constraint (13) using rules (I), (II).We simplified constraint (14) using rules (I), (II), (III) which results in the following new constraint: (16) (new_pr2F2(x556, Succ(Succ(x563)), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x563)), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Succ(x563)), Zero), x559, x560)) We solved constraint (15) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1534, x1535)=Succ(Succ(x563)) which results in the following new constraints: (17) (Succ(Succ(new_primPlusNat0(x1537, x1536)))=Succ(Succ(x563)) & Zero=Succ(x1537) & Succ(x1505)=Succ(x1536) & (\/x1538,x1539,x1540,x1541,x1542:new_primPlusNat0(x1537, x1536)=Succ(Succ(x1538)) & Zero=x1537 & Succ(x1539)=x1536 ==> new_pr2F2(x1540, Zero, Pos(Succ(x1539)), x1541, x1542)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1539)), new_sr11(x1540, x1542), new_primPlusNat0(Zero, Succ(x1539)), x1541, x1542)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1505)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1505)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1505)), x559, x560)) (18) (Succ(x1543)=Succ(Succ(x563)) & Zero=Succ(x1543) & Succ(x1505)=Zero ==> new_pr2F2(x556, Zero, Pos(Succ(x1505)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1505)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1505)), x559, x560)) (19) (Succ(x1544)=Succ(Succ(x563)) & Zero=Zero & Succ(x1505)=Succ(x1544) ==> new_pr2F2(x556, Zero, Pos(Succ(x1505)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1505)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1505)), x559, x560)) We solved constraint (17) using rules (I), (II).We solved constraint (18) using rules (I), (II).We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint: (20) (new_pr2F2(x556, Zero, Pos(Succ(Succ(x563))), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x563))), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Succ(x563))), x559, x560)) For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x580), x581, Succ(Zero), x582, x583) -> new_pr2F1(x581, Zero, new_fromInt, x582, x583), new_pr2F1(x584, x585, x586, x587, x588) -> new_pr2F34(x585, x586, x584, new_sr9(x584, x587, x588), x588) which results in the following constraint: (1) (new_pr2F1(x581, Zero, new_fromInt, x582, x583)=new_pr2F1(x584, x585, x586, x587, x588) ==> new_pr2F31(Succ(x580), x581, Succ(Zero), x582, x583)_>=_new_pr2F1(x581, Zero, new_fromInt, x582, x583)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x580), x581, Succ(Zero), x582, x583)_>=_new_pr2F1(x581, Zero, new_fromInt, x582, x583)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) the following chains were created: *We consider the chain new_pr2F0G13(x690, x691, x692, Succ(Succ(x693)), x694) -> new_pr2F0G14(x690, x691, x692, x693, x694), new_pr2F0G14(x695, x696, x697, Succ(Zero), x698) -> new_pr2F2(x696, x697, new_fromInt, x695, x698) which results in the following constraint: (1) (new_pr2F0G14(x690, x691, x692, x693, x694)=new_pr2F0G14(x695, x696, x697, Succ(Zero), x698) ==> new_pr2F0G13(x690, x691, x692, Succ(Succ(x693)), x694)_>=_new_pr2F0G14(x690, x691, x692, x693, x694)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x690, x691, x692, Succ(Succ(Succ(Zero))), x694)_>=_new_pr2F0G14(x690, x691, x692, Succ(Zero), x694)) *We consider the chain new_pr2F0G13(x699, x700, x701, Succ(Succ(x702)), x703) -> new_pr2F0G14(x699, x700, x701, x702, x703), new_pr2F0G14(x704, x705, x706, Succ(Succ(x707)), x708) -> new_pr2F0G14(x704, x705, x706, x707, x708) which results in the following constraint: (1) (new_pr2F0G14(x699, x700, x701, x702, x703)=new_pr2F0G14(x704, x705, x706, Succ(Succ(x707)), x708) ==> new_pr2F0G13(x699, x700, x701, Succ(Succ(x702)), x703)_>=_new_pr2F0G14(x699, x700, x701, x702, x703)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x699, x700, x701, Succ(Succ(Succ(Succ(x707)))), x703)_>=_new_pr2F0G14(x699, x700, x701, Succ(Succ(x707)), x703)) *We consider the chain new_pr2F0G13(x709, x710, x711, Succ(Succ(x712)), x713) -> new_pr2F0G14(x709, x710, x711, x712, x713), new_pr2F0G14(x714, x715, x716, Zero, x717) -> new_pr2F0G13(x714, new_sr10(x715, x717), new_primDivNatS1(x716), new_primDivNatS1(x716), x717) which results in the following constraint: (1) (new_pr2F0G14(x709, x710, x711, x712, x713)=new_pr2F0G14(x714, x715, x716, Zero, x717) ==> new_pr2F0G13(x709, x710, x711, Succ(Succ(x712)), x713)_>=_new_pr2F0G14(x709, x710, x711, x712, x713)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x709, x710, x711, Succ(Succ(Zero)), x713)_>=_new_pr2F0G14(x709, x710, x711, Zero, x713)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G14(x762, x763, x764, Succ(Zero), x765) -> new_pr2F2(x763, x764, new_fromInt, x762, x765), new_pr2F2(x766, x767, Pos(x768), x769, x770) -> new_pr2F31(new_primPlusNat0(x767, x768), new_sr11(x766, x770), new_primPlusNat0(x767, x768), x769, x770) which results in the following constraint: (1) (new_pr2F2(x763, x764, new_fromInt, x762, x765)=new_pr2F2(x766, x767, Pos(x768), x769, x770) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x768) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x768) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x768) ==> new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) the following chains were created: *We consider the chain new_pr2F0G14(x852, x853, x854, Succ(Succ(x855)), x856) -> new_pr2F0G14(x852, x853, x854, x855, x856), new_pr2F0G14(x857, x858, x859, Succ(Zero), x860) -> new_pr2F2(x858, x859, new_fromInt, x857, x860) which results in the following constraint: (1) (new_pr2F0G14(x852, x853, x854, x855, x856)=new_pr2F0G14(x857, x858, x859, Succ(Zero), x860) ==> new_pr2F0G14(x852, x853, x854, Succ(Succ(x855)), x856)_>=_new_pr2F0G14(x852, x853, x854, x855, x856)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x852, x853, x854, Succ(Succ(Succ(Zero))), x856)_>=_new_pr2F0G14(x852, x853, x854, Succ(Zero), x856)) *We consider the chain new_pr2F0G14(x861, x862, x863, Succ(Succ(x864)), x865) -> new_pr2F0G14(x861, x862, x863, x864, x865), new_pr2F0G14(x866, x867, x868, Succ(Succ(x869)), x870) -> new_pr2F0G14(x866, x867, x868, x869, x870) which results in the following constraint: (1) (new_pr2F0G14(x861, x862, x863, x864, x865)=new_pr2F0G14(x866, x867, x868, Succ(Succ(x869)), x870) ==> new_pr2F0G14(x861, x862, x863, Succ(Succ(x864)), x865)_>=_new_pr2F0G14(x861, x862, x863, x864, x865)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x861, x862, x863, Succ(Succ(Succ(Succ(x869)))), x865)_>=_new_pr2F0G14(x861, x862, x863, Succ(Succ(x869)), x865)) *We consider the chain new_pr2F0G14(x871, x872, x873, Succ(Succ(x874)), x875) -> new_pr2F0G14(x871, x872, x873, x874, x875), new_pr2F0G14(x876, x877, x878, Zero, x879) -> new_pr2F0G13(x876, new_sr10(x877, x879), new_primDivNatS1(x878), new_primDivNatS1(x878), x879) which results in the following constraint: (1) (new_pr2F0G14(x871, x872, x873, x874, x875)=new_pr2F0G14(x876, x877, x878, Zero, x879) ==> new_pr2F0G14(x871, x872, x873, Succ(Succ(x874)), x875)_>=_new_pr2F0G14(x871, x872, x873, x874, x875)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G14(x871, x872, x873, Succ(Succ(Zero)), x875)_>=_new_pr2F0G14(x871, x872, x873, Zero, x875)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G14(x920, x921, x922, Zero, x923) -> new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(x922), new_primDivNatS1(x922), x923), new_pr2F0G13(x924, x925, x926, Succ(Zero), x927) -> new_pr2F2(x925, x926, new_fromInt, x924, x927) which results in the following constraint: (1) (new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(x922), new_primDivNatS1(x922), x923)=new_pr2F0G13(x924, x925, x926, Succ(Zero), x927) ==> new_pr2F0G14(x920, x921, x922, Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(x922), new_primDivNatS1(x922), x923)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x922)=Succ(Zero) ==> new_pr2F0G14(x920, x921, x922, Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(x922), new_primDivNatS1(x922), x923)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x922)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1545)=Succ(Zero) ==> new_pr2F0G14(x920, x921, Succ(x1545), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(x1545)), new_primDivNatS1(Succ(x1545)), x923)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1545)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1546))=Succ(Zero) ==> new_pr2F0G14(x920, x921, Succ(Succ(Succ(x1546))), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Succ(x1546)))), new_primDivNatS1(Succ(Succ(Succ(x1546)))), x923)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G14(x920, x921, Succ(Succ(Zero)), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x923)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x920, x921, Succ(Succ(Succ(x1546))), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Succ(x1546)))), new_primDivNatS1(Succ(Succ(Succ(x1546)))), x923)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x920, x921, Succ(Succ(Zero)), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x923)) *We consider the chain new_pr2F0G14(x936, x937, x938, Zero, x939) -> new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(x938), new_primDivNatS1(x938), x939), new_pr2F0G13(x940, x941, x942, Succ(Succ(x943)), x944) -> new_pr2F0G14(x940, x941, x942, x943, x944) which results in the following constraint: (1) (new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(x938), new_primDivNatS1(x938), x939)=new_pr2F0G13(x940, x941, x942, Succ(Succ(x943)), x944) ==> new_pr2F0G14(x936, x937, x938, Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(x938), new_primDivNatS1(x938), x939)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x938)=Succ(Succ(x943)) ==> new_pr2F0G14(x936, x937, x938, Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(x938), new_primDivNatS1(x938), x939)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x938)=Succ(Succ(x943)) which results in the following new constraint: (3) (new_primDivNatS01(x1547)=Succ(Succ(x943)) ==> new_pr2F0G14(x936, x937, Succ(x1547), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(x1547)), new_primDivNatS1(Succ(x1547)), x939)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1547)=Succ(Succ(x943)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1548))=Succ(Succ(x943)) ==> new_pr2F0G14(x936, x937, Succ(Succ(Succ(x1548))), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Succ(x1548)))), new_primDivNatS1(Succ(Succ(Succ(x1548)))), x939)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x943)) ==> new_pr2F0G14(x936, x937, Succ(Succ(Zero)), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x939)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x936, x937, Succ(Succ(Succ(x1548))), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Succ(x1548)))), new_primDivNatS1(Succ(Succ(Succ(x1548)))), x939)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x936, x937, Succ(Succ(Zero)), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x939)) *We consider the chain new_pr2F0G14(x957, x958, x959, Zero, x960) -> new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(x959), new_primDivNatS1(x959), x960), new_pr2F0G13(x961, x962, x963, Zero, x964) -> new_pr2F0G13(x961, new_sr10(x962, x964), new_primDivNatS1(x963), new_primDivNatS1(x963), x964) which results in the following constraint: (1) (new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(x959), new_primDivNatS1(x959), x960)=new_pr2F0G13(x961, x962, x963, Zero, x964) ==> new_pr2F0G14(x957, x958, x959, Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(x959), new_primDivNatS1(x959), x960)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x959)=Zero ==> new_pr2F0G14(x957, x958, x959, Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(x959), new_primDivNatS1(x959), x960)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x959)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G14(x957, x958, Zero, Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x960)) (4) (new_primDivNatS01(x1549)=Zero ==> new_pr2F0G14(x957, x958, Succ(x1549), Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Succ(x1549)), new_primDivNatS1(Succ(x1549)), x960)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G14(x957, x958, Zero, Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x960)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1549)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G14(x957, x958, Succ(Zero), Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x960)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G14(x957, x958, Succ(Zero), Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x960)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G13(x997, x998, x999, Zero, x1000) -> new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(x999), new_primDivNatS1(x999), x1000), new_pr2F0G13(x1001, x1002, x1003, Succ(Zero), x1004) -> new_pr2F2(x1002, x1003, new_fromInt, x1001, x1004) which results in the following constraint: (1) (new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(x999), new_primDivNatS1(x999), x1000)=new_pr2F0G13(x1001, x1002, x1003, Succ(Zero), x1004) ==> new_pr2F0G13(x997, x998, x999, Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(x999), new_primDivNatS1(x999), x1000)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x999)=Succ(Zero) ==> new_pr2F0G13(x997, x998, x999, Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(x999), new_primDivNatS1(x999), x1000)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x999)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1551)=Succ(Zero) ==> new_pr2F0G13(x997, x998, Succ(x1551), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(x1551)), new_primDivNatS1(Succ(x1551)), x1000)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1551)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1552))=Succ(Zero) ==> new_pr2F0G13(x997, x998, Succ(Succ(Succ(x1552))), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Succ(x1552)))), new_primDivNatS1(Succ(Succ(Succ(x1552)))), x1000)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G13(x997, x998, Succ(Succ(Zero)), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1000)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x997, x998, Succ(Succ(Succ(x1552))), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Succ(x1552)))), new_primDivNatS1(Succ(Succ(Succ(x1552)))), x1000)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x997, x998, Succ(Succ(Zero)), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1000)) *We consider the chain new_pr2F0G13(x1013, x1014, x1015, Zero, x1016) -> new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(x1015), new_primDivNatS1(x1015), x1016), new_pr2F0G13(x1017, x1018, x1019, Succ(Succ(x1020)), x1021) -> new_pr2F0G14(x1017, x1018, x1019, x1020, x1021) which results in the following constraint: (1) (new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(x1015), new_primDivNatS1(x1015), x1016)=new_pr2F0G13(x1017, x1018, x1019, Succ(Succ(x1020)), x1021) ==> new_pr2F0G13(x1013, x1014, x1015, Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(x1015), new_primDivNatS1(x1015), x1016)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1015)=Succ(Succ(x1020)) ==> new_pr2F0G13(x1013, x1014, x1015, Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(x1015), new_primDivNatS1(x1015), x1016)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1015)=Succ(Succ(x1020)) which results in the following new constraint: (3) (new_primDivNatS01(x1553)=Succ(Succ(x1020)) ==> new_pr2F0G13(x1013, x1014, Succ(x1553), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(x1553)), new_primDivNatS1(Succ(x1553)), x1016)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1553)=Succ(Succ(x1020)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1554))=Succ(Succ(x1020)) ==> new_pr2F0G13(x1013, x1014, Succ(Succ(Succ(x1554))), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Succ(x1554)))), new_primDivNatS1(Succ(Succ(Succ(x1554)))), x1016)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x1020)) ==> new_pr2F0G13(x1013, x1014, Succ(Succ(Zero)), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1016)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x1013, x1014, Succ(Succ(Succ(x1554))), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Succ(x1554)))), new_primDivNatS1(Succ(Succ(Succ(x1554)))), x1016)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x1013, x1014, Succ(Succ(Zero)), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1016)) *We consider the chain new_pr2F0G13(x1034, x1035, x1036, Zero, x1037) -> new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(x1036), new_primDivNatS1(x1036), x1037), new_pr2F0G13(x1038, x1039, x1040, Zero, x1041) -> new_pr2F0G13(x1038, new_sr10(x1039, x1041), new_primDivNatS1(x1040), new_primDivNatS1(x1040), x1041) which results in the following constraint: (1) (new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(x1036), new_primDivNatS1(x1036), x1037)=new_pr2F0G13(x1038, x1039, x1040, Zero, x1041) ==> new_pr2F0G13(x1034, x1035, x1036, Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(x1036), new_primDivNatS1(x1036), x1037)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1036)=Zero ==> new_pr2F0G13(x1034, x1035, x1036, Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(x1036), new_primDivNatS1(x1036), x1037)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1036)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G13(x1034, x1035, Zero, Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1037)) (4) (new_primDivNatS01(x1555)=Zero ==> new_pr2F0G13(x1034, x1035, Succ(x1555), Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Succ(x1555)), new_primDivNatS1(Succ(x1555)), x1037)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G13(x1034, x1035, Zero, Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1037)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1555)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G13(x1034, x1035, Succ(Zero), Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1037)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G13(x1034, x1035, Succ(Zero), Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1037)) For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) the following chains were created: *We consider the chain new_pr2F31(Succ(x1124), x1125, Succ(Succ(x1126)), x1127, x1128) -> H(x1125, x1127, Succ(x1126), x1128, anew_new_pr2F0G12(x1126)), H(x1129, x1130, x1131, x1132, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(x1129, x1130, x1131, Succ(Zero), x1132) which results in the following constraint: (1) (H(x1125, x1127, Succ(x1126), x1128, anew_new_pr2F0G12(x1126))=H(x1129, x1130, x1131, x1132, cons_new_pr2F0G12(Succ(Zero))) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(x1126)), x1127, x1128)_>=_H(x1125, x1127, Succ(x1126), x1128, anew_new_pr2F0G12(x1126))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G12(x1126)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(x1126)), x1127, x1128)_>=_H(x1125, x1127, Succ(x1126), x1128, anew_new_pr2F0G12(x1126))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G12(x1126)=cons_new_pr2F0G12(Succ(Zero)) which results in the following new constraint: (3) (new_new_pr2F0G12(x1557)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(x1557)))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(x1557))), x1128, anew_new_pr2F0G12(Succ(Succ(x1557))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G12(x1557)=cons_new_pr2F0G12(Succ(Zero)) which results in the following new constraints: (4) (new_new_pr2F0G12(x1558)=cons_new_pr2F0G12(Succ(Zero)) & (\/x1559,x1560,x1561,x1562:new_new_pr2F0G12(x1558)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1559), x1560, Succ(Succ(Succ(Succ(x1558)))), x1561, x1562)_>=_H(x1560, x1561, Succ(Succ(Succ(x1558))), x1562, anew_new_pr2F0G12(Succ(Succ(x1558))))) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Succ(x1558)))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Succ(x1558))))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1558))))))) (5) (cons_new_pr2F0G12(Succ(Zero))=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Zero))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Zero)))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G12(Zero)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Zero)))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Zero))), x1128, anew_new_pr2F0G12(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1559,x1560,x1561,x1562:new_new_pr2F0G12(x1558)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1559), x1560, Succ(Succ(Succ(Succ(x1558)))), x1561, x1562)_>=_H(x1560, x1561, Succ(Succ(Succ(x1558))), x1562, anew_new_pr2F0G12(Succ(Succ(x1558))))) with sigma = [x1559 / x1124, x1560 / x1125, x1561 / x1127, x1562 / x1128] which results in the following new constraint: (7) (new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(x1558)))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(x1558))), x1128, anew_new_pr2F0G12(Succ(Succ(x1558)))) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Succ(x1558)))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Succ(x1558))))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1558))))))) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Zero))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Zero)))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_pr2F31(Succ(x1133), x1134, Succ(Succ(x1135)), x1136, x1137) -> H(x1134, x1136, Succ(x1135), x1137, anew_new_pr2F0G12(x1135)), H(x1138, x1139, x1140, x1141, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(x1138, x1139, x1140, Zero, x1141) which results in the following constraint: (1) (H(x1134, x1136, Succ(x1135), x1137, anew_new_pr2F0G12(x1135))=H(x1138, x1139, x1140, x1141, cons_new_pr2F0G12(Zero)) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(x1135)), x1136, x1137)_>=_H(x1134, x1136, Succ(x1135), x1137, anew_new_pr2F0G12(x1135))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G12(x1135)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(x1135)), x1136, x1137)_>=_H(x1134, x1136, Succ(x1135), x1137, anew_new_pr2F0G12(x1135))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G12(x1135)=cons_new_pr2F0G12(Zero) which results in the following new constraint: (3) (new_new_pr2F0G12(x1563)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(x1563)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(x1563))), x1137, anew_new_pr2F0G12(Succ(Succ(x1563))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G12(x1563)=cons_new_pr2F0G12(Zero) which results in the following new constraints: (4) (new_new_pr2F0G12(x1564)=cons_new_pr2F0G12(Zero) & (\/x1565,x1566,x1567,x1568:new_new_pr2F0G12(x1564)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1565), x1566, Succ(Succ(Succ(Succ(x1564)))), x1567, x1568)_>=_H(x1566, x1567, Succ(Succ(Succ(x1564))), x1568, anew_new_pr2F0G12(Succ(Succ(x1564))))) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Succ(Succ(x1564)))))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Succ(Succ(x1564))))), x1137, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1564))))))) (5) (cons_new_pr2F0G12(Succ(Zero))=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Succ(Zero))))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Succ(Zero)))), x1137, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G12(Zero)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Zero)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Zero))), x1137, anew_new_pr2F0G12(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1565,x1566,x1567,x1568:new_new_pr2F0G12(x1564)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1565), x1566, Succ(Succ(Succ(Succ(x1564)))), x1567, x1568)_>=_H(x1566, x1567, Succ(Succ(Succ(x1564))), x1568, anew_new_pr2F0G12(Succ(Succ(x1564))))) with sigma = [x1565 / x1133, x1566 / x1134, x1567 / x1136, x1568 / x1137] which results in the following new constraint: (7) (new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(x1564)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(x1564))), x1137, anew_new_pr2F0G12(Succ(Succ(x1564)))) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Succ(Succ(x1564)))))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Succ(Succ(x1564))))), x1137, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1564))))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Zero)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Zero))), x1137, anew_new_pr2F0G12(Succ(Succ(Zero))))) For Pair H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) the following chains were created: *We consider the chain H(x1142, x1143, x1144, x1145, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(x1142, x1143, x1144, Succ(Zero), x1145), new_pr2F0G12(x1146, x1147, x1148, Succ(Zero), x1149) -> new_pr2F1(x1146, x1148, new_fromInt, x1147, x1149) which results in the following constraint: (1) (new_pr2F0G12(x1142, x1143, x1144, Succ(Zero), x1145)=new_pr2F0G12(x1146, x1147, x1148, Succ(Zero), x1149) ==> H(x1142, x1143, x1144, x1145, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1142, x1143, x1144, Succ(Zero), x1145)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x1142, x1143, x1144, x1145, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1142, x1143, x1144, Succ(Zero), x1145)) For Pair H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) the following chains were created: *We consider the chain H(x1226, x1227, x1228, x1229, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(x1226, x1227, x1228, Zero, x1229), new_pr2F0G12(x1230, x1231, x1232, Zero, x1233) -> new_pr2F0G13(new_sr8(x1230, x1231, x1233), x1230, new_primDivNatS1(Succ(x1232)), new_primDivNatS1(Succ(x1232)), x1233) which results in the following constraint: (1) (new_pr2F0G12(x1226, x1227, x1228, Zero, x1229)=new_pr2F0G12(x1230, x1231, x1232, Zero, x1233) ==> H(x1226, x1227, x1228, x1229, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1226, x1227, x1228, Zero, x1229)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x1226, x1227, x1228, x1229, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1226, x1227, x1228, Zero, x1229)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) *(new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) *new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) *(new_pr2F1(x79, x80, Pos(x85), x82, x83)_>=_new_pr2F34(x80, Pos(x85), x79, new_sr9(x79, x82, x83), x83)) *new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) *(new_pr2F34(Succ(x176), Pos(Zero), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x176)), Zero), x171, new_primPlusNat0(Succ(Succ(x176)), Zero), x172, x173)) *(new_pr2F34(x169, Pos(Succ(x1280)), x171, x172, x173)_>=_new_pr2F31(new_primPlusNat0(Succ(x169), Succ(x1280)), x171, new_primPlusNat0(Succ(x169), Succ(x1280)), x172, x173)) *(new_pr2F34(Zero, Pos(Zero), x196, x197, x198)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x196, new_primPlusNat0(Succ(Zero), Zero), x197, x198)) *(new_pr2F34(Succ(x235), Pos(Zero), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x235)), Zero), x230, new_primPlusNat0(Succ(Succ(x235)), Zero), x231, x232)) *(new_pr2F34(x228, Pos(Succ(x1358)), x230, x231, x232)_>=_new_pr2F31(new_primPlusNat0(Succ(x228), Succ(x1358)), x230, new_primPlusNat0(Succ(x228), Succ(x1358)), x231, x232)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) *(new_pr2F31(Succ(x248), x249, Succ(Succ(Succ(Zero))), x251, x252)_>=_new_pr2F0G12(x249, x251, Succ(Succ(Zero)), Succ(Zero), x252)) *(new_pr2F31(Succ(x272), x273, Succ(Succ(Zero)), x275, x276)_>=_new_pr2F0G12(x273, x275, Succ(Zero), Zero, x276)) *new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) *(new_pr2F0G12(x356, x357, Succ(Succ(x1399)), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Succ(x1399)))), new_primDivNatS1(Succ(Succ(Succ(x1399)))), x359)) *(new_pr2F0G12(x356, x357, Succ(Zero), Zero, x359)_>=_new_pr2F0G13(new_sr8(x356, x357, x359), x356, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x359)) *(new_pr2F0G12(x372, x373, Succ(Succ(x1402)), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Succ(x1402)))), new_primDivNatS1(Succ(Succ(Succ(x1402)))), x375)) *(new_pr2F0G12(x372, x373, Succ(Zero), Zero, x375)_>=_new_pr2F0G13(new_sr8(x372, x373, x375), x372, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x375)) *(new_pr2F0G12(x393, x394, Zero, Zero, x396)_>=_new_pr2F0G13(new_sr8(x393, x394, x396), x393, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x396)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G13(x437, x438, x439, Succ(Zero), x440)_>=_new_pr2F2(x438, x439, new_fromInt, x437, x440)) *new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) *(new_pr2F2(x497, Succ(Succ(x504)), Pos(Zero), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x504)), Zero), new_sr11(x497, x501), new_primPlusNat0(Succ(Succ(x504)), Zero), x500, x501)) *(new_pr2F2(x497, Zero, Pos(Succ(Succ(x504))), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x504))), new_sr11(x497, x501), new_primPlusNat0(Zero, Succ(Succ(x504))), x500, x501)) *(new_pr2F2(x497, Succ(x1407), Pos(Succ(x1406)), x500, x501)_>=_new_pr2F31(new_primPlusNat0(Succ(x1407), Succ(x1406)), new_sr11(x497, x501), new_primPlusNat0(Succ(x1407), Succ(x1406)), x500, x501)) *(new_pr2F2(x522, Succ(Zero), Pos(Zero), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x522, x526), new_primPlusNat0(Succ(Zero), Zero), x525, x526)) *(new_pr2F2(x522, Zero, Pos(Succ(Zero)), x525, x526)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x522, x526), new_primPlusNat0(Zero, Succ(Zero)), x525, x526)) *(new_pr2F2(x556, Succ(Succ(x563)), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x563)), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Succ(x563)), Zero), x559, x560)) *(new_pr2F2(x556, Zero, Pos(Succ(Succ(x563))), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x563))), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Succ(x563))), x559, x560)) *(new_pr2F2(x556, Succ(x1498), Pos(Succ(x1497)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1498), Succ(x1497)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1498), Succ(x1497)), x559, x560)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) *(new_pr2F31(Succ(x580), x581, Succ(Zero), x582, x583)_>=_new_pr2F1(x581, Zero, new_fromInt, x582, x583)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) *(new_pr2F0G13(x690, x691, x692, Succ(Succ(Succ(Zero))), x694)_>=_new_pr2F0G14(x690, x691, x692, Succ(Zero), x694)) *(new_pr2F0G13(x699, x700, x701, Succ(Succ(Succ(Succ(x707)))), x703)_>=_new_pr2F0G14(x699, x700, x701, Succ(Succ(x707)), x703)) *(new_pr2F0G13(x709, x710, x711, Succ(Succ(Zero)), x713)_>=_new_pr2F0G14(x709, x710, x711, Zero, x713)) *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G14(x762, x763, x764, Succ(Zero), x765)_>=_new_pr2F2(x763, x764, new_fromInt, x762, x765)) *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) *(new_pr2F0G14(x852, x853, x854, Succ(Succ(Succ(Zero))), x856)_>=_new_pr2F0G14(x852, x853, x854, Succ(Zero), x856)) *(new_pr2F0G14(x861, x862, x863, Succ(Succ(Succ(Succ(x869)))), x865)_>=_new_pr2F0G14(x861, x862, x863, Succ(Succ(x869)), x865)) *(new_pr2F0G14(x871, x872, x873, Succ(Succ(Zero)), x875)_>=_new_pr2F0G14(x871, x872, x873, Zero, x875)) *new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G14(x920, x921, Succ(Succ(Succ(x1546))), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Succ(x1546)))), new_primDivNatS1(Succ(Succ(Succ(x1546)))), x923)) *(new_pr2F0G14(x920, x921, Succ(Succ(Zero)), Zero, x923)_>=_new_pr2F0G13(x920, new_sr10(x921, x923), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x923)) *(new_pr2F0G14(x936, x937, Succ(Succ(Succ(x1548))), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Succ(x1548)))), new_primDivNatS1(Succ(Succ(Succ(x1548)))), x939)) *(new_pr2F0G14(x936, x937, Succ(Succ(Zero)), Zero, x939)_>=_new_pr2F0G13(x936, new_sr10(x937, x939), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x939)) *(new_pr2F0G14(x957, x958, Zero, Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x960)) *(new_pr2F0G14(x957, x958, Succ(Zero), Zero, x960)_>=_new_pr2F0G13(x957, new_sr10(x958, x960), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x960)) *new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G13(x997, x998, Succ(Succ(Succ(x1552))), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Succ(x1552)))), new_primDivNatS1(Succ(Succ(Succ(x1552)))), x1000)) *(new_pr2F0G13(x997, x998, Succ(Succ(Zero)), Zero, x1000)_>=_new_pr2F0G13(x997, new_sr10(x998, x1000), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1000)) *(new_pr2F0G13(x1013, x1014, Succ(Succ(Succ(x1554))), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Succ(x1554)))), new_primDivNatS1(Succ(Succ(Succ(x1554)))), x1016)) *(new_pr2F0G13(x1013, x1014, Succ(Succ(Zero)), Zero, x1016)_>=_new_pr2F0G13(x1013, new_sr10(x1014, x1016), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1016)) *(new_pr2F0G13(x1034, x1035, Zero, Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1037)) *(new_pr2F0G13(x1034, x1035, Succ(Zero), Zero, x1037)_>=_new_pr2F0G13(x1034, new_sr10(x1035, x1037), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1037)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) *(new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(x1558)))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(x1558))), x1128, anew_new_pr2F0G12(Succ(Succ(x1558)))) ==> new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Succ(x1558)))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Succ(x1558))))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1558))))))) *(new_pr2F31(Succ(x1124), x1125, Succ(Succ(Succ(Succ(Succ(Zero))))), x1127, x1128)_>=_H(x1125, x1127, Succ(Succ(Succ(Succ(Zero)))), x1128, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) *(new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(x1564)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(x1564))), x1137, anew_new_pr2F0G12(Succ(Succ(x1564)))) ==> new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Succ(Succ(x1564)))))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Succ(Succ(x1564))))), x1137, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1564))))))) *(new_pr2F31(Succ(x1133), x1134, Succ(Succ(Succ(Succ(Zero)))), x1136, x1137)_>=_H(x1134, x1136, Succ(Succ(Succ(Zero))), x1137, anew_new_pr2F0G12(Succ(Succ(Zero))))) *H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) *(H(x1142, x1143, x1144, x1145, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1142, x1143, x1144, Succ(Zero), x1145)) *H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) *(H(x1226, x1227, x1228, x1229, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1226, x1227, x1228, Zero, x1229)) 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. ---------------------------------------- (44) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) new_new_pr2F0G12(Succ(Succ(x0))) anew_new_pr2F0G12(Succ(Succ(x0))) new_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) QDPPairToRuleProof (EQUIVALENT) The dependency pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) was transformed to the following new rules: anew_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Zero)) -> cons_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) -> cons_new_pr2F0G14(Zero) the following new pairs maintain the fan-in: new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) the following new pairs maintain the fan-out: H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) ---------------------------------------- (46) Complex Obligation (AND) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) anew_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Zero)) -> cons_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) -> cons_new_pr2F0G14(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) new_new_pr2F0G12(Succ(Succ(x0))) anew_new_pr2F0G12(Succ(Succ(x0))) new_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) new_new_pr2F0G14(Succ(Succ(x0))) anew_new_pr2F0G14(Succ(Succ(x0))) new_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (49) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) anew_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Zero)) -> cons_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) -> cons_new_pr2F0G14(Zero) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (50) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) the following chains were created: *We consider the chain new_pr2F0G12(x4, x5, x6, Succ(Zero), x7) -> new_pr2F1(x4, x6, new_fromInt, x5, x7), new_pr2F1(x8, x9, x10, x11, x12) -> new_pr2F34(x9, x10, x8, new_sr9(x8, x11, x12), x12) which results in the following constraint: (1) (new_pr2F1(x4, x6, new_fromInt, x5, x7)=new_pr2F1(x8, x9, x10, x11, x12) ==> new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) For Pair new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) the following chains were created: *We consider the chain new_pr2F1(x87, x88, x89, x90, x91) -> new_pr2F34(x88, x89, x87, new_sr9(x87, x90, x91), x91), new_pr2F34(x92, Pos(x93), x94, x95, x96) -> new_pr2F31(new_primPlusNat0(Succ(x92), x93), x94, new_primPlusNat0(Succ(x92), x93), x95, x96) which results in the following constraint: (1) (new_pr2F34(x88, x89, x87, new_sr9(x87, x90, x91), x91)=new_pr2F34(x92, Pos(x93), x94, x95, x96) ==> new_pr2F1(x87, x88, x89, x90, x91)_>=_new_pr2F34(x88, x89, x87, new_sr9(x87, x90, x91), x91)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F1(x87, x88, Pos(x93), x90, x91)_>=_new_pr2F34(x88, Pos(x93), x87, new_sr9(x87, x90, x91), x91)) For Pair new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) the following chains were created: *We consider the chain new_pr2F34(x187, Pos(x188), x189, x190, x191) -> new_pr2F31(new_primPlusNat0(Succ(x187), x188), x189, new_primPlusNat0(Succ(x187), x188), x190, x191), new_pr2F31(Succ(x192), x193, Succ(Succ(x194)), x195, x196) -> new_pr2F0G12(x193, x195, Succ(x194), x194, x196) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x187), x188), x189, new_primPlusNat0(Succ(x187), x188), x190, x191)=new_pr2F31(Succ(x192), x193, Succ(Succ(x194)), x195, x196) ==> new_pr2F34(x187, Pos(x188), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), x188), x189, new_primPlusNat0(Succ(x187), x188), x190, x191)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x187)=x1577 & new_primPlusNat0(x1577, x188)=Succ(x192) & Succ(x187)=x1578 & new_primPlusNat0(x1578, x188)=Succ(Succ(x194)) ==> new_pr2F34(x187, Pos(x188), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), x188), x189, new_primPlusNat0(Succ(x187), x188), x190, x191)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1577, x188)=Succ(x192) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1580, x1579)))=Succ(x192) & Succ(x187)=Succ(x1580) & Succ(x187)=x1578 & new_primPlusNat0(x1578, Succ(x1579))=Succ(Succ(x194)) & (\/x1581,x1582,x1583,x1584,x1585,x1586,x1587:new_primPlusNat0(x1580, x1579)=Succ(x1581) & Succ(x1582)=x1580 & Succ(x1582)=x1583 & new_primPlusNat0(x1583, x1579)=Succ(Succ(x1584)) ==> new_pr2F34(x1582, Pos(x1579), x1585, x1586, x1587)_>=_new_pr2F31(new_primPlusNat0(Succ(x1582), x1579), x1585, new_primPlusNat0(Succ(x1582), x1579), x1586, x1587)) ==> new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) (4) (Succ(x1588)=Succ(x192) & Succ(x187)=Succ(x1588) & Succ(x187)=x1578 & new_primPlusNat0(x1578, Zero)=Succ(Succ(x194)) ==> new_pr2F34(x187, Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Zero), x189, new_primPlusNat0(Succ(x187), Zero), x190, x191)) (5) (Succ(x1589)=Succ(x192) & Succ(x187)=Zero & Succ(x187)=x1578 & new_primPlusNat0(x1578, Succ(x1589))=Succ(Succ(x194)) ==> new_pr2F34(x187, Pos(Succ(x1589)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1589)), x189, new_primPlusNat0(Succ(x187), Succ(x1589)), x190, x191)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x187)=x1578 & Succ(x1579)=x1590 & new_primPlusNat0(x1578, x1590)=Succ(Succ(x194)) & (\/x1581,x1582,x1583,x1584,x1585,x1586,x1587:new_primPlusNat0(x187, x1579)=Succ(x1581) & Succ(x1582)=x187 & Succ(x1582)=x1583 & new_primPlusNat0(x1583, x1579)=Succ(Succ(x1584)) ==> new_pr2F34(x1582, Pos(x1579), x1585, x1586, x1587)_>=_new_pr2F31(new_primPlusNat0(Succ(x1582), x1579), x1585, new_primPlusNat0(Succ(x1582), x1579), x1586, x1587)) ==> new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x187)=x1578 & Zero=x1608 & new_primPlusNat0(x1578, x1608)=Succ(Succ(x194)) ==> new_pr2F34(x187, Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Zero), x189, new_primPlusNat0(Succ(x187), Zero), x190, x191)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1578, x1590)=Succ(Succ(x194)) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1592, x1591)))=Succ(Succ(x194)) & Succ(x187)=Succ(x1592) & Succ(x1579)=Succ(x1591) & (\/x1581,x1582,x1583,x1584,x1585,x1586,x1587:new_primPlusNat0(x187, x1579)=Succ(x1581) & Succ(x1582)=x187 & Succ(x1582)=x1583 & new_primPlusNat0(x1583, x1579)=Succ(Succ(x1584)) ==> new_pr2F34(x1582, Pos(x1579), x1585, x1586, x1587)_>=_new_pr2F31(new_primPlusNat0(Succ(x1582), x1579), x1585, new_primPlusNat0(Succ(x1582), x1579), x1586, x1587)) & (\/x1593,x1594,x1595,x1596,x1597,x1598,x1599,x1600,x1601,x1602,x1603,x1604,x1605:new_primPlusNat0(x1592, x1591)=Succ(Succ(x1593)) & Succ(x1594)=x1592 & Succ(x1595)=x1591 & (\/x1596,x1597,x1598,x1599,x1600,x1601,x1602:new_primPlusNat0(x1594, x1595)=Succ(x1596) & Succ(x1597)=x1594 & Succ(x1597)=x1598 & new_primPlusNat0(x1598, x1595)=Succ(Succ(x1599)) ==> new_pr2F34(x1597, Pos(x1595), x1600, x1601, x1602)_>=_new_pr2F31(new_primPlusNat0(Succ(x1597), x1595), x1600, new_primPlusNat0(Succ(x1597), x1595), x1601, x1602)) ==> new_pr2F34(x1594, Pos(Succ(x1595)), x1603, x1604, x1605)_>=_new_pr2F31(new_primPlusNat0(Succ(x1594), Succ(x1595)), x1603, new_primPlusNat0(Succ(x1594), Succ(x1595)), x1604, x1605)) ==> new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) (9) (Succ(x1606)=Succ(Succ(x194)) & Succ(x187)=Succ(x1606) & Succ(x1579)=Zero & (\/x1581,x1582,x1583,x1584,x1585,x1586,x1587:new_primPlusNat0(x187, x1579)=Succ(x1581) & Succ(x1582)=x187 & Succ(x1582)=x1583 & new_primPlusNat0(x1583, x1579)=Succ(Succ(x1584)) ==> new_pr2F34(x1582, Pos(x1579), x1585, x1586, x1587)_>=_new_pr2F31(new_primPlusNat0(Succ(x1582), x1579), x1585, new_primPlusNat0(Succ(x1582), x1579), x1586, x1587)) ==> new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) (10) (Succ(x1607)=Succ(Succ(x194)) & Succ(x187)=Zero & Succ(x1579)=Succ(x1607) & (\/x1581,x1582,x1583,x1584,x1585,x1586,x1587:new_primPlusNat0(x187, x1579)=Succ(x1581) & Succ(x1582)=x187 & Succ(x1582)=x1583 & new_primPlusNat0(x1583, x1579)=Succ(Succ(x1584)) ==> new_pr2F34(x1582, Pos(x1579), x1585, x1586, x1587)_>=_new_pr2F31(new_primPlusNat0(Succ(x1582), x1579), x1585, new_primPlusNat0(Succ(x1582), x1579), x1586, x1587)) ==> new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint: (11) (new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1578, x1608)=Succ(Succ(x194)) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1610, x1609)))=Succ(Succ(x194)) & Succ(x187)=Succ(x1610) & Zero=Succ(x1609) & (\/x1611,x1612,x1613,x1614,x1615:new_primPlusNat0(x1610, x1609)=Succ(Succ(x1611)) & Succ(x1612)=x1610 & Zero=x1609 ==> new_pr2F34(x1612, Pos(Zero), x1613, x1614, x1615)_>=_new_pr2F31(new_primPlusNat0(Succ(x1612), Zero), x1613, new_primPlusNat0(Succ(x1612), Zero), x1614, x1615)) ==> new_pr2F34(x187, Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Zero), x189, new_primPlusNat0(Succ(x187), Zero), x190, x191)) (13) (Succ(x1616)=Succ(Succ(x194)) & Succ(x187)=Succ(x1616) & Zero=Zero ==> new_pr2F34(x187, Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Zero), x189, new_primPlusNat0(Succ(x187), Zero), x190, x191)) (14) (Succ(x1617)=Succ(Succ(x194)) & Succ(x187)=Zero & Zero=Succ(x1617) ==> new_pr2F34(x187, Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Zero), x189, new_primPlusNat0(Succ(x187), Zero), x190, x191)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F34(Succ(x194), Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x194)), Zero), x189, new_primPlusNat0(Succ(Succ(x194)), Zero), x190, x191)) We solved constraint (14) using rules (I), (II). *We consider the chain new_pr2F34(x212, Pos(x213), x214, x215, x216) -> new_pr2F31(new_primPlusNat0(Succ(x212), x213), x214, new_primPlusNat0(Succ(x212), x213), x215, x216), new_pr2F31(Succ(x217), x218, Succ(Zero), x219, x220) -> new_pr2F1(x218, Zero, new_fromInt, x219, x220) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x212), x213), x214, new_primPlusNat0(Succ(x212), x213), x215, x216)=new_pr2F31(Succ(x217), x218, Succ(Zero), x219, x220) ==> new_pr2F34(x212, Pos(x213), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), x213), x214, new_primPlusNat0(Succ(x212), x213), x215, x216)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x212)=x1618 & new_primPlusNat0(x1618, x213)=Succ(x217) & Succ(x212)=x1619 & new_primPlusNat0(x1619, x213)=Succ(Zero) ==> new_pr2F34(x212, Pos(x213), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), x213), x214, new_primPlusNat0(Succ(x212), x213), x215, x216)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1618, x213)=Succ(x217) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1621, x1620)))=Succ(x217) & Succ(x212)=Succ(x1621) & Succ(x212)=x1619 & new_primPlusNat0(x1619, Succ(x1620))=Succ(Zero) & (\/x1622,x1623,x1624,x1625,x1626,x1627:new_primPlusNat0(x1621, x1620)=Succ(x1622) & Succ(x1623)=x1621 & Succ(x1623)=x1624 & new_primPlusNat0(x1624, x1620)=Succ(Zero) ==> new_pr2F34(x1623, Pos(x1620), x1625, x1626, x1627)_>=_new_pr2F31(new_primPlusNat0(Succ(x1623), x1620), x1625, new_primPlusNat0(Succ(x1623), x1620), x1626, x1627)) ==> new_pr2F34(x212, Pos(Succ(x1620)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1620)), x214, new_primPlusNat0(Succ(x212), Succ(x1620)), x215, x216)) (4) (Succ(x1628)=Succ(x217) & Succ(x212)=Succ(x1628) & Succ(x212)=x1619 & new_primPlusNat0(x1619, Zero)=Succ(Zero) ==> new_pr2F34(x212, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Zero), x214, new_primPlusNat0(Succ(x212), Zero), x215, x216)) (5) (Succ(x1629)=Succ(x217) & Succ(x212)=Zero & Succ(x212)=x1619 & new_primPlusNat0(x1619, Succ(x1629))=Succ(Zero) ==> new_pr2F34(x212, Pos(Succ(x1629)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1629)), x214, new_primPlusNat0(Succ(x212), Succ(x1629)), x215, x216)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x212)=x1619 & Succ(x1620)=x1630 & new_primPlusNat0(x1619, x1630)=Succ(Zero) & (\/x1622,x1623,x1624,x1625,x1626,x1627:new_primPlusNat0(x212, x1620)=Succ(x1622) & Succ(x1623)=x212 & Succ(x1623)=x1624 & new_primPlusNat0(x1624, x1620)=Succ(Zero) ==> new_pr2F34(x1623, Pos(x1620), x1625, x1626, x1627)_>=_new_pr2F31(new_primPlusNat0(Succ(x1623), x1620), x1625, new_primPlusNat0(Succ(x1623), x1620), x1626, x1627)) ==> new_pr2F34(x212, Pos(Succ(x1620)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1620)), x214, new_primPlusNat0(Succ(x212), Succ(x1620)), x215, x216)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x212)=x1619 & Zero=x1646 & new_primPlusNat0(x1619, x1646)=Succ(Zero) ==> new_pr2F34(x212, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Zero), x214, new_primPlusNat0(Succ(x212), Zero), x215, x216)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1619, x1630)=Succ(Zero) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1632, x1631)))=Succ(Zero) & Succ(x212)=Succ(x1632) & Succ(x1620)=Succ(x1631) & (\/x1622,x1623,x1624,x1625,x1626,x1627:new_primPlusNat0(x212, x1620)=Succ(x1622) & Succ(x1623)=x212 & Succ(x1623)=x1624 & new_primPlusNat0(x1624, x1620)=Succ(Zero) ==> new_pr2F34(x1623, Pos(x1620), x1625, x1626, x1627)_>=_new_pr2F31(new_primPlusNat0(Succ(x1623), x1620), x1625, new_primPlusNat0(Succ(x1623), x1620), x1626, x1627)) & (\/x1633,x1634,x1635,x1636,x1637,x1638,x1639,x1640,x1641,x1642,x1643:new_primPlusNat0(x1632, x1631)=Succ(Zero) & Succ(x1633)=x1632 & Succ(x1634)=x1631 & (\/x1635,x1636,x1637,x1638,x1639,x1640:new_primPlusNat0(x1633, x1634)=Succ(x1635) & Succ(x1636)=x1633 & Succ(x1636)=x1637 & new_primPlusNat0(x1637, x1634)=Succ(Zero) ==> new_pr2F34(x1636, Pos(x1634), x1638, x1639, x1640)_>=_new_pr2F31(new_primPlusNat0(Succ(x1636), x1634), x1638, new_primPlusNat0(Succ(x1636), x1634), x1639, x1640)) ==> new_pr2F34(x1633, Pos(Succ(x1634)), x1641, x1642, x1643)_>=_new_pr2F31(new_primPlusNat0(Succ(x1633), Succ(x1634)), x1641, new_primPlusNat0(Succ(x1633), Succ(x1634)), x1642, x1643)) ==> new_pr2F34(x212, Pos(Succ(x1620)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1620)), x214, new_primPlusNat0(Succ(x212), Succ(x1620)), x215, x216)) (9) (Succ(x1644)=Succ(Zero) & Succ(x212)=Succ(x1644) & Succ(x1620)=Zero & (\/x1622,x1623,x1624,x1625,x1626,x1627:new_primPlusNat0(x212, x1620)=Succ(x1622) & Succ(x1623)=x212 & Succ(x1623)=x1624 & new_primPlusNat0(x1624, x1620)=Succ(Zero) ==> new_pr2F34(x1623, Pos(x1620), x1625, x1626, x1627)_>=_new_pr2F31(new_primPlusNat0(Succ(x1623), x1620), x1625, new_primPlusNat0(Succ(x1623), x1620), x1626, x1627)) ==> new_pr2F34(x212, Pos(Succ(x1620)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1620)), x214, new_primPlusNat0(Succ(x212), Succ(x1620)), x215, x216)) (10) (Succ(x1645)=Succ(Zero) & Succ(x212)=Zero & Succ(x1620)=Succ(x1645) & (\/x1622,x1623,x1624,x1625,x1626,x1627:new_primPlusNat0(x212, x1620)=Succ(x1622) & Succ(x1623)=x212 & Succ(x1623)=x1624 & new_primPlusNat0(x1624, x1620)=Succ(Zero) ==> new_pr2F34(x1623, Pos(x1620), x1625, x1626, x1627)_>=_new_pr2F31(new_primPlusNat0(Succ(x1623), x1620), x1625, new_primPlusNat0(Succ(x1623), x1620), x1626, x1627)) ==> new_pr2F34(x212, Pos(Succ(x1620)), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Succ(x1620)), x214, new_primPlusNat0(Succ(x212), Succ(x1620)), x215, x216)) We solved constraint (8) using rules (I), (II).We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1619, x1646)=Succ(Zero) which results in the following new constraints: (11) (Succ(Succ(new_primPlusNat0(x1648, x1647)))=Succ(Zero) & Succ(x212)=Succ(x1648) & Zero=Succ(x1647) & (\/x1649,x1650,x1651,x1652:new_primPlusNat0(x1648, x1647)=Succ(Zero) & Succ(x1649)=x1648 & Zero=x1647 ==> new_pr2F34(x1649, Pos(Zero), x1650, x1651, x1652)_>=_new_pr2F31(new_primPlusNat0(Succ(x1649), Zero), x1650, new_primPlusNat0(Succ(x1649), Zero), x1651, x1652)) ==> new_pr2F34(x212, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Zero), x214, new_primPlusNat0(Succ(x212), Zero), x215, x216)) (12) (Succ(x1653)=Succ(Zero) & Succ(x212)=Succ(x1653) & Zero=Zero ==> new_pr2F34(x212, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Zero), x214, new_primPlusNat0(Succ(x212), Zero), x215, x216)) (13) (Succ(x1654)=Succ(Zero) & Succ(x212)=Zero & Zero=Succ(x1654) ==> new_pr2F34(x212, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(x212), Zero), x214, new_primPlusNat0(Succ(x212), Zero), x215, x216)) We solved constraint (11) using rules (I), (II).We simplified constraint (12) using rules (I), (II), (III) which results in the following new constraint: (14) (new_pr2F34(Zero, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x214, new_primPlusNat0(Succ(Zero), Zero), x215, x216)) We solved constraint (13) using rules (I), (II). *We consider the chain new_pr2F34(x241, Pos(x242), x243, x244, x245) -> new_pr2F31(new_primPlusNat0(Succ(x241), x242), x243, new_primPlusNat0(Succ(x241), x242), x244, x245), new_pr2F31(Succ(x246), x247, Succ(Succ(x248)), x249, x250) -> H(x247, x249, Succ(x248), x250, anew_new_pr2F0G12(x248)) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(Succ(x241), x242), x243, new_primPlusNat0(Succ(x241), x242), x244, x245)=new_pr2F31(Succ(x246), x247, Succ(Succ(x248)), x249, x250) ==> new_pr2F34(x241, Pos(x242), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), x242), x243, new_primPlusNat0(Succ(x241), x242), x244, x245)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x241)=x1655 & new_primPlusNat0(x1655, x242)=Succ(x246) & Succ(x241)=x1656 & new_primPlusNat0(x1656, x242)=Succ(Succ(x248)) ==> new_pr2F34(x241, Pos(x242), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), x242), x243, new_primPlusNat0(Succ(x241), x242), x244, x245)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1655, x242)=Succ(x246) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1658, x1657)))=Succ(x246) & Succ(x241)=Succ(x1658) & Succ(x241)=x1656 & new_primPlusNat0(x1656, Succ(x1657))=Succ(Succ(x248)) & (\/x1659,x1660,x1661,x1662,x1663,x1664,x1665:new_primPlusNat0(x1658, x1657)=Succ(x1659) & Succ(x1660)=x1658 & Succ(x1660)=x1661 & new_primPlusNat0(x1661, x1657)=Succ(Succ(x1662)) ==> new_pr2F34(x1660, Pos(x1657), x1663, x1664, x1665)_>=_new_pr2F31(new_primPlusNat0(Succ(x1660), x1657), x1663, new_primPlusNat0(Succ(x1660), x1657), x1664, x1665)) ==> new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) (4) (Succ(x1666)=Succ(x246) & Succ(x241)=Succ(x1666) & Succ(x241)=x1656 & new_primPlusNat0(x1656, Zero)=Succ(Succ(x248)) ==> new_pr2F34(x241, Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Zero), x243, new_primPlusNat0(Succ(x241), Zero), x244, x245)) (5) (Succ(x1667)=Succ(x246) & Succ(x241)=Zero & Succ(x241)=x1656 & new_primPlusNat0(x1656, Succ(x1667))=Succ(Succ(x248)) ==> new_pr2F34(x241, Pos(Succ(x1667)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1667)), x243, new_primPlusNat0(Succ(x241), Succ(x1667)), x244, x245)) We simplified constraint (3) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: (6) (Succ(x241)=x1656 & Succ(x1657)=x1668 & new_primPlusNat0(x1656, x1668)=Succ(Succ(x248)) & (\/x1659,x1660,x1661,x1662,x1663,x1664,x1665:new_primPlusNat0(x241, x1657)=Succ(x1659) & Succ(x1660)=x241 & Succ(x1660)=x1661 & new_primPlusNat0(x1661, x1657)=Succ(Succ(x1662)) ==> new_pr2F34(x1660, Pos(x1657), x1663, x1664, x1665)_>=_new_pr2F31(new_primPlusNat0(Succ(x1660), x1657), x1663, new_primPlusNat0(Succ(x1660), x1657), x1664, x1665)) ==> new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x241)=x1656 & Zero=x1686 & new_primPlusNat0(x1656, x1686)=Succ(Succ(x248)) ==> new_pr2F34(x241, Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Zero), x243, new_primPlusNat0(Succ(x241), Zero), x244, x245)) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1656, x1668)=Succ(Succ(x248)) which results in the following new constraints: (8) (Succ(Succ(new_primPlusNat0(x1670, x1669)))=Succ(Succ(x248)) & Succ(x241)=Succ(x1670) & Succ(x1657)=Succ(x1669) & (\/x1659,x1660,x1661,x1662,x1663,x1664,x1665:new_primPlusNat0(x241, x1657)=Succ(x1659) & Succ(x1660)=x241 & Succ(x1660)=x1661 & new_primPlusNat0(x1661, x1657)=Succ(Succ(x1662)) ==> new_pr2F34(x1660, Pos(x1657), x1663, x1664, x1665)_>=_new_pr2F31(new_primPlusNat0(Succ(x1660), x1657), x1663, new_primPlusNat0(Succ(x1660), x1657), x1664, x1665)) & (\/x1671,x1672,x1673,x1674,x1675,x1676,x1677,x1678,x1679,x1680,x1681,x1682,x1683:new_primPlusNat0(x1670, x1669)=Succ(Succ(x1671)) & Succ(x1672)=x1670 & Succ(x1673)=x1669 & (\/x1674,x1675,x1676,x1677,x1678,x1679,x1680:new_primPlusNat0(x1672, x1673)=Succ(x1674) & Succ(x1675)=x1672 & Succ(x1675)=x1676 & new_primPlusNat0(x1676, x1673)=Succ(Succ(x1677)) ==> new_pr2F34(x1675, Pos(x1673), x1678, x1679, x1680)_>=_new_pr2F31(new_primPlusNat0(Succ(x1675), x1673), x1678, new_primPlusNat0(Succ(x1675), x1673), x1679, x1680)) ==> new_pr2F34(x1672, Pos(Succ(x1673)), x1681, x1682, x1683)_>=_new_pr2F31(new_primPlusNat0(Succ(x1672), Succ(x1673)), x1681, new_primPlusNat0(Succ(x1672), Succ(x1673)), x1682, x1683)) ==> new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) (9) (Succ(x1684)=Succ(Succ(x248)) & Succ(x241)=Succ(x1684) & Succ(x1657)=Zero & (\/x1659,x1660,x1661,x1662,x1663,x1664,x1665:new_primPlusNat0(x241, x1657)=Succ(x1659) & Succ(x1660)=x241 & Succ(x1660)=x1661 & new_primPlusNat0(x1661, x1657)=Succ(Succ(x1662)) ==> new_pr2F34(x1660, Pos(x1657), x1663, x1664, x1665)_>=_new_pr2F31(new_primPlusNat0(Succ(x1660), x1657), x1663, new_primPlusNat0(Succ(x1660), x1657), x1664, x1665)) ==> new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) (10) (Succ(x1685)=Succ(Succ(x248)) & Succ(x241)=Zero & Succ(x1657)=Succ(x1685) & (\/x1659,x1660,x1661,x1662,x1663,x1664,x1665:new_primPlusNat0(x241, x1657)=Succ(x1659) & Succ(x1660)=x241 & Succ(x1660)=x1661 & new_primPlusNat0(x1661, x1657)=Succ(Succ(x1662)) ==> new_pr2F34(x1660, Pos(x1657), x1663, x1664, x1665)_>=_new_pr2F31(new_primPlusNat0(Succ(x1660), x1657), x1663, new_primPlusNat0(Succ(x1660), x1657), x1664, x1665)) ==> new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) We simplified constraint (8) using rules (I), (II), (III), (IV) which results in the following new constraint: (11) (new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1656, x1686)=Succ(Succ(x248)) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1688, x1687)))=Succ(Succ(x248)) & Succ(x241)=Succ(x1688) & Zero=Succ(x1687) & (\/x1689,x1690,x1691,x1692,x1693:new_primPlusNat0(x1688, x1687)=Succ(Succ(x1689)) & Succ(x1690)=x1688 & Zero=x1687 ==> new_pr2F34(x1690, Pos(Zero), x1691, x1692, x1693)_>=_new_pr2F31(new_primPlusNat0(Succ(x1690), Zero), x1691, new_primPlusNat0(Succ(x1690), Zero), x1692, x1693)) ==> new_pr2F34(x241, Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Zero), x243, new_primPlusNat0(Succ(x241), Zero), x244, x245)) (13) (Succ(x1694)=Succ(Succ(x248)) & Succ(x241)=Succ(x1694) & Zero=Zero ==> new_pr2F34(x241, Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Zero), x243, new_primPlusNat0(Succ(x241), Zero), x244, x245)) (14) (Succ(x1695)=Succ(Succ(x248)) & Succ(x241)=Zero & Zero=Succ(x1695) ==> new_pr2F34(x241, Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Zero), x243, new_primPlusNat0(Succ(x241), Zero), x244, x245)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F34(Succ(x248), Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x248)), Zero), x243, new_primPlusNat0(Succ(Succ(x248)), Zero), x244, x245)) We solved constraint (14) using rules (I), (II). For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x276), x277, Succ(Succ(x278)), x279, x280) -> new_pr2F0G12(x277, x279, Succ(x278), x278, x280), new_pr2F0G12(x281, x282, x283, Succ(Zero), x284) -> new_pr2F1(x281, x283, new_fromInt, x282, x284) which results in the following constraint: (1) (new_pr2F0G12(x277, x279, Succ(x278), x278, x280)=new_pr2F0G12(x281, x282, x283, Succ(Zero), x284) ==> new_pr2F31(Succ(x276), x277, Succ(Succ(x278)), x279, x280)_>=_new_pr2F0G12(x277, x279, Succ(x278), x278, x280)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x276), x277, Succ(Succ(Succ(Zero))), x279, x280)_>=_new_pr2F0G12(x277, x279, Succ(Succ(Zero)), Succ(Zero), x280)) *We consider the chain new_pr2F31(Succ(x300), x301, Succ(Succ(x302)), x303, x304) -> new_pr2F0G12(x301, x303, Succ(x302), x302, x304), new_pr2F0G12(x305, x306, x307, Zero, x308) -> new_pr2F0G13(new_sr8(x305, x306, x308), x305, new_primDivNatS1(Succ(x307)), new_primDivNatS1(Succ(x307)), x308) which results in the following constraint: (1) (new_pr2F0G12(x301, x303, Succ(x302), x302, x304)=new_pr2F0G12(x305, x306, x307, Zero, x308) ==> new_pr2F31(Succ(x300), x301, Succ(Succ(x302)), x303, x304)_>=_new_pr2F0G12(x301, x303, Succ(x302), x302, x304)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x300), x301, Succ(Succ(Zero)), x303, x304)_>=_new_pr2F0G12(x301, x303, Succ(Zero), Zero, x304)) For Pair new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) the following chains were created: *We consider the chain new_pr2F0G12(x394, x395, x396, Zero, x397) -> new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x396)), new_primDivNatS1(Succ(x396)), x397), new_pr2F0G13(x398, x399, x400, Succ(Zero), x401) -> new_pr2F2(x399, x400, new_fromInt, x398, x401) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x396)), new_primDivNatS1(Succ(x396)), x397)=new_pr2F0G13(x398, x399, x400, Succ(Zero), x401) ==> new_pr2F0G12(x394, x395, x396, Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x396)), new_primDivNatS1(Succ(x396)), x397)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x396)=x1696 & new_primDivNatS1(x1696)=Succ(Zero) ==> new_pr2F0G12(x394, x395, x396, Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x396)), new_primDivNatS1(Succ(x396)), x397)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1696)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1697)=Succ(Zero) & Succ(x396)=Succ(x1697) ==> new_pr2F0G12(x394, x395, x396, Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x396)), new_primDivNatS1(Succ(x396)), x397)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1697)=Succ(Zero) ==> new_pr2F0G12(x394, x395, x1697, Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(x1697)), new_primDivNatS1(Succ(x1697)), x397)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1697)=Succ(Zero) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1698))=Succ(Zero) ==> new_pr2F0G12(x394, x395, Succ(Succ(x1698)), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Succ(x1698)))), new_primDivNatS1(Succ(Succ(Succ(x1698)))), x397)) (6) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G12(x394, x395, Succ(Zero), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x397)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x394, x395, Succ(Succ(x1698)), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Succ(x1698)))), new_primDivNatS1(Succ(Succ(Succ(x1698)))), x397)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x394, x395, Succ(Zero), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x397)) *We consider the chain new_pr2F0G12(x410, x411, x412, Zero, x413) -> new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x412)), new_primDivNatS1(Succ(x412)), x413), new_pr2F0G13(x414, x415, x416, Succ(Succ(x417)), x418) -> new_pr2F0G14(x414, x415, x416, x417, x418) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x412)), new_primDivNatS1(Succ(x412)), x413)=new_pr2F0G13(x414, x415, x416, Succ(Succ(x417)), x418) ==> new_pr2F0G12(x410, x411, x412, Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x412)), new_primDivNatS1(Succ(x412)), x413)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x412)=x1699 & new_primDivNatS1(x1699)=Succ(Succ(x417)) ==> new_pr2F0G12(x410, x411, x412, Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x412)), new_primDivNatS1(Succ(x412)), x413)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1699)=Succ(Succ(x417)) which results in the following new constraint: (3) (new_primDivNatS01(x1700)=Succ(Succ(x417)) & Succ(x412)=Succ(x1700) ==> new_pr2F0G12(x410, x411, x412, Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x412)), new_primDivNatS1(Succ(x412)), x413)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1700)=Succ(Succ(x417)) ==> new_pr2F0G12(x410, x411, x1700, Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(x1700)), new_primDivNatS1(Succ(x1700)), x413)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1700)=Succ(Succ(x417)) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1701))=Succ(Succ(x417)) ==> new_pr2F0G12(x410, x411, Succ(Succ(x1701)), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Succ(x1701)))), new_primDivNatS1(Succ(Succ(Succ(x1701)))), x413)) (6) (Succ(new_primDivNatS2)=Succ(Succ(x417)) ==> new_pr2F0G12(x410, x411, Succ(Zero), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x413)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x410, x411, Succ(Succ(x1701)), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Succ(x1701)))), new_primDivNatS1(Succ(Succ(Succ(x1701)))), x413)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x410, x411, Succ(Zero), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x413)) *We consider the chain new_pr2F0G12(x427, x428, x429, Zero, x430) -> new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430), new_pr2F0G13(x431, x432, x433, Zero, x434) -> new_pr2F0G13(x431, new_sr10(x432, x434), new_primDivNatS1(x433), new_primDivNatS1(x433), x434) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430)=new_pr2F0G13(x431, x432, x433, Zero, x434) ==> new_pr2F0G12(x427, x428, x429, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x429)=x1702 & new_primDivNatS1(x1702)=Zero ==> new_pr2F0G12(x427, x428, x429, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1702)=Zero which results in the following new constraints: (3) (Zero=Zero & Succ(x429)=Zero ==> new_pr2F0G12(x427, x428, x429, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430)) (4) (new_primDivNatS01(x1703)=Zero & Succ(x429)=Succ(x1703) ==> new_pr2F0G12(x427, x428, x429, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x429)), new_primDivNatS1(Succ(x429)), x430)) We solved constraint (3) using rules (I), (II).We simplified constraint (4) using rules (I), (II), (III) which results in the following new constraint: (5) (new_primDivNatS01(x1703)=Zero ==> new_pr2F0G12(x427, x428, x1703, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(x1703)), new_primDivNatS1(Succ(x1703)), x430)) We simplified constraint (5) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1703)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G12(x427, x428, Zero, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x430)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G12(x427, x428, Zero, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x430)) *We consider the chain new_pr2F0G12(x447, x448, x449, Zero, x450) -> new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x449)), new_primDivNatS1(Succ(x449)), x450), new_pr2F0G13(x451, x452, x453, Succ(Succ(x454)), x455) -> H'(x451, x452, x453, x455, anew_new_pr2F0G14(x454)) which results in the following constraint: (1) (new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x449)), new_primDivNatS1(Succ(x449)), x450)=new_pr2F0G13(x451, x452, x453, Succ(Succ(x454)), x455) ==> new_pr2F0G12(x447, x448, x449, Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x449)), new_primDivNatS1(Succ(x449)), x450)) We simplified constraint (1) using rules (I), (II), (IV), (VII) which results in the following new constraint: (2) (Succ(x449)=x1705 & new_primDivNatS1(x1705)=Succ(Succ(x454)) ==> new_pr2F0G12(x447, x448, x449, Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x449)), new_primDivNatS1(Succ(x449)), x450)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1705)=Succ(Succ(x454)) which results in the following new constraint: (3) (new_primDivNatS01(x1706)=Succ(Succ(x454)) & Succ(x449)=Succ(x1706) ==> new_pr2F0G12(x447, x448, x449, Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x449)), new_primDivNatS1(Succ(x449)), x450)) We simplified constraint (3) using rules (I), (II), (III) which results in the following new constraint: (4) (new_primDivNatS01(x1706)=Succ(Succ(x454)) ==> new_pr2F0G12(x447, x448, x1706, Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(x1706)), new_primDivNatS1(Succ(x1706)), x450)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1706)=Succ(Succ(x454)) which results in the following new constraints: (5) (Succ(new_primDivNatS4(x1707))=Succ(Succ(x454)) ==> new_pr2F0G12(x447, x448, Succ(Succ(x1707)), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Succ(x1707)))), new_primDivNatS1(Succ(Succ(Succ(x1707)))), x450)) (6) (Succ(new_primDivNatS2)=Succ(Succ(x454)) ==> new_pr2F0G12(x447, x448, Succ(Zero), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x450)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G12(x447, x448, Succ(Succ(x1707)), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Succ(x1707)))), new_primDivNatS1(Succ(Succ(Succ(x1707)))), x450)) We simplified constraint (6) using rules (I), (II), (IV) which results in the following new constraint: (8) (new_pr2F0G12(x447, x448, Succ(Zero), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x450)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G13(x488, x489, x490, Succ(Zero), x491) -> new_pr2F2(x489, x490, new_fromInt, x488, x491), new_pr2F2(x492, x493, Pos(x494), x495, x496) -> new_pr2F31(new_primPlusNat0(x493, x494), new_sr11(x492, x496), new_primPlusNat0(x493, x494), x495, x496) which results in the following constraint: (1) (new_pr2F2(x489, x490, new_fromInt, x488, x491)=new_pr2F2(x492, x493, Pos(x494), x495, x496) ==> new_pr2F0G13(x488, x489, x490, Succ(Zero), x491)_>=_new_pr2F2(x489, x490, new_fromInt, x488, x491)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x494) ==> new_pr2F0G13(x488, x489, x490, Succ(Zero), x491)_>=_new_pr2F2(x489, x490, new_fromInt, x488, x491)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x494) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x494) ==> new_pr2F0G13(x488, x489, x490, Succ(Zero), x491)_>=_new_pr2F2(x489, x490, new_fromInt, x488, x491)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G13(x488, x489, x490, Succ(Zero), x491)_>=_new_pr2F2(x489, x490, new_fromInt, x488, x491)) For Pair new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) the following chains were created: *We consider the chain new_pr2F2(x556, x557, Pos(x558), x559, x560) -> new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560), new_pr2F31(Succ(x561), x562, Succ(Succ(x563)), x564, x565) -> new_pr2F0G12(x562, x564, Succ(x563), x563, x565) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)=new_pr2F31(Succ(x561), x562, Succ(Succ(x563)), x564, x565) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x557, x558)=Succ(x561) & new_primPlusNat0(x557, x558)=Succ(Succ(x563)) ==> new_pr2F2(x556, x557, Pos(x558), x559, x560)_>=_new_pr2F31(new_primPlusNat0(x557, x558), new_sr11(x556, x560), new_primPlusNat0(x557, x558), x559, x560)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x557, x558)=Succ(x561) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1709, x1708)))=Succ(x561) & new_primPlusNat0(Succ(x1709), Succ(x1708))=Succ(Succ(x563)) & (\/x1710,x1711,x1712,x1713,x1714:new_primPlusNat0(x1709, x1708)=Succ(x1710) & new_primPlusNat0(x1709, x1708)=Succ(Succ(x1711)) ==> new_pr2F2(x1712, x1709, Pos(x1708), x1713, x1714)_>=_new_pr2F31(new_primPlusNat0(x1709, x1708), new_sr11(x1712, x1714), new_primPlusNat0(x1709, x1708), x1713, x1714)) ==> new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) (4) (Succ(x1715)=Succ(x561) & new_primPlusNat0(Succ(x1715), Zero)=Succ(Succ(x563)) ==> new_pr2F2(x556, Succ(x1715), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1715), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1715), Zero), x559, x560)) (5) (Succ(x1716)=Succ(x561) & new_primPlusNat0(Zero, Succ(x1716))=Succ(Succ(x563)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1716)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1716)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1716)), x559, x560)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1709)=x1717 & Succ(x1708)=x1718 & new_primPlusNat0(x1717, x1718)=Succ(Succ(x563)) & (\/x1710,x1711,x1712,x1713,x1714:new_primPlusNat0(x1709, x1708)=Succ(x1710) & new_primPlusNat0(x1709, x1708)=Succ(Succ(x1711)) ==> new_pr2F2(x1712, x1709, Pos(x1708), x1713, x1714)_>=_new_pr2F31(new_primPlusNat0(x1709, x1708), new_sr11(x1712, x1714), new_primPlusNat0(x1709, x1708), x1713, x1714)) ==> new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1715)=x1734 & Zero=x1735 & new_primPlusNat0(x1734, x1735)=Succ(Succ(x563)) ==> new_pr2F2(x556, Succ(x1715), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1715), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1715), Zero), x559, x560)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1745 & Succ(x1716)=x1746 & new_primPlusNat0(x1745, x1746)=Succ(Succ(x563)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1716)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1716)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1716)), x559, x560)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1717, x1718)=Succ(Succ(x563)) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1720, x1719)))=Succ(Succ(x563)) & Succ(x1709)=Succ(x1720) & Succ(x1708)=Succ(x1719) & (\/x1710,x1711,x1712,x1713,x1714:new_primPlusNat0(x1709, x1708)=Succ(x1710) & new_primPlusNat0(x1709, x1708)=Succ(Succ(x1711)) ==> new_pr2F2(x1712, x1709, Pos(x1708), x1713, x1714)_>=_new_pr2F31(new_primPlusNat0(x1709, x1708), new_sr11(x1712, x1714), new_primPlusNat0(x1709, x1708), x1713, x1714)) & (\/x1721,x1722,x1723,x1724,x1725,x1726,x1727,x1728,x1729,x1730,x1731:new_primPlusNat0(x1720, x1719)=Succ(Succ(x1721)) & Succ(x1722)=x1720 & Succ(x1723)=x1719 & (\/x1724,x1725,x1726,x1727,x1728:new_primPlusNat0(x1722, x1723)=Succ(x1724) & new_primPlusNat0(x1722, x1723)=Succ(Succ(x1725)) ==> new_pr2F2(x1726, x1722, Pos(x1723), x1727, x1728)_>=_new_pr2F31(new_primPlusNat0(x1722, x1723), new_sr11(x1726, x1728), new_primPlusNat0(x1722, x1723), x1727, x1728)) ==> new_pr2F2(x1729, Succ(x1722), Pos(Succ(x1723)), x1730, x1731)_>=_new_pr2F31(new_primPlusNat0(Succ(x1722), Succ(x1723)), new_sr11(x1729, x1731), new_primPlusNat0(Succ(x1722), Succ(x1723)), x1730, x1731)) ==> new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) (10) (Succ(x1732)=Succ(Succ(x563)) & Succ(x1709)=Succ(x1732) & Succ(x1708)=Zero & (\/x1710,x1711,x1712,x1713,x1714:new_primPlusNat0(x1709, x1708)=Succ(x1710) & new_primPlusNat0(x1709, x1708)=Succ(Succ(x1711)) ==> new_pr2F2(x1712, x1709, Pos(x1708), x1713, x1714)_>=_new_pr2F31(new_primPlusNat0(x1709, x1708), new_sr11(x1712, x1714), new_primPlusNat0(x1709, x1708), x1713, x1714)) ==> new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) (11) (Succ(x1733)=Succ(Succ(x563)) & Succ(x1709)=Zero & Succ(x1708)=Succ(x1733) & (\/x1710,x1711,x1712,x1713,x1714:new_primPlusNat0(x1709, x1708)=Succ(x1710) & new_primPlusNat0(x1709, x1708)=Succ(Succ(x1711)) ==> new_pr2F2(x1712, x1709, Pos(x1708), x1713, x1714)_>=_new_pr2F31(new_primPlusNat0(x1709, x1708), new_sr11(x1712, x1714), new_primPlusNat0(x1709, x1708), x1713, x1714)) ==> new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: (12) (new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1734, x1735)=Succ(Succ(x563)) which results in the following new constraints: (13) (Succ(Succ(new_primPlusNat0(x1737, x1736)))=Succ(Succ(x563)) & Succ(x1715)=Succ(x1737) & Zero=Succ(x1736) & (\/x1738,x1739,x1740,x1741,x1742:new_primPlusNat0(x1737, x1736)=Succ(Succ(x1738)) & Succ(x1739)=x1737 & Zero=x1736 ==> new_pr2F2(x1740, Succ(x1739), Pos(Zero), x1741, x1742)_>=_new_pr2F31(new_primPlusNat0(Succ(x1739), Zero), new_sr11(x1740, x1742), new_primPlusNat0(Succ(x1739), Zero), x1741, x1742)) ==> new_pr2F2(x556, Succ(x1715), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1715), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1715), Zero), x559, x560)) (14) (Succ(x1743)=Succ(Succ(x563)) & Succ(x1715)=Succ(x1743) & Zero=Zero ==> new_pr2F2(x556, Succ(x1715), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1715), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1715), Zero), x559, x560)) (15) (Succ(x1744)=Succ(Succ(x563)) & Succ(x1715)=Zero & Zero=Succ(x1744) ==> new_pr2F2(x556, Succ(x1715), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1715), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(x1715), Zero), x559, x560)) We solved constraint (13) using rules (I), (II).We simplified constraint (14) using rules (I), (II), (III) which results in the following new constraint: (16) (new_pr2F2(x556, Succ(Succ(x563)), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x563)), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Succ(x563)), Zero), x559, x560)) We solved constraint (15) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1745, x1746)=Succ(Succ(x563)) which results in the following new constraints: (17) (Succ(Succ(new_primPlusNat0(x1748, x1747)))=Succ(Succ(x563)) & Zero=Succ(x1748) & Succ(x1716)=Succ(x1747) & (\/x1749,x1750,x1751,x1752,x1753:new_primPlusNat0(x1748, x1747)=Succ(Succ(x1749)) & Zero=x1748 & Succ(x1750)=x1747 ==> new_pr2F2(x1751, Zero, Pos(Succ(x1750)), x1752, x1753)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1750)), new_sr11(x1751, x1753), new_primPlusNat0(Zero, Succ(x1750)), x1752, x1753)) ==> new_pr2F2(x556, Zero, Pos(Succ(x1716)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1716)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1716)), x559, x560)) (18) (Succ(x1754)=Succ(Succ(x563)) & Zero=Succ(x1754) & Succ(x1716)=Zero ==> new_pr2F2(x556, Zero, Pos(Succ(x1716)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1716)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1716)), x559, x560)) (19) (Succ(x1755)=Succ(Succ(x563)) & Zero=Zero & Succ(x1716)=Succ(x1755) ==> new_pr2F2(x556, Zero, Pos(Succ(x1716)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1716)), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(x1716)), x559, x560)) We solved constraint (17) using rules (I), (II).We solved constraint (18) using rules (I), (II).We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint: (20) (new_pr2F2(x556, Zero, Pos(Succ(Succ(x563))), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x563))), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Succ(x563))), x559, x560)) *We consider the chain new_pr2F2(x581, x582, Pos(x583), x584, x585) -> new_pr2F31(new_primPlusNat0(x582, x583), new_sr11(x581, x585), new_primPlusNat0(x582, x583), x584, x585), new_pr2F31(Succ(x586), x587, Succ(Zero), x588, x589) -> new_pr2F1(x587, Zero, new_fromInt, x588, x589) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x582, x583), new_sr11(x581, x585), new_primPlusNat0(x582, x583), x584, x585)=new_pr2F31(Succ(x586), x587, Succ(Zero), x588, x589) ==> new_pr2F2(x581, x582, Pos(x583), x584, x585)_>=_new_pr2F31(new_primPlusNat0(x582, x583), new_sr11(x581, x585), new_primPlusNat0(x582, x583), x584, x585)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x582, x583)=Succ(x586) & new_primPlusNat0(x582, x583)=Succ(Zero) ==> new_pr2F2(x581, x582, Pos(x583), x584, x585)_>=_new_pr2F31(new_primPlusNat0(x582, x583), new_sr11(x581, x585), new_primPlusNat0(x582, x583), x584, x585)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x582, x583)=Succ(x586) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1757, x1756)))=Succ(x586) & new_primPlusNat0(Succ(x1757), Succ(x1756))=Succ(Zero) & (\/x1758,x1759,x1760,x1761:new_primPlusNat0(x1757, x1756)=Succ(x1758) & new_primPlusNat0(x1757, x1756)=Succ(Zero) ==> new_pr2F2(x1759, x1757, Pos(x1756), x1760, x1761)_>=_new_pr2F31(new_primPlusNat0(x1757, x1756), new_sr11(x1759, x1761), new_primPlusNat0(x1757, x1756), x1760, x1761)) ==> new_pr2F2(x581, Succ(x1757), Pos(Succ(x1756)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1757), Succ(x1756)), new_sr11(x581, x585), new_primPlusNat0(Succ(x1757), Succ(x1756)), x584, x585)) (4) (Succ(x1762)=Succ(x586) & new_primPlusNat0(Succ(x1762), Zero)=Succ(Zero) ==> new_pr2F2(x581, Succ(x1762), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1762), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(x1762), Zero), x584, x585)) (5) (Succ(x1763)=Succ(x586) & new_primPlusNat0(Zero, Succ(x1763))=Succ(Zero) ==> new_pr2F2(x581, Zero, Pos(Succ(x1763)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1763)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(x1763)), x584, x585)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1757)=x1764 & Succ(x1756)=x1765 & new_primPlusNat0(x1764, x1765)=Succ(Zero) & (\/x1758,x1759,x1760,x1761:new_primPlusNat0(x1757, x1756)=Succ(x1758) & new_primPlusNat0(x1757, x1756)=Succ(Zero) ==> new_pr2F2(x1759, x1757, Pos(x1756), x1760, x1761)_>=_new_pr2F31(new_primPlusNat0(x1757, x1756), new_sr11(x1759, x1761), new_primPlusNat0(x1757, x1756), x1760, x1761)) ==> new_pr2F2(x581, Succ(x1757), Pos(Succ(x1756)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1757), Succ(x1756)), new_sr11(x581, x585), new_primPlusNat0(Succ(x1757), Succ(x1756)), x584, x585)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1762)=x1779 & Zero=x1780 & new_primPlusNat0(x1779, x1780)=Succ(Zero) ==> new_pr2F2(x581, Succ(x1762), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1762), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(x1762), Zero), x584, x585)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1789 & Succ(x1763)=x1790 & new_primPlusNat0(x1789, x1790)=Succ(Zero) ==> new_pr2F2(x581, Zero, Pos(Succ(x1763)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1763)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(x1763)), x584, x585)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1764, x1765)=Succ(Zero) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1767, x1766)))=Succ(Zero) & Succ(x1757)=Succ(x1767) & Succ(x1756)=Succ(x1766) & (\/x1758,x1759,x1760,x1761:new_primPlusNat0(x1757, x1756)=Succ(x1758) & new_primPlusNat0(x1757, x1756)=Succ(Zero) ==> new_pr2F2(x1759, x1757, Pos(x1756), x1760, x1761)_>=_new_pr2F31(new_primPlusNat0(x1757, x1756), new_sr11(x1759, x1761), new_primPlusNat0(x1757, x1756), x1760, x1761)) & (\/x1768,x1769,x1770,x1771,x1772,x1773,x1774,x1775,x1776:new_primPlusNat0(x1767, x1766)=Succ(Zero) & Succ(x1768)=x1767 & Succ(x1769)=x1766 & (\/x1770,x1771,x1772,x1773:new_primPlusNat0(x1768, x1769)=Succ(x1770) & new_primPlusNat0(x1768, x1769)=Succ(Zero) ==> new_pr2F2(x1771, x1768, Pos(x1769), x1772, x1773)_>=_new_pr2F31(new_primPlusNat0(x1768, x1769), new_sr11(x1771, x1773), new_primPlusNat0(x1768, x1769), x1772, x1773)) ==> new_pr2F2(x1774, Succ(x1768), Pos(Succ(x1769)), x1775, x1776)_>=_new_pr2F31(new_primPlusNat0(Succ(x1768), Succ(x1769)), new_sr11(x1774, x1776), new_primPlusNat0(Succ(x1768), Succ(x1769)), x1775, x1776)) ==> new_pr2F2(x581, Succ(x1757), Pos(Succ(x1756)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1757), Succ(x1756)), new_sr11(x581, x585), new_primPlusNat0(Succ(x1757), Succ(x1756)), x584, x585)) (10) (Succ(x1777)=Succ(Zero) & Succ(x1757)=Succ(x1777) & Succ(x1756)=Zero & (\/x1758,x1759,x1760,x1761:new_primPlusNat0(x1757, x1756)=Succ(x1758) & new_primPlusNat0(x1757, x1756)=Succ(Zero) ==> new_pr2F2(x1759, x1757, Pos(x1756), x1760, x1761)_>=_new_pr2F31(new_primPlusNat0(x1757, x1756), new_sr11(x1759, x1761), new_primPlusNat0(x1757, x1756), x1760, x1761)) ==> new_pr2F2(x581, Succ(x1757), Pos(Succ(x1756)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1757), Succ(x1756)), new_sr11(x581, x585), new_primPlusNat0(Succ(x1757), Succ(x1756)), x584, x585)) (11) (Succ(x1778)=Succ(Zero) & Succ(x1757)=Zero & Succ(x1756)=Succ(x1778) & (\/x1758,x1759,x1760,x1761:new_primPlusNat0(x1757, x1756)=Succ(x1758) & new_primPlusNat0(x1757, x1756)=Succ(Zero) ==> new_pr2F2(x1759, x1757, Pos(x1756), x1760, x1761)_>=_new_pr2F31(new_primPlusNat0(x1757, x1756), new_sr11(x1759, x1761), new_primPlusNat0(x1757, x1756), x1760, x1761)) ==> new_pr2F2(x581, Succ(x1757), Pos(Succ(x1756)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1757), Succ(x1756)), new_sr11(x581, x585), new_primPlusNat0(Succ(x1757), Succ(x1756)), x584, x585)) We solved constraint (9) using rules (I), (II).We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1779, x1780)=Succ(Zero) which results in the following new constraints: (12) (Succ(Succ(new_primPlusNat0(x1782, x1781)))=Succ(Zero) & Succ(x1762)=Succ(x1782) & Zero=Succ(x1781) & (\/x1783,x1784,x1785,x1786:new_primPlusNat0(x1782, x1781)=Succ(Zero) & Succ(x1783)=x1782 & Zero=x1781 ==> new_pr2F2(x1784, Succ(x1783), Pos(Zero), x1785, x1786)_>=_new_pr2F31(new_primPlusNat0(Succ(x1783), Zero), new_sr11(x1784, x1786), new_primPlusNat0(Succ(x1783), Zero), x1785, x1786)) ==> new_pr2F2(x581, Succ(x1762), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1762), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(x1762), Zero), x584, x585)) (13) (Succ(x1787)=Succ(Zero) & Succ(x1762)=Succ(x1787) & Zero=Zero ==> new_pr2F2(x581, Succ(x1762), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1762), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(x1762), Zero), x584, x585)) (14) (Succ(x1788)=Succ(Zero) & Succ(x1762)=Zero & Zero=Succ(x1788) ==> new_pr2F2(x581, Succ(x1762), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(x1762), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(x1762), Zero), x584, x585)) We solved constraint (12) using rules (I), (II).We simplified constraint (13) using rules (I), (II), (III) which results in the following new constraint: (15) (new_pr2F2(x581, Succ(Zero), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(Zero), Zero), x584, x585)) We solved constraint (14) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1789, x1790)=Succ(Zero) which results in the following new constraints: (16) (Succ(Succ(new_primPlusNat0(x1792, x1791)))=Succ(Zero) & Zero=Succ(x1792) & Succ(x1763)=Succ(x1791) & (\/x1793,x1794,x1795,x1796:new_primPlusNat0(x1792, x1791)=Succ(Zero) & Zero=x1792 & Succ(x1793)=x1791 ==> new_pr2F2(x1794, Zero, Pos(Succ(x1793)), x1795, x1796)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1793)), new_sr11(x1794, x1796), new_primPlusNat0(Zero, Succ(x1793)), x1795, x1796)) ==> new_pr2F2(x581, Zero, Pos(Succ(x1763)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1763)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(x1763)), x584, x585)) (17) (Succ(x1797)=Succ(Zero) & Zero=Succ(x1797) & Succ(x1763)=Zero ==> new_pr2F2(x581, Zero, Pos(Succ(x1763)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1763)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(x1763)), x584, x585)) (18) (Succ(x1798)=Succ(Zero) & Zero=Zero & Succ(x1763)=Succ(x1798) ==> new_pr2F2(x581, Zero, Pos(Succ(x1763)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1763)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(x1763)), x584, x585)) We solved constraint (16) using rules (I), (II).We solved constraint (17) using rules (I), (II).We simplified constraint (18) using rules (I), (II), (III) which results in the following new constraint: (19) (new_pr2F2(x581, Zero, Pos(Succ(Zero)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(Zero)), x584, x585)) *We consider the chain new_pr2F2(x610, x611, Pos(x612), x613, x614) -> new_pr2F31(new_primPlusNat0(x611, x612), new_sr11(x610, x614), new_primPlusNat0(x611, x612), x613, x614), new_pr2F31(Succ(x615), x616, Succ(Succ(x617)), x618, x619) -> H(x616, x618, Succ(x617), x619, anew_new_pr2F0G12(x617)) which results in the following constraint: (1) (new_pr2F31(new_primPlusNat0(x611, x612), new_sr11(x610, x614), new_primPlusNat0(x611, x612), x613, x614)=new_pr2F31(Succ(x615), x616, Succ(Succ(x617)), x618, x619) ==> new_pr2F2(x610, x611, Pos(x612), x613, x614)_>=_new_pr2F31(new_primPlusNat0(x611, x612), new_sr11(x610, x614), new_primPlusNat0(x611, x612), x613, x614)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primPlusNat0(x611, x612)=Succ(x615) & new_primPlusNat0(x611, x612)=Succ(Succ(x617)) ==> new_pr2F2(x610, x611, Pos(x612), x613, x614)_>=_new_pr2F31(new_primPlusNat0(x611, x612), new_sr11(x610, x614), new_primPlusNat0(x611, x612), x613, x614)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x611, x612)=Succ(x615) which results in the following new constraints: (3) (Succ(Succ(new_primPlusNat0(x1800, x1799)))=Succ(x615) & new_primPlusNat0(Succ(x1800), Succ(x1799))=Succ(Succ(x617)) & (\/x1801,x1802,x1803,x1804,x1805:new_primPlusNat0(x1800, x1799)=Succ(x1801) & new_primPlusNat0(x1800, x1799)=Succ(Succ(x1802)) ==> new_pr2F2(x1803, x1800, Pos(x1799), x1804, x1805)_>=_new_pr2F31(new_primPlusNat0(x1800, x1799), new_sr11(x1803, x1805), new_primPlusNat0(x1800, x1799), x1804, x1805)) ==> new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) (4) (Succ(x1806)=Succ(x615) & new_primPlusNat0(Succ(x1806), Zero)=Succ(Succ(x617)) ==> new_pr2F2(x610, Succ(x1806), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1806), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(x1806), Zero), x613, x614)) (5) (Succ(x1807)=Succ(x615) & new_primPlusNat0(Zero, Succ(x1807))=Succ(Succ(x617)) ==> new_pr2F2(x610, Zero, Pos(Succ(x1807)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1807)), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(x1807)), x613, x614)) We simplified constraint (3) using rules (I), (II), (IV), (VII) which results in the following new constraint: (6) (Succ(x1800)=x1808 & Succ(x1799)=x1809 & new_primPlusNat0(x1808, x1809)=Succ(Succ(x617)) & (\/x1801,x1802,x1803,x1804,x1805:new_primPlusNat0(x1800, x1799)=Succ(x1801) & new_primPlusNat0(x1800, x1799)=Succ(Succ(x1802)) ==> new_pr2F2(x1803, x1800, Pos(x1799), x1804, x1805)_>=_new_pr2F31(new_primPlusNat0(x1800, x1799), new_sr11(x1803, x1805), new_primPlusNat0(x1800, x1799), x1804, x1805)) ==> new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) We simplified constraint (4) using rules (I), (II), (IV), (VII) which results in the following new constraint: (7) (Succ(x1806)=x1825 & Zero=x1826 & new_primPlusNat0(x1825, x1826)=Succ(Succ(x617)) ==> new_pr2F2(x610, Succ(x1806), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1806), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(x1806), Zero), x613, x614)) We simplified constraint (5) using rules (I), (II), (IV), (VII) which results in the following new constraint: (8) (Zero=x1836 & Succ(x1807)=x1837 & new_primPlusNat0(x1836, x1837)=Succ(Succ(x617)) ==> new_pr2F2(x610, Zero, Pos(Succ(x1807)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1807)), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(x1807)), x613, x614)) We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1808, x1809)=Succ(Succ(x617)) which results in the following new constraints: (9) (Succ(Succ(new_primPlusNat0(x1811, x1810)))=Succ(Succ(x617)) & Succ(x1800)=Succ(x1811) & Succ(x1799)=Succ(x1810) & (\/x1801,x1802,x1803,x1804,x1805:new_primPlusNat0(x1800, x1799)=Succ(x1801) & new_primPlusNat0(x1800, x1799)=Succ(Succ(x1802)) ==> new_pr2F2(x1803, x1800, Pos(x1799), x1804, x1805)_>=_new_pr2F31(new_primPlusNat0(x1800, x1799), new_sr11(x1803, x1805), new_primPlusNat0(x1800, x1799), x1804, x1805)) & (\/x1812,x1813,x1814,x1815,x1816,x1817,x1818,x1819,x1820,x1821,x1822:new_primPlusNat0(x1811, x1810)=Succ(Succ(x1812)) & Succ(x1813)=x1811 & Succ(x1814)=x1810 & (\/x1815,x1816,x1817,x1818,x1819:new_primPlusNat0(x1813, x1814)=Succ(x1815) & new_primPlusNat0(x1813, x1814)=Succ(Succ(x1816)) ==> new_pr2F2(x1817, x1813, Pos(x1814), x1818, x1819)_>=_new_pr2F31(new_primPlusNat0(x1813, x1814), new_sr11(x1817, x1819), new_primPlusNat0(x1813, x1814), x1818, x1819)) ==> new_pr2F2(x1820, Succ(x1813), Pos(Succ(x1814)), x1821, x1822)_>=_new_pr2F31(new_primPlusNat0(Succ(x1813), Succ(x1814)), new_sr11(x1820, x1822), new_primPlusNat0(Succ(x1813), Succ(x1814)), x1821, x1822)) ==> new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) (10) (Succ(x1823)=Succ(Succ(x617)) & Succ(x1800)=Succ(x1823) & Succ(x1799)=Zero & (\/x1801,x1802,x1803,x1804,x1805:new_primPlusNat0(x1800, x1799)=Succ(x1801) & new_primPlusNat0(x1800, x1799)=Succ(Succ(x1802)) ==> new_pr2F2(x1803, x1800, Pos(x1799), x1804, x1805)_>=_new_pr2F31(new_primPlusNat0(x1800, x1799), new_sr11(x1803, x1805), new_primPlusNat0(x1800, x1799), x1804, x1805)) ==> new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) (11) (Succ(x1824)=Succ(Succ(x617)) & Succ(x1800)=Zero & Succ(x1799)=Succ(x1824) & (\/x1801,x1802,x1803,x1804,x1805:new_primPlusNat0(x1800, x1799)=Succ(x1801) & new_primPlusNat0(x1800, x1799)=Succ(Succ(x1802)) ==> new_pr2F2(x1803, x1800, Pos(x1799), x1804, x1805)_>=_new_pr2F31(new_primPlusNat0(x1800, x1799), new_sr11(x1803, x1805), new_primPlusNat0(x1800, x1799), x1804, x1805)) ==> new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: (12) (new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) We solved constraint (10) using rules (I), (II).We solved constraint (11) using rules (I), (II).We simplified constraint (7) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1825, x1826)=Succ(Succ(x617)) which results in the following new constraints: (13) (Succ(Succ(new_primPlusNat0(x1828, x1827)))=Succ(Succ(x617)) & Succ(x1806)=Succ(x1828) & Zero=Succ(x1827) & (\/x1829,x1830,x1831,x1832,x1833:new_primPlusNat0(x1828, x1827)=Succ(Succ(x1829)) & Succ(x1830)=x1828 & Zero=x1827 ==> new_pr2F2(x1831, Succ(x1830), Pos(Zero), x1832, x1833)_>=_new_pr2F31(new_primPlusNat0(Succ(x1830), Zero), new_sr11(x1831, x1833), new_primPlusNat0(Succ(x1830), Zero), x1832, x1833)) ==> new_pr2F2(x610, Succ(x1806), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1806), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(x1806), Zero), x613, x614)) (14) (Succ(x1834)=Succ(Succ(x617)) & Succ(x1806)=Succ(x1834) & Zero=Zero ==> new_pr2F2(x610, Succ(x1806), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1806), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(x1806), Zero), x613, x614)) (15) (Succ(x1835)=Succ(Succ(x617)) & Succ(x1806)=Zero & Zero=Succ(x1835) ==> new_pr2F2(x610, Succ(x1806), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1806), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(x1806), Zero), x613, x614)) We solved constraint (13) using rules (I), (II).We simplified constraint (14) using rules (I), (II), (III) which results in the following new constraint: (16) (new_pr2F2(x610, Succ(Succ(x617)), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x617)), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(Succ(x617)), Zero), x613, x614)) We solved constraint (15) using rules (I), (II).We simplified constraint (8) using rule (V) (with possible (I) afterwards) using induction on new_primPlusNat0(x1836, x1837)=Succ(Succ(x617)) which results in the following new constraints: (17) (Succ(Succ(new_primPlusNat0(x1839, x1838)))=Succ(Succ(x617)) & Zero=Succ(x1839) & Succ(x1807)=Succ(x1838) & (\/x1840,x1841,x1842,x1843,x1844:new_primPlusNat0(x1839, x1838)=Succ(Succ(x1840)) & Zero=x1839 & Succ(x1841)=x1838 ==> new_pr2F2(x1842, Zero, Pos(Succ(x1841)), x1843, x1844)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1841)), new_sr11(x1842, x1844), new_primPlusNat0(Zero, Succ(x1841)), x1843, x1844)) ==> new_pr2F2(x610, Zero, Pos(Succ(x1807)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1807)), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(x1807)), x613, x614)) (18) (Succ(x1845)=Succ(Succ(x617)) & Zero=Succ(x1845) & Succ(x1807)=Zero ==> new_pr2F2(x610, Zero, Pos(Succ(x1807)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1807)), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(x1807)), x613, x614)) (19) (Succ(x1846)=Succ(Succ(x617)) & Zero=Zero & Succ(x1807)=Succ(x1846) ==> new_pr2F2(x610, Zero, Pos(Succ(x1807)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(x1807)), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(x1807)), x613, x614)) We solved constraint (17) using rules (I), (II).We solved constraint (18) using rules (I), (II).We simplified constraint (19) using rules (I), (II), (III) which results in the following new constraint: (20) (new_pr2F2(x610, Zero, Pos(Succ(Succ(x617))), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x617))), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(Succ(x617))), x613, x614)) For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) the following chains were created: *We consider the chain new_pr2F31(Succ(x649), x650, Succ(Zero), x651, x652) -> new_pr2F1(x650, Zero, new_fromInt, x651, x652), new_pr2F1(x653, x654, x655, x656, x657) -> new_pr2F34(x654, x655, x653, new_sr9(x653, x656, x657), x657) which results in the following constraint: (1) (new_pr2F1(x650, Zero, new_fromInt, x651, x652)=new_pr2F1(x653, x654, x655, x656, x657) ==> new_pr2F31(Succ(x649), x650, Succ(Zero), x651, x652)_>=_new_pr2F1(x650, Zero, new_fromInt, x651, x652)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_pr2F31(Succ(x649), x650, Succ(Zero), x651, x652)_>=_new_pr2F1(x650, Zero, new_fromInt, x651, x652)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) the following chains were created: *We consider the chain new_pr2F0G13(x767, x768, x769, Succ(Succ(x770)), x771) -> new_pr2F0G14(x767, x768, x769, x770, x771), new_pr2F0G14(x772, x773, x774, Succ(Zero), x775) -> new_pr2F2(x773, x774, new_fromInt, x772, x775) which results in the following constraint: (1) (new_pr2F0G14(x767, x768, x769, x770, x771)=new_pr2F0G14(x772, x773, x774, Succ(Zero), x775) ==> new_pr2F0G13(x767, x768, x769, Succ(Succ(x770)), x771)_>=_new_pr2F0G14(x767, x768, x769, x770, x771)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x767, x768, x769, Succ(Succ(Succ(Zero))), x771)_>=_new_pr2F0G14(x767, x768, x769, Succ(Zero), x771)) *We consider the chain new_pr2F0G13(x776, x777, x778, Succ(Succ(x779)), x780) -> new_pr2F0G14(x776, x777, x778, x779, x780), new_pr2F0G14(x781, x782, x783, Zero, x784) -> new_pr2F0G13(x781, new_sr10(x782, x784), new_primDivNatS1(x783), new_primDivNatS1(x783), x784) which results in the following constraint: (1) (new_pr2F0G14(x776, x777, x778, x779, x780)=new_pr2F0G14(x781, x782, x783, Zero, x784) ==> new_pr2F0G13(x776, x777, x778, Succ(Succ(x779)), x780)_>=_new_pr2F0G14(x776, x777, x778, x779, x780)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G13(x776, x777, x778, Succ(Succ(Zero)), x780)_>=_new_pr2F0G14(x776, x777, x778, Zero, x780)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) the following chains were created: *We consider the chain new_pr2F0G14(x844, x845, x846, Succ(Zero), x847) -> new_pr2F2(x845, x846, new_fromInt, x844, x847), new_pr2F2(x848, x849, Pos(x850), x851, x852) -> new_pr2F31(new_primPlusNat0(x849, x850), new_sr11(x848, x852), new_primPlusNat0(x849, x850), x851, x852) which results in the following constraint: (1) (new_pr2F2(x845, x846, new_fromInt, x844, x847)=new_pr2F2(x848, x849, Pos(x850), x851, x852) ==> new_pr2F0G14(x844, x845, x846, Succ(Zero), x847)_>=_new_pr2F2(x845, x846, new_fromInt, x844, x847)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_fromInt=Pos(x850) ==> new_pr2F0G14(x844, x845, x846, Succ(Zero), x847)_>=_new_pr2F2(x845, x846, new_fromInt, x844, x847)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_fromInt=Pos(x850) which results in the following new constraint: (3) (Pos(Succ(Zero))=Pos(x850) ==> new_pr2F0G14(x844, x845, x846, Succ(Zero), x847)_>=_new_pr2F2(x845, x846, new_fromInt, x844, x847)) We simplified constraint (3) using rules (I), (II), (IV) which results in the following new constraint: (4) (new_pr2F0G14(x844, x845, x846, Succ(Zero), x847)_>=_new_pr2F2(x845, x846, new_fromInt, x844, x847)) For Pair new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G14(x917, x918, x919, Zero, x920) -> new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(x919), new_primDivNatS1(x919), x920), new_pr2F0G13(x921, x922, x923, Succ(Zero), x924) -> new_pr2F2(x922, x923, new_fromInt, x921, x924) which results in the following constraint: (1) (new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(x919), new_primDivNatS1(x919), x920)=new_pr2F0G13(x921, x922, x923, Succ(Zero), x924) ==> new_pr2F0G14(x917, x918, x919, Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(x919), new_primDivNatS1(x919), x920)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x919)=Succ(Zero) ==> new_pr2F0G14(x917, x918, x919, Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(x919), new_primDivNatS1(x919), x920)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x919)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1847)=Succ(Zero) ==> new_pr2F0G14(x917, x918, Succ(x1847), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(x1847)), new_primDivNatS1(Succ(x1847)), x920)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1847)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1848))=Succ(Zero) ==> new_pr2F0G14(x917, x918, Succ(Succ(Succ(x1848))), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Succ(x1848)))), new_primDivNatS1(Succ(Succ(Succ(x1848)))), x920)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G14(x917, x918, Succ(Succ(Zero)), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x920)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x917, x918, Succ(Succ(Succ(x1848))), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Succ(x1848)))), new_primDivNatS1(Succ(Succ(Succ(x1848)))), x920)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x917, x918, Succ(Succ(Zero)), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x920)) *We consider the chain new_pr2F0G14(x933, x934, x935, Zero, x936) -> new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(x935), new_primDivNatS1(x935), x936), new_pr2F0G13(x937, x938, x939, Succ(Succ(x940)), x941) -> new_pr2F0G14(x937, x938, x939, x940, x941) which results in the following constraint: (1) (new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(x935), new_primDivNatS1(x935), x936)=new_pr2F0G13(x937, x938, x939, Succ(Succ(x940)), x941) ==> new_pr2F0G14(x933, x934, x935, Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(x935), new_primDivNatS1(x935), x936)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x935)=Succ(Succ(x940)) ==> new_pr2F0G14(x933, x934, x935, Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(x935), new_primDivNatS1(x935), x936)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x935)=Succ(Succ(x940)) which results in the following new constraint: (3) (new_primDivNatS01(x1849)=Succ(Succ(x940)) ==> new_pr2F0G14(x933, x934, Succ(x1849), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(x1849)), new_primDivNatS1(Succ(x1849)), x936)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1849)=Succ(Succ(x940)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1850))=Succ(Succ(x940)) ==> new_pr2F0G14(x933, x934, Succ(Succ(Succ(x1850))), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Succ(x1850)))), new_primDivNatS1(Succ(Succ(Succ(x1850)))), x936)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x940)) ==> new_pr2F0G14(x933, x934, Succ(Succ(Zero)), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x936)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x933, x934, Succ(Succ(Succ(x1850))), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Succ(x1850)))), new_primDivNatS1(Succ(Succ(Succ(x1850)))), x936)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x933, x934, Succ(Succ(Zero)), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x936)) *We consider the chain new_pr2F0G14(x950, x951, x952, Zero, x953) -> new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(x952), new_primDivNatS1(x952), x953), new_pr2F0G13(x954, x955, x956, Zero, x957) -> new_pr2F0G13(x954, new_sr10(x955, x957), new_primDivNatS1(x956), new_primDivNatS1(x956), x957) which results in the following constraint: (1) (new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(x952), new_primDivNatS1(x952), x953)=new_pr2F0G13(x954, x955, x956, Zero, x957) ==> new_pr2F0G14(x950, x951, x952, Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(x952), new_primDivNatS1(x952), x953)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x952)=Zero ==> new_pr2F0G14(x950, x951, x952, Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(x952), new_primDivNatS1(x952), x953)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x952)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G14(x950, x951, Zero, Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x953)) (4) (new_primDivNatS01(x1851)=Zero ==> new_pr2F0G14(x950, x951, Succ(x1851), Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Succ(x1851)), new_primDivNatS1(Succ(x1851)), x953)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G14(x950, x951, Zero, Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x953)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1851)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G14(x950, x951, Succ(Zero), Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x953)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G14(x950, x951, Succ(Zero), Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x953)) *We consider the chain new_pr2F0G14(x970, x971, x972, Zero, x973) -> new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(x972), new_primDivNatS1(x972), x973), new_pr2F0G13(x974, x975, x976, Succ(Succ(x977)), x978) -> H'(x974, x975, x976, x978, anew_new_pr2F0G14(x977)) which results in the following constraint: (1) (new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(x972), new_primDivNatS1(x972), x973)=new_pr2F0G13(x974, x975, x976, Succ(Succ(x977)), x978) ==> new_pr2F0G14(x970, x971, x972, Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(x972), new_primDivNatS1(x972), x973)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x972)=Succ(Succ(x977)) ==> new_pr2F0G14(x970, x971, x972, Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(x972), new_primDivNatS1(x972), x973)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x972)=Succ(Succ(x977)) which results in the following new constraint: (3) (new_primDivNatS01(x1853)=Succ(Succ(x977)) ==> new_pr2F0G14(x970, x971, Succ(x1853), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(x1853)), new_primDivNatS1(Succ(x1853)), x973)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1853)=Succ(Succ(x977)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1854))=Succ(Succ(x977)) ==> new_pr2F0G14(x970, x971, Succ(Succ(Succ(x1854))), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Succ(x1854)))), new_primDivNatS1(Succ(Succ(Succ(x1854)))), x973)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x977)) ==> new_pr2F0G14(x970, x971, Succ(Succ(Zero)), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x973)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G14(x970, x971, Succ(Succ(Succ(x1854))), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Succ(x1854)))), new_primDivNatS1(Succ(Succ(Succ(x1854)))), x973)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G14(x970, x971, Succ(Succ(Zero)), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x973)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) the following chains were created: *We consider the chain new_pr2F0G13(x1007, x1008, x1009, Zero, x1010) -> new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(x1009), new_primDivNatS1(x1009), x1010), new_pr2F0G13(x1011, x1012, x1013, Succ(Zero), x1014) -> new_pr2F2(x1012, x1013, new_fromInt, x1011, x1014) which results in the following constraint: (1) (new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(x1009), new_primDivNatS1(x1009), x1010)=new_pr2F0G13(x1011, x1012, x1013, Succ(Zero), x1014) ==> new_pr2F0G13(x1007, x1008, x1009, Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(x1009), new_primDivNatS1(x1009), x1010)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1009)=Succ(Zero) ==> new_pr2F0G13(x1007, x1008, x1009, Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(x1009), new_primDivNatS1(x1009), x1010)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1009)=Succ(Zero) which results in the following new constraint: (3) (new_primDivNatS01(x1855)=Succ(Zero) ==> new_pr2F0G13(x1007, x1008, Succ(x1855), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(x1855)), new_primDivNatS1(Succ(x1855)), x1010)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1855)=Succ(Zero) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1856))=Succ(Zero) ==> new_pr2F0G13(x1007, x1008, Succ(Succ(Succ(x1856))), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Succ(x1856)))), new_primDivNatS1(Succ(Succ(Succ(x1856)))), x1010)) (5) (Succ(new_primDivNatS2)=Succ(Zero) ==> new_pr2F0G13(x1007, x1008, Succ(Succ(Zero)), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1010)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x1007, x1008, Succ(Succ(Succ(x1856))), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Succ(x1856)))), new_primDivNatS1(Succ(Succ(Succ(x1856)))), x1010)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x1007, x1008, Succ(Succ(Zero)), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1010)) *We consider the chain new_pr2F0G13(x1023, x1024, x1025, Zero, x1026) -> new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(x1025), new_primDivNatS1(x1025), x1026), new_pr2F0G13(x1027, x1028, x1029, Succ(Succ(x1030)), x1031) -> new_pr2F0G14(x1027, x1028, x1029, x1030, x1031) which results in the following constraint: (1) (new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(x1025), new_primDivNatS1(x1025), x1026)=new_pr2F0G13(x1027, x1028, x1029, Succ(Succ(x1030)), x1031) ==> new_pr2F0G13(x1023, x1024, x1025, Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(x1025), new_primDivNatS1(x1025), x1026)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1025)=Succ(Succ(x1030)) ==> new_pr2F0G13(x1023, x1024, x1025, Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(x1025), new_primDivNatS1(x1025), x1026)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1025)=Succ(Succ(x1030)) which results in the following new constraint: (3) (new_primDivNatS01(x1857)=Succ(Succ(x1030)) ==> new_pr2F0G13(x1023, x1024, Succ(x1857), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(x1857)), new_primDivNatS1(Succ(x1857)), x1026)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1857)=Succ(Succ(x1030)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1858))=Succ(Succ(x1030)) ==> new_pr2F0G13(x1023, x1024, Succ(Succ(Succ(x1858))), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Succ(x1858)))), new_primDivNatS1(Succ(Succ(Succ(x1858)))), x1026)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x1030)) ==> new_pr2F0G13(x1023, x1024, Succ(Succ(Zero)), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1026)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x1023, x1024, Succ(Succ(Succ(x1858))), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Succ(x1858)))), new_primDivNatS1(Succ(Succ(Succ(x1858)))), x1026)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x1023, x1024, Succ(Succ(Zero)), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1026)) *We consider the chain new_pr2F0G13(x1040, x1041, x1042, Zero, x1043) -> new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(x1042), new_primDivNatS1(x1042), x1043), new_pr2F0G13(x1044, x1045, x1046, Zero, x1047) -> new_pr2F0G13(x1044, new_sr10(x1045, x1047), new_primDivNatS1(x1046), new_primDivNatS1(x1046), x1047) which results in the following constraint: (1) (new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(x1042), new_primDivNatS1(x1042), x1043)=new_pr2F0G13(x1044, x1045, x1046, Zero, x1047) ==> new_pr2F0G13(x1040, x1041, x1042, Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(x1042), new_primDivNatS1(x1042), x1043)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1042)=Zero ==> new_pr2F0G13(x1040, x1041, x1042, Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(x1042), new_primDivNatS1(x1042), x1043)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1042)=Zero which results in the following new constraints: (3) (Zero=Zero ==> new_pr2F0G13(x1040, x1041, Zero, Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1043)) (4) (new_primDivNatS01(x1859)=Zero ==> new_pr2F0G13(x1040, x1041, Succ(x1859), Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Succ(x1859)), new_primDivNatS1(Succ(x1859)), x1043)) We simplified constraint (3) using rules (I), (II) which results in the following new constraint: (5) (new_pr2F0G13(x1040, x1041, Zero, Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1043)) We simplified constraint (4) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1859)=Zero which results in the following new constraint: (6) (Zero=Zero ==> new_pr2F0G13(x1040, x1041, Succ(Zero), Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1043)) We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G13(x1040, x1041, Succ(Zero), Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1043)) *We consider the chain new_pr2F0G13(x1060, x1061, x1062, Zero, x1063) -> new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(x1062), new_primDivNatS1(x1062), x1063), new_pr2F0G13(x1064, x1065, x1066, Succ(Succ(x1067)), x1068) -> H'(x1064, x1065, x1066, x1068, anew_new_pr2F0G14(x1067)) which results in the following constraint: (1) (new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(x1062), new_primDivNatS1(x1062), x1063)=new_pr2F0G13(x1064, x1065, x1066, Succ(Succ(x1067)), x1068) ==> new_pr2F0G13(x1060, x1061, x1062, Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(x1062), new_primDivNatS1(x1062), x1063)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x1062)=Succ(Succ(x1067)) ==> new_pr2F0G13(x1060, x1061, x1062, Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(x1062), new_primDivNatS1(x1062), x1063)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x1062)=Succ(Succ(x1067)) which results in the following new constraint: (3) (new_primDivNatS01(x1861)=Succ(Succ(x1067)) ==> new_pr2F0G13(x1060, x1061, Succ(x1861), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(x1861)), new_primDivNatS1(Succ(x1861)), x1063)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x1861)=Succ(Succ(x1067)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x1862))=Succ(Succ(x1067)) ==> new_pr2F0G13(x1060, x1061, Succ(Succ(Succ(x1862))), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Succ(x1862)))), new_primDivNatS1(Succ(Succ(Succ(x1862)))), x1063)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x1067)) ==> new_pr2F0G13(x1060, x1061, Succ(Succ(Zero)), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1063)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G13(x1060, x1061, Succ(Succ(Succ(x1862))), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Succ(x1862)))), new_primDivNatS1(Succ(Succ(Succ(x1862)))), x1063)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G13(x1060, x1061, Succ(Succ(Zero)), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1063)) For Pair new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) the following chains were created: *We consider the chain new_pr2F31(Succ(x1142), x1143, Succ(Succ(x1144)), x1145, x1146) -> H(x1143, x1145, Succ(x1144), x1146, anew_new_pr2F0G12(x1144)), H(x1147, x1148, x1149, x1150, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(x1147, x1148, x1149, Succ(Zero), x1150) which results in the following constraint: (1) (H(x1143, x1145, Succ(x1144), x1146, anew_new_pr2F0G12(x1144))=H(x1147, x1148, x1149, x1150, cons_new_pr2F0G12(Succ(Zero))) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(x1144)), x1145, x1146)_>=_H(x1143, x1145, Succ(x1144), x1146, anew_new_pr2F0G12(x1144))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G12(x1144)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(x1144)), x1145, x1146)_>=_H(x1143, x1145, Succ(x1144), x1146, anew_new_pr2F0G12(x1144))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G12(x1144)=cons_new_pr2F0G12(Succ(Zero)) which results in the following new constraint: (3) (new_new_pr2F0G12(x1863)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(x1863)))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(x1863))), x1146, anew_new_pr2F0G12(Succ(Succ(x1863))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G12(x1863)=cons_new_pr2F0G12(Succ(Zero)) which results in the following new constraints: (4) (new_new_pr2F0G12(x1864)=cons_new_pr2F0G12(Succ(Zero)) & (\/x1865,x1866,x1867,x1868:new_new_pr2F0G12(x1864)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1865), x1866, Succ(Succ(Succ(Succ(x1864)))), x1867, x1868)_>=_H(x1866, x1867, Succ(Succ(Succ(x1864))), x1868, anew_new_pr2F0G12(Succ(Succ(x1864))))) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Succ(x1864)))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Succ(x1864))))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1864))))))) (5) (cons_new_pr2F0G12(Succ(Zero))=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Zero))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Zero)))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G12(Zero)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Zero)))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Zero))), x1146, anew_new_pr2F0G12(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1865,x1866,x1867,x1868:new_new_pr2F0G12(x1864)=cons_new_pr2F0G12(Succ(Zero)) ==> new_pr2F31(Succ(x1865), x1866, Succ(Succ(Succ(Succ(x1864)))), x1867, x1868)_>=_H(x1866, x1867, Succ(Succ(Succ(x1864))), x1868, anew_new_pr2F0G12(Succ(Succ(x1864))))) with sigma = [x1865 / x1142, x1866 / x1143, x1867 / x1145, x1868 / x1146] which results in the following new constraint: (7) (new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(x1864)))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(x1864))), x1146, anew_new_pr2F0G12(Succ(Succ(x1864)))) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Succ(x1864)))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Succ(x1864))))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1864))))))) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Zero))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Zero)))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_pr2F31(Succ(x1151), x1152, Succ(Succ(x1153)), x1154, x1155) -> H(x1152, x1154, Succ(x1153), x1155, anew_new_pr2F0G12(x1153)), H(x1156, x1157, x1158, x1159, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(x1156, x1157, x1158, Zero, x1159) which results in the following constraint: (1) (H(x1152, x1154, Succ(x1153), x1155, anew_new_pr2F0G12(x1153))=H(x1156, x1157, x1158, x1159, cons_new_pr2F0G12(Zero)) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(x1153)), x1154, x1155)_>=_H(x1152, x1154, Succ(x1153), x1155, anew_new_pr2F0G12(x1153))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G12(x1153)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(x1153)), x1154, x1155)_>=_H(x1152, x1154, Succ(x1153), x1155, anew_new_pr2F0G12(x1153))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G12(x1153)=cons_new_pr2F0G12(Zero) which results in the following new constraint: (3) (new_new_pr2F0G12(x1869)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(x1869)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(x1869))), x1155, anew_new_pr2F0G12(Succ(Succ(x1869))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G12(x1869)=cons_new_pr2F0G12(Zero) which results in the following new constraints: (4) (new_new_pr2F0G12(x1870)=cons_new_pr2F0G12(Zero) & (\/x1871,x1872,x1873,x1874:new_new_pr2F0G12(x1870)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1871), x1872, Succ(Succ(Succ(Succ(x1870)))), x1873, x1874)_>=_H(x1872, x1873, Succ(Succ(Succ(x1870))), x1874, anew_new_pr2F0G12(Succ(Succ(x1870))))) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Succ(Succ(x1870)))))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Succ(Succ(x1870))))), x1155, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1870))))))) (5) (cons_new_pr2F0G12(Succ(Zero))=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Succ(Zero))))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Succ(Zero)))), x1155, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G12(Zero)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Zero)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Zero))), x1155, anew_new_pr2F0G12(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1871,x1872,x1873,x1874:new_new_pr2F0G12(x1870)=cons_new_pr2F0G12(Zero) ==> new_pr2F31(Succ(x1871), x1872, Succ(Succ(Succ(Succ(x1870)))), x1873, x1874)_>=_H(x1872, x1873, Succ(Succ(Succ(x1870))), x1874, anew_new_pr2F0G12(Succ(Succ(x1870))))) with sigma = [x1871 / x1151, x1872 / x1152, x1873 / x1154, x1874 / x1155] which results in the following new constraint: (7) (new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(x1870)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(x1870))), x1155, anew_new_pr2F0G12(Succ(Succ(x1870)))) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Succ(Succ(x1870)))))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Succ(Succ(x1870))))), x1155, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1870))))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Zero)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Zero))), x1155, anew_new_pr2F0G12(Succ(Succ(Zero))))) For Pair H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) the following chains were created: *We consider the chain H(x1175, x1176, x1177, x1178, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(x1175, x1176, x1177, Succ(Zero), x1178), new_pr2F0G12(x1179, x1180, x1181, Succ(Zero), x1182) -> new_pr2F1(x1179, x1181, new_fromInt, x1180, x1182) which results in the following constraint: (1) (new_pr2F0G12(x1175, x1176, x1177, Succ(Zero), x1178)=new_pr2F0G12(x1179, x1180, x1181, Succ(Zero), x1182) ==> H(x1175, x1176, x1177, x1178, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1175, x1176, x1177, Succ(Zero), x1178)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x1175, x1176, x1177, x1178, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1175, x1176, x1177, Succ(Zero), x1178)) For Pair H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) the following chains were created: *We consider the chain H(x1267, x1268, x1269, x1270, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(x1267, x1268, x1269, Zero, x1270), new_pr2F0G12(x1271, x1272, x1273, Zero, x1274) -> new_pr2F0G13(new_sr8(x1271, x1272, x1274), x1271, new_primDivNatS1(Succ(x1273)), new_primDivNatS1(Succ(x1273)), x1274) which results in the following constraint: (1) (new_pr2F0G12(x1267, x1268, x1269, Zero, x1270)=new_pr2F0G12(x1271, x1272, x1273, Zero, x1274) ==> H(x1267, x1268, x1269, x1270, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1267, x1268, x1269, Zero, x1270)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x1267, x1268, x1269, x1270, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1267, x1268, x1269, Zero, x1270)) For Pair new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) the following chains were created: *We consider the chain new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(x1410)), x1411) -> H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(x1410)), H'(x1412, x1413, x1414, x1415, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(x1412, x1413, x1414, Succ(Zero), x1415) which results in the following constraint: (1) (H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(x1410))=H'(x1412, x1413, x1414, x1415, cons_new_pr2F0G14(Succ(Zero))) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(x1410)), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(x1410))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G14(x1410)=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(x1410)), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(x1410))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G14(x1410)=cons_new_pr2F0G14(Succ(Zero)) which results in the following new constraint: (3) (new_new_pr2F0G14(x1875)=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(x1875)))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(x1875))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G14(x1875)=cons_new_pr2F0G14(Succ(Zero)) which results in the following new constraints: (4) (new_new_pr2F0G14(x1876)=cons_new_pr2F0G14(Succ(Zero)) & (\/x1877,x1878,x1879,x1880:new_new_pr2F0G14(x1876)=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1877, x1878, x1879, Succ(Succ(Succ(Succ(x1876)))), x1880)_>=_H'(x1877, x1878, x1879, x1880, anew_new_pr2F0G14(Succ(Succ(x1876))))) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Succ(x1876)))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1876))))))) (5) (cons_new_pr2F0G14(Succ(Zero))=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Zero))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G14(Zero)=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Zero)))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1877,x1878,x1879,x1880:new_new_pr2F0G14(x1876)=cons_new_pr2F0G14(Succ(Zero)) ==> new_pr2F0G13(x1877, x1878, x1879, Succ(Succ(Succ(Succ(x1876)))), x1880)_>=_H'(x1877, x1878, x1879, x1880, anew_new_pr2F0G14(Succ(Succ(x1876))))) with sigma = [x1877 / x1407, x1878 / x1408, x1879 / x1409, x1880 / x1411] which results in the following new constraint: (7) (new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(x1876)))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(x1876)))) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Succ(x1876)))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1876))))))) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Zero))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Zero)))))) We solved constraint (6) using rules (I), (II). *We consider the chain new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(x1419)), x1420) -> H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(x1419)), H'(x1421, x1422, x1423, x1424, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(x1421, x1422, x1423, Zero, x1424) which results in the following constraint: (1) (H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(x1419))=H'(x1421, x1422, x1423, x1424, cons_new_pr2F0G14(Zero)) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(x1419)), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(x1419))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G14(x1419)=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(x1419)), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(x1419))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G14(x1419)=cons_new_pr2F0G14(Zero) which results in the following new constraint: (3) (new_new_pr2F0G14(x1881)=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(x1881)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(x1881))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G14(x1881)=cons_new_pr2F0G14(Zero) which results in the following new constraints: (4) (new_new_pr2F0G14(x1882)=cons_new_pr2F0G14(Zero) & (\/x1883,x1884,x1885,x1886:new_new_pr2F0G14(x1882)=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1883, x1884, x1885, Succ(Succ(Succ(Succ(x1882)))), x1886)_>=_H'(x1883, x1884, x1885, x1886, anew_new_pr2F0G14(Succ(Succ(x1882))))) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Succ(Succ(x1882)))))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1882))))))) (5) (cons_new_pr2F0G14(Succ(Zero))=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Succ(Zero))))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Succ(Zero)))))) (6) (cons_new_pr2F0G14(Zero)=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Zero)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x1883,x1884,x1885,x1886:new_new_pr2F0G14(x1882)=cons_new_pr2F0G14(Zero) ==> new_pr2F0G13(x1883, x1884, x1885, Succ(Succ(Succ(Succ(x1882)))), x1886)_>=_H'(x1883, x1884, x1885, x1886, anew_new_pr2F0G14(Succ(Succ(x1882))))) with sigma = [x1883 / x1416, x1884 / x1417, x1885 / x1418, x1886 / x1420] which results in the following new constraint: (7) (new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(x1882)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(x1882)))) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Succ(Succ(x1882)))))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1882))))))) We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rules (I), (II) which results in the following new constraint: (8) (new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Zero)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Zero))))) For Pair H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) the following chains were created: *We consider the chain H'(x1461, x1462, x1463, x1464, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(x1461, x1462, x1463, Succ(Zero), x1464), new_pr2F0G14(x1465, x1466, x1467, Succ(Zero), x1468) -> new_pr2F2(x1466, x1467, new_fromInt, x1465, x1468) which results in the following constraint: (1) (new_pr2F0G14(x1461, x1462, x1463, Succ(Zero), x1464)=new_pr2F0G14(x1465, x1466, x1467, Succ(Zero), x1468) ==> H'(x1461, x1462, x1463, x1464, cons_new_pr2F0G14(Succ(Zero)))_>=_new_pr2F0G14(x1461, x1462, x1463, Succ(Zero), x1464)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H'(x1461, x1462, x1463, x1464, cons_new_pr2F0G14(Succ(Zero)))_>=_new_pr2F0G14(x1461, x1462, x1463, Succ(Zero), x1464)) For Pair H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) the following chains were created: *We consider the chain H'(x1541, x1542, x1543, x1544, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(x1541, x1542, x1543, Zero, x1544), new_pr2F0G14(x1545, x1546, x1547, Zero, x1548) -> new_pr2F0G13(x1545, new_sr10(x1546, x1548), new_primDivNatS1(x1547), new_primDivNatS1(x1547), x1548) which results in the following constraint: (1) (new_pr2F0G14(x1541, x1542, x1543, Zero, x1544)=new_pr2F0G14(x1545, x1546, x1547, Zero, x1548) ==> H'(x1541, x1542, x1543, x1544, cons_new_pr2F0G14(Zero))_>=_new_pr2F0G14(x1541, x1542, x1543, Zero, x1544)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H'(x1541, x1542, x1543, x1544, cons_new_pr2F0G14(Zero))_>=_new_pr2F0G14(x1541, x1542, x1543, Zero, x1544)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) *(new_pr2F0G12(x4, x5, x6, Succ(Zero), x7)_>=_new_pr2F1(x4, x6, new_fromInt, x5, x7)) *new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) *(new_pr2F1(x87, x88, Pos(x93), x90, x91)_>=_new_pr2F34(x88, Pos(x93), x87, new_sr9(x87, x90, x91), x91)) *new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) *(new_pr2F34(Succ(x194), Pos(Zero), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x194)), Zero), x189, new_primPlusNat0(Succ(Succ(x194)), Zero), x190, x191)) *(new_pr2F34(x187, Pos(Succ(x1579)), x189, x190, x191)_>=_new_pr2F31(new_primPlusNat0(Succ(x187), Succ(x1579)), x189, new_primPlusNat0(Succ(x187), Succ(x1579)), x190, x191)) *(new_pr2F34(Zero, Pos(Zero), x214, x215, x216)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), x214, new_primPlusNat0(Succ(Zero), Zero), x215, x216)) *(new_pr2F34(Succ(x248), Pos(Zero), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x248)), Zero), x243, new_primPlusNat0(Succ(Succ(x248)), Zero), x244, x245)) *(new_pr2F34(x241, Pos(Succ(x1657)), x243, x244, x245)_>=_new_pr2F31(new_primPlusNat0(Succ(x241), Succ(x1657)), x243, new_primPlusNat0(Succ(x241), Succ(x1657)), x244, x245)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) *(new_pr2F31(Succ(x276), x277, Succ(Succ(Succ(Zero))), x279, x280)_>=_new_pr2F0G12(x277, x279, Succ(Succ(Zero)), Succ(Zero), x280)) *(new_pr2F31(Succ(x300), x301, Succ(Succ(Zero)), x303, x304)_>=_new_pr2F0G12(x301, x303, Succ(Zero), Zero, x304)) *new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) *(new_pr2F0G12(x394, x395, Succ(Succ(x1698)), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Succ(x1698)))), new_primDivNatS1(Succ(Succ(Succ(x1698)))), x397)) *(new_pr2F0G12(x394, x395, Succ(Zero), Zero, x397)_>=_new_pr2F0G13(new_sr8(x394, x395, x397), x394, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x397)) *(new_pr2F0G12(x410, x411, Succ(Succ(x1701)), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Succ(x1701)))), new_primDivNatS1(Succ(Succ(Succ(x1701)))), x413)) *(new_pr2F0G12(x410, x411, Succ(Zero), Zero, x413)_>=_new_pr2F0G13(new_sr8(x410, x411, x413), x410, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x413)) *(new_pr2F0G12(x427, x428, Zero, Zero, x430)_>=_new_pr2F0G13(new_sr8(x427, x428, x430), x427, new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x430)) *(new_pr2F0G12(x447, x448, Succ(Succ(x1707)), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Succ(x1707)))), new_primDivNatS1(Succ(Succ(Succ(x1707)))), x450)) *(new_pr2F0G12(x447, x448, Succ(Zero), Zero, x450)_>=_new_pr2F0G13(new_sr8(x447, x448, x450), x447, new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x450)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G13(x488, x489, x490, Succ(Zero), x491)_>=_new_pr2F2(x489, x490, new_fromInt, x488, x491)) *new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) *(new_pr2F2(x556, Succ(Succ(x563)), Pos(Zero), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x563)), Zero), new_sr11(x556, x560), new_primPlusNat0(Succ(Succ(x563)), Zero), x559, x560)) *(new_pr2F2(x556, Zero, Pos(Succ(Succ(x563))), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x563))), new_sr11(x556, x560), new_primPlusNat0(Zero, Succ(Succ(x563))), x559, x560)) *(new_pr2F2(x556, Succ(x1709), Pos(Succ(x1708)), x559, x560)_>=_new_pr2F31(new_primPlusNat0(Succ(x1709), Succ(x1708)), new_sr11(x556, x560), new_primPlusNat0(Succ(x1709), Succ(x1708)), x559, x560)) *(new_pr2F2(x581, Succ(Zero), Pos(Zero), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Succ(Zero), Zero), new_sr11(x581, x585), new_primPlusNat0(Succ(Zero), Zero), x584, x585)) *(new_pr2F2(x581, Zero, Pos(Succ(Zero)), x584, x585)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Zero)), new_sr11(x581, x585), new_primPlusNat0(Zero, Succ(Zero)), x584, x585)) *(new_pr2F2(x610, Succ(Succ(x617)), Pos(Zero), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(Succ(x617)), Zero), new_sr11(x610, x614), new_primPlusNat0(Succ(Succ(x617)), Zero), x613, x614)) *(new_pr2F2(x610, Zero, Pos(Succ(Succ(x617))), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Zero, Succ(Succ(x617))), new_sr11(x610, x614), new_primPlusNat0(Zero, Succ(Succ(x617))), x613, x614)) *(new_pr2F2(x610, Succ(x1800), Pos(Succ(x1799)), x613, x614)_>=_new_pr2F31(new_primPlusNat0(Succ(x1800), Succ(x1799)), new_sr11(x610, x614), new_primPlusNat0(Succ(x1800), Succ(x1799)), x613, x614)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) *(new_pr2F31(Succ(x649), x650, Succ(Zero), x651, x652)_>=_new_pr2F1(x650, Zero, new_fromInt, x651, x652)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) *(new_pr2F0G13(x767, x768, x769, Succ(Succ(Succ(Zero))), x771)_>=_new_pr2F0G14(x767, x768, x769, Succ(Zero), x771)) *(new_pr2F0G13(x776, x777, x778, Succ(Succ(Zero)), x780)_>=_new_pr2F0G14(x776, x777, x778, Zero, x780)) *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) *(new_pr2F0G14(x844, x845, x846, Succ(Zero), x847)_>=_new_pr2F2(x845, x846, new_fromInt, x844, x847)) *new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G14(x917, x918, Succ(Succ(Succ(x1848))), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Succ(x1848)))), new_primDivNatS1(Succ(Succ(Succ(x1848)))), x920)) *(new_pr2F0G14(x917, x918, Succ(Succ(Zero)), Zero, x920)_>=_new_pr2F0G13(x917, new_sr10(x918, x920), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x920)) *(new_pr2F0G14(x933, x934, Succ(Succ(Succ(x1850))), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Succ(x1850)))), new_primDivNatS1(Succ(Succ(Succ(x1850)))), x936)) *(new_pr2F0G14(x933, x934, Succ(Succ(Zero)), Zero, x936)_>=_new_pr2F0G13(x933, new_sr10(x934, x936), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x936)) *(new_pr2F0G14(x950, x951, Zero, Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x953)) *(new_pr2F0G14(x950, x951, Succ(Zero), Zero, x953)_>=_new_pr2F0G13(x950, new_sr10(x951, x953), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x953)) *(new_pr2F0G14(x970, x971, Succ(Succ(Succ(x1854))), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Succ(x1854)))), new_primDivNatS1(Succ(Succ(Succ(x1854)))), x973)) *(new_pr2F0G14(x970, x971, Succ(Succ(Zero)), Zero, x973)_>=_new_pr2F0G13(x970, new_sr10(x971, x973), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x973)) *new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) *(new_pr2F0G13(x1007, x1008, Succ(Succ(Succ(x1856))), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Succ(x1856)))), new_primDivNatS1(Succ(Succ(Succ(x1856)))), x1010)) *(new_pr2F0G13(x1007, x1008, Succ(Succ(Zero)), Zero, x1010)_>=_new_pr2F0G13(x1007, new_sr10(x1008, x1010), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1010)) *(new_pr2F0G13(x1023, x1024, Succ(Succ(Succ(x1858))), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Succ(x1858)))), new_primDivNatS1(Succ(Succ(Succ(x1858)))), x1026)) *(new_pr2F0G13(x1023, x1024, Succ(Succ(Zero)), Zero, x1026)_>=_new_pr2F0G13(x1023, new_sr10(x1024, x1026), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1026)) *(new_pr2F0G13(x1040, x1041, Zero, Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x1043)) *(new_pr2F0G13(x1040, x1041, Succ(Zero), Zero, x1043)_>=_new_pr2F0G13(x1040, new_sr10(x1041, x1043), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x1043)) *(new_pr2F0G13(x1060, x1061, Succ(Succ(Succ(x1862))), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Succ(x1862)))), new_primDivNatS1(Succ(Succ(Succ(x1862)))), x1063)) *(new_pr2F0G13(x1060, x1061, Succ(Succ(Zero)), Zero, x1063)_>=_new_pr2F0G13(x1060, new_sr10(x1061, x1063), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x1063)) *new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) *(new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(x1864)))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(x1864))), x1146, anew_new_pr2F0G12(Succ(Succ(x1864)))) ==> new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Succ(x1864)))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Succ(x1864))))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1864))))))) *(new_pr2F31(Succ(x1142), x1143, Succ(Succ(Succ(Succ(Succ(Zero))))), x1145, x1146)_>=_H(x1143, x1145, Succ(Succ(Succ(Succ(Zero)))), x1146, anew_new_pr2F0G12(Succ(Succ(Succ(Zero)))))) *(new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(x1870)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(x1870))), x1155, anew_new_pr2F0G12(Succ(Succ(x1870)))) ==> new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Succ(Succ(x1870)))))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Succ(Succ(x1870))))), x1155, anew_new_pr2F0G12(Succ(Succ(Succ(Succ(x1870))))))) *(new_pr2F31(Succ(x1151), x1152, Succ(Succ(Succ(Succ(Zero)))), x1154, x1155)_>=_H(x1152, x1154, Succ(Succ(Succ(Zero))), x1155, anew_new_pr2F0G12(Succ(Succ(Zero))))) *H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) *(H(x1175, x1176, x1177, x1178, cons_new_pr2F0G12(Succ(Zero)))_>=_new_pr2F0G12(x1175, x1176, x1177, Succ(Zero), x1178)) *H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) *(H(x1267, x1268, x1269, x1270, cons_new_pr2F0G12(Zero))_>=_new_pr2F0G12(x1267, x1268, x1269, Zero, x1270)) *new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) *(new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(x1876)))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(x1876)))) ==> new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Succ(x1876)))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1876))))))) *(new_pr2F0G13(x1407, x1408, x1409, Succ(Succ(Succ(Succ(Succ(Zero))))), x1411)_>=_H'(x1407, x1408, x1409, x1411, anew_new_pr2F0G14(Succ(Succ(Succ(Zero)))))) *(new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(x1882)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(x1882)))) ==> new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Succ(Succ(x1882)))))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Succ(Succ(x1882))))))) *(new_pr2F0G13(x1416, x1417, x1418, Succ(Succ(Succ(Succ(Zero)))), x1420)_>=_H'(x1416, x1417, x1418, x1420, anew_new_pr2F0G14(Succ(Succ(Zero))))) *H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) *(H'(x1461, x1462, x1463, x1464, cons_new_pr2F0G14(Succ(Zero)))_>=_new_pr2F0G14(x1461, x1462, x1463, Succ(Zero), x1464)) *H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) *(H'(x1541, x1542, x1543, x1544, cons_new_pr2F0G14(Zero))_>=_new_pr2F0G14(x1541, x1542, x1543, Zero, x1544)) 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. ---------------------------------------- (51) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) -> new_pr2F1(vuz228, vuz230, new_fromInt, vuz229, bd) new_pr2F1(vuz228, vuz230, vuz233, vuz229, bd) -> new_pr2F34(vuz230, vuz233, vuz228, new_sr9(vuz228, vuz229, bd), bd) new_pr2F34(vuz214, Pos(vuz2150), vuz216, vuz217, bc) -> new_pr2F31(new_primPlusNat0(Succ(vuz214), vuz2150), vuz216, new_primPlusNat0(Succ(vuz214), vuz2150), vuz217, bc) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> new_pr2F0G12(vuz216, vuz217, Succ(vuz21800), vuz21800, bc) new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) -> new_pr2F0G13(new_sr8(vuz228, vuz229, bd), vuz228, new_primDivNatS1(Succ(vuz230)), new_primDivNatS1(Succ(vuz230)), bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F2(vuz111, vuz113, Pos(vuz1160), vuz110, be) -> new_pr2F31(new_primPlusNat0(vuz113, vuz1160), new_sr11(vuz111, be), new_primPlusNat0(vuz113, vuz1160), vuz110, be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Zero), vuz217, bc) -> new_pr2F1(vuz216, Zero, new_fromInt, vuz217, bc) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) -> new_pr2F2(vuz111, vuz113, new_fromInt, vuz110, be) new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F0G13(vuz110, vuz111, vuz113, Zero, be) -> new_pr2F0G13(vuz110, new_sr10(vuz111, be), new_primDivNatS1(vuz113), new_primDivNatS1(vuz113), be) new_pr2F31(Succ(vuz2190), vuz216, Succ(Succ(vuz21800)), vuz217, bc) -> H(vuz216, vuz217, Succ(vuz21800), bc, anew_new_pr2F0G12(vuz21800)) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Succ(Zero))) -> new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Zero), bd) H(vuz228, vuz229, vuz230, bd, cons_new_pr2F0G12(Zero)) -> new_pr2F0G12(vuz228, vuz229, vuz230, Zero, bd) new_pr2F0G13(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> H'(vuz110, vuz111, vuz113, be, anew_new_pr2F0G14(vuz11400)) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Succ(Zero))) -> new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Zero), be) H'(vuz110, vuz111, vuz113, be, cons_new_pr2F0G14(Zero)) -> new_pr2F0G14(vuz110, vuz111, vuz113, Zero, be) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) anew_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Succ(vuz23100))) -> new_new_pr2F0G12(vuz23100) new_new_pr2F0G12(Succ(Zero)) -> cons_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) -> cons_new_pr2F0G12(Zero) anew_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Succ(vuz11400))) -> new_new_pr2F0G14(vuz11400) new_new_pr2F0G14(Succ(Zero)) -> cons_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) -> cons_new_pr2F0G14(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) new_new_pr2F0G12(Succ(Succ(x0))) anew_new_pr2F0G12(Succ(Succ(x0))) new_new_pr2F0G12(Succ(Zero)) new_new_pr2F0G12(Zero) new_new_pr2F0G14(Succ(Succ(x0))) anew_new_pr2F0G14(Succ(Succ(x0))) new_new_pr2F0G14(Succ(Zero)) new_new_pr2F0G14(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (52) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (53) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G14(vuz110, vuz111, vuz113, Succ(Succ(vuz11400)), be) -> new_pr2F0G14(vuz110, vuz111, vuz113, vuz11400, be) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5 ---------------------------------------- (54) YES ---------------------------------------- (55) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (56) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G12(vuz228, vuz229, vuz230, Succ(Succ(vuz23100)), bd) -> new_pr2F0G12(vuz228, vuz229, vuz230, vuz23100, bd) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5 ---------------------------------------- (57) YES ---------------------------------------- (58) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F30(Succ(vuz2120), vuz204, Succ(Zero), vuz205, h) -> new_pr2F(vuz204, Zero, new_fromInt, vuz205, h) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (59) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_pr2F30(Succ(vuz2120), vuz204, Succ(Zero), vuz205, h) -> new_pr2F(vuz204, Zero, new_fromInt, vuz205, h) we obtained the following new rules [LPAR04]: (new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3),new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3)) ---------------------------------------- (60) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) The TRS R consists of the following rules: new_sr11(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr6(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_primPlusNat0(Zero, Zero) -> Zero new_sr5(vuz103, ty_Float) -> new_sr21(vuz103) new_sr11(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr(vuz204, vuz205, ty_Double) -> new_sr13(vuz204, vuz205) new_sr8(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_sr6(vuz103, ty_Float) -> new_sr21(vuz103) new_sr10(vuz111, ty_Int) -> new_sr19(vuz111) new_sr6(vuz103, ty_Double) -> new_sr12(vuz103) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr6(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr5(vuz103, ty_Int) -> new_sr19(vuz103) new_sr10(vuz111, ty_Float) -> new_sr21(vuz111) new_sr10(vuz111, ty_Integer) -> new_sr20(vuz111) new_sr10(vuz111, ty_Double) -> new_sr12(vuz111) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr5(vuz103, ty_Double) -> new_sr12(vuz103) new_sr9(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr9(vuz228, vuz229, ty_Integer) -> new_sr16(vuz228, vuz229) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_sr6(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr7(vuz216, vuz217, app(ty_Ratio, ce)) -> new_sr14(vuz216, vuz217, ce) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr8(vuz228, vuz229, ty_Double) -> new_sr13(vuz228, vuz229) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr4(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_sr7(vuz216, vuz217, ty_Int) -> new_sr15(vuz216, vuz217) new_sr(vuz204, vuz205, ty_Float) -> new_sr17(vuz204, vuz205) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr7(vuz216, vuz217, ty_Integer) -> new_sr16(vuz216, vuz217) new_sr13(vuz72, vuz20) -> error([]) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primMulNat0(Zero, Zero) -> Zero new_sr9(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_primDivNatS01(Zero) -> Zero new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_primDivNatS1(Zero) -> Zero new_sr9(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_primDivNatS3 -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr(vuz204, vuz205, app(ty_Ratio, bg)) -> new_sr14(vuz204, vuz205, bg) new_sr16(vuz71, vuz20) -> error([]) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr8(vuz228, vuz229, app(ty_Ratio, bh)) -> new_sr14(vuz228, vuz229, bh) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr8(vuz228, vuz229, ty_Float) -> new_sr17(vuz228, vuz229) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_fromInt -> Pos(Succ(Zero)) new_sr9(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr11(vuz111, ty_Float) -> new_sr21(vuz111) new_sr4(vuz103, ty_Int) -> new_sr19(vuz103) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr11(vuz111, ty_Double) -> new_sr12(vuz111) new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr7(vuz216, vuz217, ty_Float) -> new_sr17(vuz216, vuz217) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr(vuz204, vuz205, ty_Integer) -> new_sr16(vuz204, vuz205) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr8(vuz228, vuz229, ty_Int) -> new_sr15(vuz228, vuz229) new_sr4(vuz103, ty_Double) -> new_sr12(vuz103) new_sr5(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr4(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_sr4(vuz103, ty_Float) -> new_sr21(vuz103) new_primDivNatS2 -> new_primDivNatS3 new_sr(vuz204, vuz205, ty_Int) -> new_sr15(vuz204, vuz205) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr10(vuz111, app(ty_Ratio, cc)) -> new_sr18(vuz111, cc) new_sr7(vuz216, vuz217, ty_Double) -> new_sr13(vuz216, vuz217) new_sr11(vuz111, ty_Int) -> new_sr19(vuz111) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_sr17(vuz73, vuz20) -> error([]) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (61) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr17(x0, x1) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_primMulNat0(Succ(x0), Succ(x1)) new_sr6(x0, app(ty_Ratio, x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr4(x0, ty_Integer) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr6(x0, ty_Integer) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_sr8(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_sr9(x0, x1, ty_Int) new_primPlusNat0(Succ(x0), Zero) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr2(x0, ty_Int) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr16(x0, x1) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_primPlusNat0(Zero, Zero) new_sr8(x0, x1, ty_Int) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_sr11(x0, ty_Int) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_sr(x0, x1, ty_Integer) new_sr6(x0, ty_Int) new_sr7(x0, x1, ty_Int) new_sr9(x0, x1, ty_Float) new_sr5(x0, ty_Integer) new_sr(x0, x1, app(ty_Ratio, x2)) new_sr6(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Integer) new_sr(x0, x1, ty_Int) new_sr5(x0, ty_Int) new_sr6(x0, ty_Integer) new_sr4(x0, app(ty_Ratio, x1)) new_sr4(x0, ty_Float) new_sr11(x0, ty_Float) new_sr11(x0, ty_Double) new_sr8(x0, x1, ty_Double) new_sr(x0, x1, ty_Float) new_sr10(x0, ty_Int) new_sr4(x0, ty_Double) new_sr6(x0, ty_Double) new_sr8(x0, x1, ty_Float) new_sr11(x0, ty_Integer) new_sr7(x0, x1, ty_Float) new_sr7(x0, x1, ty_Integer) new_sr9(x0, x1, ty_Int) new_sr8(x0, x1, ty_Integer) new_sr6(x0, ty_Float) new_sr11(x0, app(ty_Ratio, x1)) new_sr7(x0, x1, app(ty_Ratio, x2)) new_sr9(x0, x1, ty_Integer) new_sr7(x0, x1, ty_Double) new_sr10(x0, ty_Double) new_sr5(x0, ty_Float) new_sr9(x0, x1, app(ty_Ratio, x2)) new_sr8(x0, x1, app(ty_Ratio, x2)) new_sr(x0, x1, ty_Double) new_sr5(x0, app(ty_Ratio, x1)) new_sr9(x0, x1, ty_Double) new_sr10(x0, ty_Float) new_sr10(x0, ty_Integer) new_sr4(x0, ty_Int) new_sr5(x0, ty_Double) new_sr10(x0, app(ty_Ratio, x1)) new_sr8(x0, x1, ty_Int) new_sr11(x0, ty_Int) ---------------------------------------- (64) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (65) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, new_fromInt, vuz223, ba) at position [2] we obtained the following new rules [LPAR04]: (new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba),new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba)) ---------------------------------------- (66) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (67) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS1(Succ(vuz224)), new_primDivNatS1(Succ(vuz224)), ba) at position [2] we obtained the following new rules [LPAR04]: (new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba),new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba)) ---------------------------------------- (68) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (69) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) at position [2] we obtained the following new rules [LPAR04]: (new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb),new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb)) ---------------------------------------- (70) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (71) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, new_fromInt, vuz102, bb) at position [2] we obtained the following new rules [LPAR04]: (new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb),new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb)) ---------------------------------------- (72) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (73) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, new_fromInt, z2, z3) at position [2] we obtained the following new rules [LPAR04]: (new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3),new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3)) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) The TRS R consists of the following rules: new_fromInt -> Pos(Succ(Zero)) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr13(vuz72, vuz20) -> error([]) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr17(vuz73, vuz20) -> error([]) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr16(vuz71, vuz20) -> error([]) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (75) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (76) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_fromInt new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (77) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_fromInt ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) TransformationProof (EQUIVALENT) By rewriting [LPAR04] the rule new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS1(Succ(vuz224)), ba) at position [3] we obtained the following new rules [LPAR04]: (new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba),new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba)) ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_pr2F0(vuz103, Succ(vuz1050), Pos(Succ(vuz11500)), vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) we obtained the following new rules [LPAR04]: (new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3),new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3)) ---------------------------------------- (82) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (83) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (84) Complex Obligation (AND) ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_pr2F30(Succ(vuz2120), vuz204, Succ(Succ(vuz21100)), vuz205, h) -> new_pr2F0G1(vuz204, vuz205, Succ(vuz21100), vuz21100, h) we obtained the following new rules [LPAR04]: (new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3),new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3)) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_pr2F(vuz222, vuz224, vuz232, vuz223, ba) -> new_pr2F32(vuz224, vuz232, vuz222, new_sr1(vuz222, vuz223, ba), ba) we obtained the following new rules [LPAR04]: (new_pr2F(z0, z2, Pos(Succ(Zero)), z1, z3) -> new_pr2F32(z2, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z3), z3),new_pr2F(z0, z2, Pos(Succ(Zero)), z1, z3) -> new_pr2F32(z2, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z3), z3)) (new_pr2F(z0, Zero, Pos(Succ(Zero)), z1, z2) -> new_pr2F32(Zero, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z2), z2),new_pr2F(z0, Zero, Pos(Succ(Zero)), z1, z2) -> new_pr2F32(Zero, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z2), z2)) ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F32(Succ(vuz2020), Pos(Succ(Succ(vuz203000))), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) new_pr2F3(Succ(vuz2020), Zero, vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F32(vuz202, Pos(Zero), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz202), vuz204, Succ(vuz202), vuz205, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3) new_pr2F(z0, z2, Pos(Succ(Zero)), z1, z3) -> new_pr2F32(z2, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z3), z3) new_pr2F(z0, Zero, Pos(Succ(Zero)), z1, z2) -> new_pr2F32(Zero, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z2), z2) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (91) Complex Obligation (AND) ---------------------------------------- (92) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F(z0, z2, Pos(Succ(Zero)), z1, z3) -> new_pr2F32(z2, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z3), z3) new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (93) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_pr2F32(Succ(vuz2020), Pos(Succ(Zero)), vuz204, vuz205, h) -> new_pr2F30(Succ(vuz2020), vuz204, Succ(vuz2020), vuz205, h) new_pr2F0G1(vuz222, vuz223, vuz224, Zero, ba) -> new_pr2F0G10(new_sr0(vuz222, vuz223, ba), vuz222, new_primDivNatS01(vuz224), new_primDivNatS01(vuz224), ba) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 1 POL(Pos(x_1)) = 1 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL([]) = 1 POL(app(x_1, x_2)) = 1 + x_1 + x_2 POL(error(x_1)) = 0 POL(new_pr2F(x_1, x_2, x_3, x_4, x_5)) = 1 + x_1 + x_2 + x_5 POL(new_pr2F0(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_5 POL(new_pr2F0G1(x_1, x_2, x_3, x_4, x_5)) = 1 + x_1 + x_3 + x_5 POL(new_pr2F0G10(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_3 + x_5 POL(new_pr2F0G11(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_3 + x_5 POL(new_pr2F30(x_1, x_2, x_3, x_4, x_5)) = x_2 + x_3 + x_5 POL(new_pr2F32(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_5 POL(new_pr2F33(x_1, x_2, x_3, x_4, x_5)) = 1 + x_1 + x_2 + x_3 + x_5 POL(new_primDivNatS01(x_1)) = x_1 POL(new_primDivNatS1(x_1)) = x_1 POL(new_primDivNatS2) = 0 POL(new_primDivNatS3) = 0 POL(new_primDivNatS4(x_1)) = x_1 POL(new_primDivNatS5(x_1)) = x_1 POL(new_primMulNat0(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = 0 POL(new_sr0(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_sr1(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(new_sr12(x_1)) = x_1 POL(new_sr13(x_1, x_2)) = x_1 POL(new_sr14(x_1, x_2, x_3)) = x_1 POL(new_sr15(x_1, x_2)) = x_1 POL(new_sr16(x_1, x_2)) = x_1 POL(new_sr17(x_1, x_2)) = x_1 POL(new_sr18(x_1, x_2)) = x_1 POL(new_sr19(x_1)) = x_1 POL(new_sr2(x_1, x_2)) = x_1 POL(new_sr20(x_1)) = x_1 POL(new_sr21(x_1)) = x_1 POL(new_sr3(x_1, x_2)) = 1 + x_1 POL(ty_Double) = 1 POL(ty_Float) = 1 POL(ty_Int) = 1 POL(ty_Integer) = 1 POL(ty_Ratio) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS1(Zero) -> Zero new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr13(vuz72, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr16(vuz71, vuz20) -> error([]) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr12(vuz12) -> new_sr13(vuz12, vuz12) ---------------------------------------- (94) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Zero), ba) -> new_pr2F(vuz222, vuz224, Pos(Succ(Zero)), vuz223, ba) new_pr2F(z0, z2, Pos(Succ(Zero)), z1, z3) -> new_pr2F32(z2, Pos(Succ(Zero)), z0, new_sr1(z0, z1, z3), z3) new_pr2F30(Succ(Zero), y_0, Succ(Zero), z2, z3) -> new_pr2F(y_0, Zero, Pos(Succ(Zero)), z2, z3) new_pr2F30(Succ(Succ(x2)), z1, Succ(Succ(x2)), z2, z3) -> new_pr2F0G1(z1, z2, Succ(x2), x2, z3) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) new_pr2F0(z1, Succ(x1), Pos(Succ(Zero)), z0, z3) -> new_pr2F33(x1, Zero, z1, z0, z3) new_pr2F33(Succ(vuz1050), Zero, vuz103, vuz102, bb) -> new_pr2F30(Succ(vuz1050), new_sr3(vuz103, bb), Succ(vuz1050), vuz102, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Zero), bb) -> new_pr2F0(vuz103, vuz105, Pos(Succ(Zero)), vuz102, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (95) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 8 less nodes. ---------------------------------------- (96) Complex Obligation (AND) ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (98) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G1(vuz222, vuz223, vuz224, Succ(Succ(vuz22500)), ba) -> new_pr2F0G1(vuz222, vuz223, vuz224, vuz22500, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5 ---------------------------------------- (99) YES ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (102) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (103) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) the following chains were created: *We consider the chain new_pr2F0G10(x5, x6, x7, Succ(Succ(x8)), x9) -> new_pr2F0G11(x5, x6, x7, x8, x9), new_pr2F0G11(x10, x11, x12, Succ(Succ(x13)), x14) -> new_pr2F0G11(x10, x11, x12, x13, x14) which results in the following constraint: (1) (new_pr2F0G11(x5, x6, x7, x8, x9)=new_pr2F0G11(x10, x11, x12, Succ(Succ(x13)), x14) ==> new_pr2F0G10(x5, x6, x7, Succ(Succ(x8)), x9)_>=_new_pr2F0G11(x5, x6, x7, x8, x9)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G10(x5, x6, x7, Succ(Succ(Succ(Succ(x13)))), x9)_>=_new_pr2F0G11(x5, x6, x7, Succ(Succ(x13)), x9)) *We consider the chain new_pr2F0G10(x15, x16, x17, Succ(Succ(x18)), x19) -> new_pr2F0G11(x15, x16, x17, x18, x19), new_pr2F0G11(x20, x21, x22, Zero, x23) -> new_pr2F0G10(x20, new_sr2(x21, x23), new_primDivNatS1(x22), new_primDivNatS1(x22), x23) which results in the following constraint: (1) (new_pr2F0G11(x15, x16, x17, x18, x19)=new_pr2F0G11(x20, x21, x22, Zero, x23) ==> new_pr2F0G10(x15, x16, x17, Succ(Succ(x18)), x19)_>=_new_pr2F0G11(x15, x16, x17, x18, x19)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G10(x15, x16, x17, Succ(Succ(Zero)), x19)_>=_new_pr2F0G11(x15, x16, x17, Zero, x19)) For Pair new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) the following chains were created: *We consider the chain new_pr2F0G11(x34, x35, x36, Succ(Succ(x37)), x38) -> new_pr2F0G11(x34, x35, x36, x37, x38), new_pr2F0G11(x39, x40, x41, Succ(Succ(x42)), x43) -> new_pr2F0G11(x39, x40, x41, x42, x43) which results in the following constraint: (1) (new_pr2F0G11(x34, x35, x36, x37, x38)=new_pr2F0G11(x39, x40, x41, Succ(Succ(x42)), x43) ==> new_pr2F0G11(x34, x35, x36, Succ(Succ(x37)), x38)_>=_new_pr2F0G11(x34, x35, x36, x37, x38)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G11(x34, x35, x36, Succ(Succ(Succ(Succ(x42)))), x38)_>=_new_pr2F0G11(x34, x35, x36, Succ(Succ(x42)), x38)) *We consider the chain new_pr2F0G11(x44, x45, x46, Succ(Succ(x47)), x48) -> new_pr2F0G11(x44, x45, x46, x47, x48), new_pr2F0G11(x49, x50, x51, Zero, x52) -> new_pr2F0G10(x49, new_sr2(x50, x52), new_primDivNatS1(x51), new_primDivNatS1(x51), x52) which results in the following constraint: (1) (new_pr2F0G11(x44, x45, x46, x47, x48)=new_pr2F0G11(x49, x50, x51, Zero, x52) ==> new_pr2F0G11(x44, x45, x46, Succ(Succ(x47)), x48)_>=_new_pr2F0G11(x44, x45, x46, x47, x48)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G11(x44, x45, x46, Succ(Succ(Zero)), x48)_>=_new_pr2F0G11(x44, x45, x46, Zero, x48)) For Pair new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) the following chains were created: *We consider the chain new_pr2F0G11(x58, x59, x60, Zero, x61) -> new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(x60), new_primDivNatS1(x60), x61), new_pr2F0G10(x62, x63, x64, Succ(Succ(x65)), x66) -> new_pr2F0G11(x62, x63, x64, x65, x66) which results in the following constraint: (1) (new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(x60), new_primDivNatS1(x60), x61)=new_pr2F0G10(x62, x63, x64, Succ(Succ(x65)), x66) ==> new_pr2F0G11(x58, x59, x60, Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(x60), new_primDivNatS1(x60), x61)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x60)=Succ(Succ(x65)) ==> new_pr2F0G11(x58, x59, x60, Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(x60), new_primDivNatS1(x60), x61)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x60)=Succ(Succ(x65)) which results in the following new constraint: (3) (new_primDivNatS01(x108)=Succ(Succ(x65)) ==> new_pr2F0G11(x58, x59, Succ(x108), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(x108)), new_primDivNatS1(Succ(x108)), x61)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x108)=Succ(Succ(x65)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x109))=Succ(Succ(x65)) ==> new_pr2F0G11(x58, x59, Succ(Succ(Succ(x109))), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Succ(x109)))), new_primDivNatS1(Succ(Succ(Succ(x109)))), x61)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x65)) ==> new_pr2F0G11(x58, x59, Succ(Succ(Zero)), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x61)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G11(x58, x59, Succ(Succ(Succ(x109))), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Succ(x109)))), new_primDivNatS1(Succ(Succ(Succ(x109)))), x61)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G11(x58, x59, Succ(Succ(Zero)), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x61)) *We consider the chain new_pr2F0G11(x75, x76, x77, Zero, x78) -> new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(x77), new_primDivNatS1(x77), x78), new_pr2F0G10(x79, x80, x81, Zero, x82) -> new_pr2F0G10(x79, new_sr2(x80, x82), new_primDivNatS1(x81), new_primDivNatS1(x81), x82) which results in the following constraint: (1) (new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(x77), new_primDivNatS1(x77), x78)=new_pr2F0G10(x79, x80, x81, Zero, x82) ==> new_pr2F0G11(x75, x76, x77, Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(x77), new_primDivNatS1(x77), x78)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x77)=Zero ==> new_pr2F0G11(x75, x76, x77, Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(x77), new_primDivNatS1(x77), x78)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x77)=Zero which results in the following new constraints: (3) (new_primDivNatS01(x110)=Zero ==> new_pr2F0G11(x75, x76, Succ(x110), Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Succ(x110)), new_primDivNatS1(Succ(x110)), x78)) (4) (Zero=Zero ==> new_pr2F0G11(x75, x76, Zero, Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x78)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x110)=Zero which results in the following new constraint: (5) (Zero=Zero ==> new_pr2F0G11(x75, x76, Succ(Zero), Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x78)) We simplified constraint (4) using rules (I), (II) which results in the following new constraint: (6) (new_pr2F0G11(x75, x76, Zero, Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x78)) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G11(x75, x76, Succ(Zero), Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x78)) For Pair new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) the following chains were created: *We consider the chain new_pr2F0G10(x83, x84, x85, Zero, x86) -> new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(x85), new_primDivNatS1(x85), x86), new_pr2F0G10(x87, x88, x89, Succ(Succ(x90)), x91) -> new_pr2F0G11(x87, x88, x89, x90, x91) which results in the following constraint: (1) (new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(x85), new_primDivNatS1(x85), x86)=new_pr2F0G10(x87, x88, x89, Succ(Succ(x90)), x91) ==> new_pr2F0G10(x83, x84, x85, Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(x85), new_primDivNatS1(x85), x86)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x85)=Succ(Succ(x90)) ==> new_pr2F0G10(x83, x84, x85, Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(x85), new_primDivNatS1(x85), x86)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x85)=Succ(Succ(x90)) which results in the following new constraint: (3) (new_primDivNatS01(x112)=Succ(Succ(x90)) ==> new_pr2F0G10(x83, x84, Succ(x112), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(x112)), new_primDivNatS1(Succ(x112)), x86)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x112)=Succ(Succ(x90)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x113))=Succ(Succ(x90)) ==> new_pr2F0G10(x83, x84, Succ(Succ(Succ(x113))), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Succ(x113)))), new_primDivNatS1(Succ(Succ(Succ(x113)))), x86)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x90)) ==> new_pr2F0G10(x83, x84, Succ(Succ(Zero)), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x86)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G10(x83, x84, Succ(Succ(Succ(x113))), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Succ(x113)))), new_primDivNatS1(Succ(Succ(Succ(x113)))), x86)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G10(x83, x84, Succ(Succ(Zero)), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x86)) *We consider the chain new_pr2F0G10(x100, x101, x102, Zero, x103) -> new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(x102), new_primDivNatS1(x102), x103), new_pr2F0G10(x104, x105, x106, Zero, x107) -> new_pr2F0G10(x104, new_sr2(x105, x107), new_primDivNatS1(x106), new_primDivNatS1(x106), x107) which results in the following constraint: (1) (new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(x102), new_primDivNatS1(x102), x103)=new_pr2F0G10(x104, x105, x106, Zero, x107) ==> new_pr2F0G10(x100, x101, x102, Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(x102), new_primDivNatS1(x102), x103)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x102)=Zero ==> new_pr2F0G10(x100, x101, x102, Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(x102), new_primDivNatS1(x102), x103)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x102)=Zero which results in the following new constraints: (3) (new_primDivNatS01(x114)=Zero ==> new_pr2F0G10(x100, x101, Succ(x114), Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Succ(x114)), new_primDivNatS1(Succ(x114)), x103)) (4) (Zero=Zero ==> new_pr2F0G10(x100, x101, Zero, Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x103)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x114)=Zero which results in the following new constraint: (5) (Zero=Zero ==> new_pr2F0G10(x100, x101, Succ(Zero), Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x103)) We simplified constraint (4) using rules (I), (II) which results in the following new constraint: (6) (new_pr2F0G10(x100, x101, Zero, Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x103)) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G10(x100, x101, Succ(Zero), Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x103)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) *(new_pr2F0G10(x5, x6, x7, Succ(Succ(Succ(Succ(x13)))), x9)_>=_new_pr2F0G11(x5, x6, x7, Succ(Succ(x13)), x9)) *(new_pr2F0G10(x15, x16, x17, Succ(Succ(Zero)), x19)_>=_new_pr2F0G11(x15, x16, x17, Zero, x19)) *new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) *(new_pr2F0G11(x34, x35, x36, Succ(Succ(Succ(Succ(x42)))), x38)_>=_new_pr2F0G11(x34, x35, x36, Succ(Succ(x42)), x38)) *(new_pr2F0G11(x44, x45, x46, Succ(Succ(Zero)), x48)_>=_new_pr2F0G11(x44, x45, x46, Zero, x48)) *new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) *(new_pr2F0G11(x58, x59, Succ(Succ(Succ(x109))), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Succ(x109)))), new_primDivNatS1(Succ(Succ(Succ(x109)))), x61)) *(new_pr2F0G11(x58, x59, Succ(Succ(Zero)), Zero, x61)_>=_new_pr2F0G10(x58, new_sr2(x59, x61), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x61)) *(new_pr2F0G11(x75, x76, Succ(Zero), Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x78)) *(new_pr2F0G11(x75, x76, Zero, Zero, x78)_>=_new_pr2F0G10(x75, new_sr2(x76, x78), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x78)) *new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) *(new_pr2F0G10(x83, x84, Succ(Succ(Succ(x113))), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Succ(x113)))), new_primDivNatS1(Succ(Succ(Succ(x113)))), x86)) *(new_pr2F0G10(x83, x84, Succ(Succ(Zero)), Zero, x86)_>=_new_pr2F0G10(x83, new_sr2(x84, x86), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x86)) *(new_pr2F0G10(x100, x101, Succ(Zero), Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x103)) *(new_pr2F0G10(x100, x101, Zero, Zero, x103)_>=_new_pr2F0G10(x100, new_sr2(x101, x103), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x103)) 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. ---------------------------------------- (104) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (105) QDPPairToRuleProof (EQUIVALENT) The dependency pair new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) was transformed to the following new rules: anew_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Zero) -> cons_new_pr2F0G11(Zero) the following new pairs maintain the fan-in: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) the following new pairs maintain the fan-out: H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) ---------------------------------------- (106) Complex Obligation (AND) ---------------------------------------- (107) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) anew_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Zero) -> cons_new_pr2F0G11(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_new_pr2F0G11(Succ(Succ(x0))) anew_new_pr2F0G11(Succ(Succ(x0))) new_new_pr2F0G11(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (108) MNOCProof (EQUIVALENT) We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. ---------------------------------------- (109) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) anew_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Zero) -> cons_new_pr2F0G11(Zero) Q is empty. We have to consider all (P,Q,R)-chains. ---------------------------------------- (110) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) the following chains were created: *We consider the chain new_pr2F0G10(x5, x6, x7, Succ(Succ(x8)), x9) -> new_pr2F0G11(x5, x6, x7, x8, x9), new_pr2F0G11(x10, x11, x12, Zero, x13) -> new_pr2F0G10(x10, new_sr2(x11, x13), new_primDivNatS1(x12), new_primDivNatS1(x12), x13) which results in the following constraint: (1) (new_pr2F0G11(x5, x6, x7, x8, x9)=new_pr2F0G11(x10, x11, x12, Zero, x13) ==> new_pr2F0G10(x5, x6, x7, Succ(Succ(x8)), x9)_>=_new_pr2F0G11(x5, x6, x7, x8, x9)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_pr2F0G10(x5, x6, x7, Succ(Succ(Zero)), x9)_>=_new_pr2F0G11(x5, x6, x7, Zero, x9)) For Pair new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) the following chains were created: *We consider the chain new_pr2F0G11(x29, x30, x31, Zero, x32) -> new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(x31), new_primDivNatS1(x31), x32), new_pr2F0G10(x33, x34, x35, Succ(Succ(x36)), x37) -> new_pr2F0G11(x33, x34, x35, x36, x37) which results in the following constraint: (1) (new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(x31), new_primDivNatS1(x31), x32)=new_pr2F0G10(x33, x34, x35, Succ(Succ(x36)), x37) ==> new_pr2F0G11(x29, x30, x31, Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(x31), new_primDivNatS1(x31), x32)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x31)=Succ(Succ(x36)) ==> new_pr2F0G11(x29, x30, x31, Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(x31), new_primDivNatS1(x31), x32)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x31)=Succ(Succ(x36)) which results in the following new constraint: (3) (new_primDivNatS01(x150)=Succ(Succ(x36)) ==> new_pr2F0G11(x29, x30, Succ(x150), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(x150)), new_primDivNatS1(Succ(x150)), x32)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x150)=Succ(Succ(x36)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x151))=Succ(Succ(x36)) ==> new_pr2F0G11(x29, x30, Succ(Succ(Succ(x151))), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Succ(x151)))), new_primDivNatS1(Succ(Succ(Succ(x151)))), x32)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x36)) ==> new_pr2F0G11(x29, x30, Succ(Succ(Zero)), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x32)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G11(x29, x30, Succ(Succ(Succ(x151))), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Succ(x151)))), new_primDivNatS1(Succ(Succ(Succ(x151)))), x32)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G11(x29, x30, Succ(Succ(Zero)), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x32)) *We consider the chain new_pr2F0G11(x42, x43, x44, Zero, x45) -> new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(x44), new_primDivNatS1(x44), x45), new_pr2F0G10(x46, x47, x48, Zero, x49) -> new_pr2F0G10(x46, new_sr2(x47, x49), new_primDivNatS1(x48), new_primDivNatS1(x48), x49) which results in the following constraint: (1) (new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(x44), new_primDivNatS1(x44), x45)=new_pr2F0G10(x46, x47, x48, Zero, x49) ==> new_pr2F0G11(x42, x43, x44, Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(x44), new_primDivNatS1(x44), x45)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x44)=Zero ==> new_pr2F0G11(x42, x43, x44, Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(x44), new_primDivNatS1(x44), x45)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x44)=Zero which results in the following new constraints: (3) (new_primDivNatS01(x152)=Zero ==> new_pr2F0G11(x42, x43, Succ(x152), Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Succ(x152)), new_primDivNatS1(Succ(x152)), x45)) (4) (Zero=Zero ==> new_pr2F0G11(x42, x43, Zero, Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x45)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x152)=Zero which results in the following new constraint: (5) (Zero=Zero ==> new_pr2F0G11(x42, x43, Succ(Zero), Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x45)) We simplified constraint (4) using rules (I), (II) which results in the following new constraint: (6) (new_pr2F0G11(x42, x43, Zero, Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x45)) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G11(x42, x43, Succ(Zero), Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x45)) *We consider the chain new_pr2F0G11(x50, x51, x52, Zero, x53) -> new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(x52), new_primDivNatS1(x52), x53), new_pr2F0G10(x54, x55, x56, Succ(Succ(x57)), x58) -> H(x54, x55, x56, x58, anew_new_pr2F0G11(x57)) which results in the following constraint: (1) (new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(x52), new_primDivNatS1(x52), x53)=new_pr2F0G10(x54, x55, x56, Succ(Succ(x57)), x58) ==> new_pr2F0G11(x50, x51, x52, Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(x52), new_primDivNatS1(x52), x53)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x52)=Succ(Succ(x57)) ==> new_pr2F0G11(x50, x51, x52, Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(x52), new_primDivNatS1(x52), x53)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x52)=Succ(Succ(x57)) which results in the following new constraint: (3) (new_primDivNatS01(x154)=Succ(Succ(x57)) ==> new_pr2F0G11(x50, x51, Succ(x154), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(x154)), new_primDivNatS1(Succ(x154)), x53)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x154)=Succ(Succ(x57)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x155))=Succ(Succ(x57)) ==> new_pr2F0G11(x50, x51, Succ(Succ(Succ(x155))), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Succ(x155)))), new_primDivNatS1(Succ(Succ(Succ(x155)))), x53)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x57)) ==> new_pr2F0G11(x50, x51, Succ(Succ(Zero)), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x53)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G11(x50, x51, Succ(Succ(Succ(x155))), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Succ(x155)))), new_primDivNatS1(Succ(Succ(Succ(x155)))), x53)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G11(x50, x51, Succ(Succ(Zero)), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x53)) For Pair new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) the following chains were created: *We consider the chain new_pr2F0G10(x63, x64, x65, Zero, x66) -> new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(x65), new_primDivNatS1(x65), x66), new_pr2F0G10(x67, x68, x69, Succ(Succ(x70)), x71) -> new_pr2F0G11(x67, x68, x69, x70, x71) which results in the following constraint: (1) (new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(x65), new_primDivNatS1(x65), x66)=new_pr2F0G10(x67, x68, x69, Succ(Succ(x70)), x71) ==> new_pr2F0G10(x63, x64, x65, Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(x65), new_primDivNatS1(x65), x66)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x65)=Succ(Succ(x70)) ==> new_pr2F0G10(x63, x64, x65, Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(x65), new_primDivNatS1(x65), x66)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x65)=Succ(Succ(x70)) which results in the following new constraint: (3) (new_primDivNatS01(x156)=Succ(Succ(x70)) ==> new_pr2F0G10(x63, x64, Succ(x156), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(x156)), new_primDivNatS1(Succ(x156)), x66)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x156)=Succ(Succ(x70)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x157))=Succ(Succ(x70)) ==> new_pr2F0G10(x63, x64, Succ(Succ(Succ(x157))), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Succ(x157)))), new_primDivNatS1(Succ(Succ(Succ(x157)))), x66)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x70)) ==> new_pr2F0G10(x63, x64, Succ(Succ(Zero)), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x66)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G10(x63, x64, Succ(Succ(Succ(x157))), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Succ(x157)))), new_primDivNatS1(Succ(Succ(Succ(x157)))), x66)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G10(x63, x64, Succ(Succ(Zero)), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x66)) *We consider the chain new_pr2F0G10(x76, x77, x78, Zero, x79) -> new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(x78), new_primDivNatS1(x78), x79), new_pr2F0G10(x80, x81, x82, Zero, x83) -> new_pr2F0G10(x80, new_sr2(x81, x83), new_primDivNatS1(x82), new_primDivNatS1(x82), x83) which results in the following constraint: (1) (new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(x78), new_primDivNatS1(x78), x79)=new_pr2F0G10(x80, x81, x82, Zero, x83) ==> new_pr2F0G10(x76, x77, x78, Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(x78), new_primDivNatS1(x78), x79)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x78)=Zero ==> new_pr2F0G10(x76, x77, x78, Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(x78), new_primDivNatS1(x78), x79)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x78)=Zero which results in the following new constraints: (3) (new_primDivNatS01(x158)=Zero ==> new_pr2F0G10(x76, x77, Succ(x158), Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Succ(x158)), new_primDivNatS1(Succ(x158)), x79)) (4) (Zero=Zero ==> new_pr2F0G10(x76, x77, Zero, Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x79)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x158)=Zero which results in the following new constraint: (5) (Zero=Zero ==> new_pr2F0G10(x76, x77, Succ(Zero), Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x79)) We simplified constraint (4) using rules (I), (II) which results in the following new constraint: (6) (new_pr2F0G10(x76, x77, Zero, Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x79)) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G10(x76, x77, Succ(Zero), Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x79)) *We consider the chain new_pr2F0G10(x84, x85, x86, Zero, x87) -> new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(x86), new_primDivNatS1(x86), x87), new_pr2F0G10(x88, x89, x90, Succ(Succ(x91)), x92) -> H(x88, x89, x90, x92, anew_new_pr2F0G11(x91)) which results in the following constraint: (1) (new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(x86), new_primDivNatS1(x86), x87)=new_pr2F0G10(x88, x89, x90, Succ(Succ(x91)), x92) ==> new_pr2F0G10(x84, x85, x86, Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(x86), new_primDivNatS1(x86), x87)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_primDivNatS1(x86)=Succ(Succ(x91)) ==> new_pr2F0G10(x84, x85, x86, Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(x86), new_primDivNatS1(x86), x87)) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS1(x86)=Succ(Succ(x91)) which results in the following new constraint: (3) (new_primDivNatS01(x160)=Succ(Succ(x91)) ==> new_pr2F0G10(x84, x85, Succ(x160), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(x160)), new_primDivNatS1(Succ(x160)), x87)) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x160)=Succ(Succ(x91)) which results in the following new constraints: (4) (Succ(new_primDivNatS4(x161))=Succ(Succ(x91)) ==> new_pr2F0G10(x84, x85, Succ(Succ(Succ(x161))), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Succ(x161)))), new_primDivNatS1(Succ(Succ(Succ(x161)))), x87)) (5) (Succ(new_primDivNatS2)=Succ(Succ(x91)) ==> new_pr2F0G10(x84, x85, Succ(Succ(Zero)), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x87)) We simplified constraint (4) using rules (I), (II), (IV) which results in the following new constraint: (6) (new_pr2F0G10(x84, x85, Succ(Succ(Succ(x161))), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Succ(x161)))), new_primDivNatS1(Succ(Succ(Succ(x161)))), x87)) We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: (7) (new_pr2F0G10(x84, x85, Succ(Succ(Zero)), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x87)) For Pair new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) the following chains were created: *We consider the chain new_pr2F0G10(x117, x118, x119, Succ(Succ(x120)), x121) -> H(x117, x118, x119, x121, anew_new_pr2F0G11(x120)), H(x122, x123, x124, x125, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(x122, x123, x124, Zero, x125) which results in the following constraint: (1) (H(x117, x118, x119, x121, anew_new_pr2F0G11(x120))=H(x122, x123, x124, x125, cons_new_pr2F0G11(Zero)) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(x120)), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(x120))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (anew_new_pr2F0G11(x120)=cons_new_pr2F0G11(Zero) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(x120)), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(x120))) We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on anew_new_pr2F0G11(x120)=cons_new_pr2F0G11(Zero) which results in the following new constraint: (3) (new_new_pr2F0G11(x162)=cons_new_pr2F0G11(Zero) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(x162)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(x162))))) We simplified constraint (3) using rule (V) (with possible (I) afterwards) using induction on new_new_pr2F0G11(x162)=cons_new_pr2F0G11(Zero) which results in the following new constraints: (4) (new_new_pr2F0G11(x163)=cons_new_pr2F0G11(Zero) & (\/x164,x165,x166,x167:new_new_pr2F0G11(x163)=cons_new_pr2F0G11(Zero) ==> new_pr2F0G10(x164, x165, x166, Succ(Succ(Succ(Succ(x163)))), x167)_>=_H(x164, x165, x166, x167, anew_new_pr2F0G11(Succ(Succ(x163))))) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Succ(Succ(x163)))))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Succ(Succ(x163))))))) (5) (cons_new_pr2F0G11(Zero)=cons_new_pr2F0G11(Zero) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Zero)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Zero))))) We simplified constraint (4) using rule (VI) where we applied the induction hypothesis (\/x164,x165,x166,x167:new_new_pr2F0G11(x163)=cons_new_pr2F0G11(Zero) ==> new_pr2F0G10(x164, x165, x166, Succ(Succ(Succ(Succ(x163)))), x167)_>=_H(x164, x165, x166, x167, anew_new_pr2F0G11(Succ(Succ(x163))))) with sigma = [x164 / x117, x165 / x118, x166 / x119, x167 / x121] which results in the following new constraint: (6) (new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(x163)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(x163)))) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Succ(Succ(x163)))))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Succ(Succ(x163))))))) We simplified constraint (5) using rules (I), (II) which results in the following new constraint: (7) (new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Zero)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Zero))))) For Pair H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) the following chains were created: *We consider the chain H(x130, x131, x132, x133, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(x130, x131, x132, Zero, x133), new_pr2F0G11(x134, x135, x136, Zero, x137) -> new_pr2F0G10(x134, new_sr2(x135, x137), new_primDivNatS1(x136), new_primDivNatS1(x136), x137) which results in the following constraint: (1) (new_pr2F0G11(x130, x131, x132, Zero, x133)=new_pr2F0G11(x134, x135, x136, Zero, x137) ==> H(x130, x131, x132, x133, cons_new_pr2F0G11(Zero))_>=_new_pr2F0G11(x130, x131, x132, Zero, x133)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (H(x130, x131, x132, x133, cons_new_pr2F0G11(Zero))_>=_new_pr2F0G11(x130, x131, x132, Zero, x133)) To summarize, we get the following constraints P__>=_ for the following pairs. *new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) *(new_pr2F0G10(x5, x6, x7, Succ(Succ(Zero)), x9)_>=_new_pr2F0G11(x5, x6, x7, Zero, x9)) *new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) *(new_pr2F0G11(x29, x30, Succ(Succ(Succ(x151))), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Succ(x151)))), new_primDivNatS1(Succ(Succ(Succ(x151)))), x32)) *(new_pr2F0G11(x29, x30, Succ(Succ(Zero)), Zero, x32)_>=_new_pr2F0G10(x29, new_sr2(x30, x32), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x32)) *(new_pr2F0G11(x42, x43, Succ(Zero), Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x45)) *(new_pr2F0G11(x42, x43, Zero, Zero, x45)_>=_new_pr2F0G10(x42, new_sr2(x43, x45), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x45)) *(new_pr2F0G11(x50, x51, Succ(Succ(Succ(x155))), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Succ(x155)))), new_primDivNatS1(Succ(Succ(Succ(x155)))), x53)) *(new_pr2F0G11(x50, x51, Succ(Succ(Zero)), Zero, x53)_>=_new_pr2F0G10(x50, new_sr2(x51, x53), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x53)) *new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) *(new_pr2F0G10(x63, x64, Succ(Succ(Succ(x157))), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Succ(x157)))), new_primDivNatS1(Succ(Succ(Succ(x157)))), x66)) *(new_pr2F0G10(x63, x64, Succ(Succ(Zero)), Zero, x66)_>=_new_pr2F0G10(x63, new_sr2(x64, x66), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x66)) *(new_pr2F0G10(x76, x77, Succ(Zero), Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Succ(Zero)), new_primDivNatS1(Succ(Zero)), x79)) *(new_pr2F0G10(x76, x77, Zero, Zero, x79)_>=_new_pr2F0G10(x76, new_sr2(x77, x79), new_primDivNatS1(Zero), new_primDivNatS1(Zero), x79)) *(new_pr2F0G10(x84, x85, Succ(Succ(Succ(x161))), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Succ(x161)))), new_primDivNatS1(Succ(Succ(Succ(x161)))), x87)) *(new_pr2F0G10(x84, x85, Succ(Succ(Zero)), Zero, x87)_>=_new_pr2F0G10(x84, new_sr2(x85, x87), new_primDivNatS1(Succ(Succ(Zero))), new_primDivNatS1(Succ(Succ(Zero))), x87)) *new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) *(new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(x163)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(x163)))) ==> new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Succ(Succ(x163)))))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Succ(Succ(x163))))))) *(new_pr2F0G10(x117, x118, x119, Succ(Succ(Succ(Succ(Zero)))), x121)_>=_H(x117, x118, x119, x121, anew_new_pr2F0G11(Succ(Succ(Zero))))) *H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) *(H(x130, x131, x132, x133, cons_new_pr2F0G11(Zero))_>=_new_pr2F0G11(x130, x131, x132, Zero, x133)) 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. ---------------------------------------- (111) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Zero, bb) -> new_pr2F0G10(vuz102, new_sr2(vuz103, bb), new_primDivNatS1(vuz105), new_primDivNatS1(vuz105), bb) new_pr2F0G10(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> H(vuz102, vuz103, vuz105, bb, anew_new_pr2F0G11(vuz10600)) H(vuz102, vuz103, vuz105, bb, cons_new_pr2F0G11(Zero)) -> new_pr2F0G11(vuz102, vuz103, vuz105, Zero, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) anew_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Succ(Succ(vuz10600))) -> new_new_pr2F0G11(vuz10600) new_new_pr2F0G11(Zero) -> cons_new_pr2F0G11(Zero) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) new_new_pr2F0G11(Succ(Succ(x0))) anew_new_pr2F0G11(Succ(Succ(x0))) new_new_pr2F0G11(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (112) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (113) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G11(vuz102, vuz103, vuz105, Succ(Succ(vuz10600)), bb) -> new_pr2F0G11(vuz102, vuz103, vuz105, vuz10600, bb) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 > 4, 5 >= 5 ---------------------------------------- (114) YES ---------------------------------------- (115) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (116) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (117) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) R is empty. The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (118) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) ---------------------------------------- (119) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (120) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F3(Succ(vuz2020), Succ(vuz203000), vuz204, vuz205, h) -> new_pr2F3(vuz2020, vuz203000, vuz204, vuz205, h) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 ---------------------------------------- (121) YES ---------------------------------------- (122) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) The TRS R consists of the following rules: new_sr0(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr0(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr0(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr0(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_primDivNatS01(Succ(Succ(vuz130000))) -> Succ(new_primDivNatS4(vuz130000)) new_primDivNatS01(Zero) -> Zero new_primDivNatS01(Succ(Zero)) -> Succ(new_primDivNatS2) new_primDivNatS1(Succ(vuz550)) -> new_primDivNatS01(vuz550) new_primDivNatS2 -> new_primDivNatS3 new_primDivNatS3 -> Zero new_primDivNatS4(vuz1300) -> new_primDivNatS5(vuz1300) new_primDivNatS5(vuz1300) -> new_primDivNatS01(vuz1300) new_sr15(Pos(vuz700), Pos(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Neg(vuz200)) -> Pos(new_primMulNat0(vuz700, vuz200)) new_sr15(Pos(vuz700), Neg(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_sr15(Neg(vuz700), Pos(vuz200)) -> Neg(new_primMulNat0(vuz700, vuz200)) new_primMulNat0(Succ(vuz7000), Zero) -> Zero new_primMulNat0(Zero, Succ(vuz2000)) -> Zero new_primMulNat0(Zero, Zero) -> Zero new_primMulNat0(Succ(vuz7000), Succ(vuz2000)) -> new_primPlusNat0(new_primMulNat0(vuz7000, Succ(vuz2000)), Succ(vuz2000)) new_primPlusNat0(Succ(vuz4000), Succ(vuz5000)) -> Succ(Succ(new_primPlusNat0(vuz4000, vuz5000))) new_primPlusNat0(Zero, Succ(vuz5000)) -> Succ(vuz5000) new_primPlusNat0(Zero, Zero) -> Zero new_primPlusNat0(Succ(vuz4000), Zero) -> Succ(vuz4000) new_sr16(vuz71, vuz20) -> error([]) new_sr17(vuz73, vuz20) -> error([]) new_sr14(vuz69, vuz20, cd) -> error([]) new_sr13(vuz72, vuz20) -> error([]) new_sr2(vuz103, ty_Int) -> new_sr19(vuz103) new_sr2(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) new_sr2(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr2(vuz103, ty_Float) -> new_sr21(vuz103) new_sr2(vuz103, ty_Double) -> new_sr12(vuz103) new_primDivNatS1(Zero) -> Zero new_sr12(vuz12) -> new_sr13(vuz12, vuz12) new_sr21(vuz12) -> new_sr17(vuz12, vuz12) new_sr20(vuz12) -> new_sr16(vuz12, vuz12) new_sr18(vuz12, cb) -> new_sr14(vuz12, vuz12, cb) new_sr19(vuz12) -> new_sr15(vuz12, vuz12) new_sr1(vuz222, vuz223, ty_Double) -> new_sr13(vuz222, vuz223) new_sr1(vuz222, vuz223, app(ty_Ratio, bf)) -> new_sr14(vuz222, vuz223, bf) new_sr1(vuz222, vuz223, ty_Int) -> new_sr15(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Integer) -> new_sr16(vuz222, vuz223) new_sr1(vuz222, vuz223, ty_Float) -> new_sr17(vuz222, vuz223) new_sr3(vuz103, ty_Float) -> new_sr21(vuz103) new_sr3(vuz103, ty_Int) -> new_sr19(vuz103) new_sr3(vuz103, ty_Double) -> new_sr12(vuz103) new_sr3(vuz103, ty_Integer) -> new_sr20(vuz103) new_sr3(vuz103, app(ty_Ratio, ca)) -> new_sr18(vuz103, ca) The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (123) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (124) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) R is empty. The set Q consists of the following terms: new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (125) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_sr1(x0, x1, ty_Integer) new_sr17(x0, x1) new_primMulNat0(Succ(x0), Succ(x1)) new_sr13(x0, x1) new_sr14(x0, x1, x2) new_sr0(x0, x1, ty_Integer) new_sr2(x0, ty_Double) new_sr2(x0, ty_Float) new_sr12(x0) new_primMulNat0(Succ(x0), Zero) new_primDivNatS1(Zero) new_sr15(Pos(x0), Pos(x1)) new_sr3(x0, ty_Double) new_sr0(x0, x1, ty_Int) new_primMulNat0(Zero, Zero) new_sr20(x0) new_sr3(x0, ty_Int) new_sr0(x0, x1, ty_Double) new_primDivNatS4(x0) new_sr2(x0, ty_Integer) new_primMulNat0(Zero, Succ(x0)) new_primDivNatS2 new_primDivNatS1(Succ(x0)) new_sr1(x0, x1, ty_Float) new_primDivNatS01(Succ(Zero)) new_primPlusNat0(Succ(x0), Zero) new_sr2(x0, ty_Int) new_sr15(Neg(x0), Neg(x1)) new_primPlusNat0(Zero, Succ(x0)) new_sr1(x0, x1, app(ty_Ratio, x2)) new_sr3(x0, app(ty_Ratio, x1)) new_sr0(x0, x1, app(ty_Ratio, x2)) new_sr2(x0, app(ty_Ratio, x1)) new_sr1(x0, x1, ty_Double) new_primDivNatS01(Succ(Succ(x0))) new_sr19(x0) new_primPlusNat0(Succ(x0), Succ(x1)) new_primDivNatS5(x0) new_primDivNatS3 new_sr18(x0, x1) new_sr0(x0, x1, ty_Float) new_sr1(x0, x1, ty_Int) new_sr21(x0) new_primDivNatS01(Zero) new_sr16(x0, x1) new_primPlusNat0(Zero, Zero) new_sr3(x0, ty_Integer) new_sr15(Pos(x0), Neg(x1)) new_sr15(Neg(x0), Pos(x1)) new_sr3(x0, ty_Float) ---------------------------------------- (126) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (127) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F33(Succ(vuz1050), Succ(vuz11500), vuz103, vuz102, bb) -> new_pr2F33(vuz1050, vuz11500, vuz103, vuz102, bb) The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 ---------------------------------------- (128) YES ---------------------------------------- (129) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMulNat(Succ(vuz7000), Succ(vuz2000)) -> new_primMulNat(vuz7000, Succ(vuz2000)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (130) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primMulNat(Succ(vuz7000), Succ(vuz2000)) -> new_primMulNat(vuz7000, Succ(vuz2000)) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (131) YES ---------------------------------------- (132) Obligation: Q DP problem: The TRS P consists of the following rules: new_pr2F0G15(vuz12, vuz13, Succ(Succ(vuz1400)), h) -> new_pr2F0G15(vuz12, vuz13, vuz1400, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (133) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_pr2F0G15(vuz12, vuz13, Succ(Succ(vuz1400)), h) -> new_pr2F0G15(vuz12, vuz13, vuz1400, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4 ---------------------------------------- (134) YES ---------------------------------------- (135) Obligation: Q DP problem: The TRS P consists of the following rules: new_primPlusNat(Succ(vuz4000), Succ(vuz5000)) -> new_primPlusNat(vuz4000, vuz5000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (136) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primPlusNat(Succ(vuz4000), Succ(vuz5000)) -> new_primPlusNat(vuz4000, vuz5000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (137) YES ---------------------------------------- (138) Narrow (COMPLETE) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="(^)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="(^) vuz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="(^) vuz3 vuz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="pr4 vuz3 vuz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="pr3 (vuz4 == fromInt (Pos Zero)) vuz3 vuz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="pr3 (primEqInt vuz4 (fromInt (Pos Zero))) vuz3 vuz4",fontsize=16,color="burlywood",shape="box"];4738[label="vuz4/Pos vuz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 4738[label="",style="solid", color="burlywood", weight=9]; 4738 -> 8[label="",style="solid", color="burlywood", weight=3]; 4739[label="vuz4/Neg vuz40",fontsize=10,color="white",style="solid",shape="box"];7 -> 4739[label="",style="solid", color="burlywood", weight=9]; 4739 -> 9[label="",style="solid", color="burlywood", weight=3]; 8[label="pr3 (primEqInt (Pos vuz40) (fromInt (Pos Zero))) vuz3 (Pos vuz40)",fontsize=16,color="burlywood",shape="box"];4740[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];8 -> 4740[label="",style="solid", color="burlywood", weight=9]; 4740 -> 10[label="",style="solid", color="burlywood", weight=3]; 4741[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 4741[label="",style="solid", color="burlywood", weight=9]; 4741 -> 11[label="",style="solid", color="burlywood", weight=3]; 9[label="pr3 (primEqInt (Neg vuz40) (fromInt (Pos Zero))) vuz3 (Neg vuz40)",fontsize=16,color="burlywood",shape="box"];4742[label="vuz40/Succ vuz400",fontsize=10,color="white",style="solid",shape="box"];9 -> 4742[label="",style="solid", color="burlywood", weight=9]; 4742 -> 12[label="",style="solid", color="burlywood", weight=3]; 4743[label="vuz40/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 4743[label="",style="solid", color="burlywood", weight=9]; 4743 -> 13[label="",style="solid", color="burlywood", weight=3]; 10[label="pr3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3]; 11[label="pr3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 12[label="pr3 (primEqInt (Neg (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 13[label="pr3 (primEqInt (Neg Zero) (fromInt (Pos Zero))) vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 14[label="pr3 (primEqInt (Pos (Succ vuz400)) (Pos Zero)) vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 15[label="pr3 (primEqInt (Pos Zero) (Pos Zero)) vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 16[label="pr3 (primEqInt (Neg (Succ vuz400)) (Pos Zero)) vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 17[label="pr3 (primEqInt (Neg Zero) (Pos Zero)) vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 18[label="pr3 False vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 19[label="pr3 True vuz3 (Pos Zero)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 20[label="pr3 False vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 21[label="pr3 True vuz3 (Neg Zero)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 22[label="pr2 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 23[label="fromInt (Pos (Succ Zero))",fontsize=16,color="black",shape="triangle"];23 -> 27[label="",style="solid", color="black", weight=3]; 24[label="pr2 vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 25 -> 23[label="",style="dashed", color="red", weight=0]; 25[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];26[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (Pos (Succ vuz400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];26 -> 29[label="",style="solid", color="black", weight=3]; 27[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];28[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (Neg (Succ vuz400) > fromInt (Pos Zero))",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3]; 29[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (compare (Pos (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3]; 30[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (compare (Neg (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3]; 31[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpInt (Pos (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3]; 32[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (primCmpInt (Neg (Succ vuz400)) (fromInt (Pos Zero)) == GT)",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3]; 33[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpInt (Pos (Succ vuz400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3]; 34[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (primCmpInt (Neg (Succ vuz400)) (Pos Zero) == GT)",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3]; 35[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (primCmpNat (Succ vuz400) Zero == GT)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 36[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) (LT == GT)",fontsize=16,color="black",shape="box"];36 -> 38[label="",style="solid", color="black", weight=3]; 37[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) (GT == GT)",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3]; 38[label="pr2Pr1 vuz3 (Neg (Succ vuz400)) False",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3]; 39[label="pr2Pr1 vuz3 (Pos (Succ vuz400)) True",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3]; 40[label="pr0 vuz3 (Neg (Succ vuz400))",fontsize=16,color="black",shape="box"];40 -> 42[label="",style="solid", color="black", weight=3]; 41 -> 43[label="",style="dashed", color="red", weight=0]; 41[label="pr2F vuz3 (Pos (Succ vuz400) - fromInt (Pos (Succ Zero))) vuz3",fontsize=16,color="magenta"];41 -> 44[label="",style="dashed", color="magenta", weight=3]; 42[label="error []",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3]; 44 -> 23[label="",style="dashed", color="red", weight=0]; 44[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];43[label="pr2F vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="triangle"];43 -> 46[label="",style="solid", color="black", weight=3]; 45[label="error []",fontsize=16,color="red",shape="box"];46[label="pr2F4 vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];46 -> 47[label="",style="solid", color="black", weight=3]; 47[label="pr2F3 (Pos (Succ vuz400) - vuz5 == fromInt (Pos Zero)) vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];47 -> 48[label="",style="solid", color="black", weight=3]; 48[label="pr2F3 (primEqInt (Pos (Succ vuz400) - vuz5) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400) - vuz5) vuz3",fontsize=16,color="black",shape="box"];48 -> 49[label="",style="solid", color="black", weight=3]; 49[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) vuz5) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) vuz5) vuz3",fontsize=16,color="burlywood",shape="box"];4744[label="vuz5/Pos vuz50",fontsize=10,color="white",style="solid",shape="box"];49 -> 4744[label="",style="solid", color="burlywood", weight=9]; 4744 -> 50[label="",style="solid", color="burlywood", weight=3]; 4745[label="vuz5/Neg vuz50",fontsize=10,color="white",style="solid",shape="box"];49 -> 4745[label="",style="solid", color="burlywood", weight=9]; 4745 -> 51[label="",style="solid", color="burlywood", weight=3]; 50[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) (Pos vuz50)) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) (Pos vuz50)) vuz3",fontsize=16,color="black",shape="box"];50 -> 52[label="",style="solid", color="black", weight=3]; 51[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz400)) (Neg vuz50)) (fromInt (Pos Zero))) vuz3 (primMinusInt (Pos (Succ vuz400)) (Neg vuz50)) vuz3",fontsize=16,color="black",shape="box"];51 -> 53[label="",style="solid", color="black", weight=3]; 52[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) vuz50) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) vuz50) vuz3",fontsize=16,color="burlywood",shape="box"];4746[label="vuz50/Succ vuz500",fontsize=10,color="white",style="solid",shape="box"];52 -> 4746[label="",style="solid", color="burlywood", weight=9]; 4746 -> 54[label="",style="solid", color="burlywood", weight=3]; 4747[label="vuz50/Zero",fontsize=10,color="white",style="solid",shape="box"];52 -> 4747[label="",style="solid", color="burlywood", weight=9]; 4747 -> 55[label="",style="solid", color="burlywood", weight=3]; 53[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) vuz50)) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) vuz50)) vuz3",fontsize=16,color="burlywood",shape="box"];4748[label="vuz50/Succ vuz500",fontsize=10,color="white",style="solid",shape="box"];53 -> 4748[label="",style="solid", color="burlywood", weight=9]; 4748 -> 56[label="",style="solid", color="burlywood", weight=3]; 4749[label="vuz50/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 4749[label="",style="solid", color="burlywood", weight=9]; 4749 -> 57[label="",style="solid", color="burlywood", weight=3]; 54[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) (Succ vuz500)) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) (Succ vuz500)) vuz3",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3]; 55[label="pr2F3 (primEqInt (primMinusNat (Succ vuz400) Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz400) Zero) vuz3",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 56[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) (Succ vuz500))) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) (Succ vuz500))) vuz3",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 57[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz400) Zero)) (fromInt (Pos Zero))) vuz3 (Pos (primPlusNat (Succ vuz400) Zero)) vuz3",fontsize=16,color="black",shape="box"];57 -> 61[label="",style="solid", color="black", weight=3]; 58[label="pr2F3 (primEqInt (primMinusNat vuz400 vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat vuz400 vuz500) vuz3",fontsize=16,color="burlywood",shape="triangle"];4750[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];58 -> 4750[label="",style="solid", color="burlywood", weight=9]; 4750 -> 62[label="",style="solid", color="burlywood", weight=3]; 4751[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 4751[label="",style="solid", color="burlywood", weight=9]; 4751 -> 63[label="",style="solid", color="burlywood", weight=3]; 59[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="triangle"];59 -> 64[label="",style="solid", color="black", weight=3]; 60 -> 59[label="",style="dashed", color="red", weight=0]; 60[label="pr2F3 (primEqInt (Pos (Succ (Succ (primPlusNat vuz400 vuz500)))) (fromInt (Pos Zero))) vuz3 (Pos (Succ (Succ (primPlusNat vuz400 vuz500)))) vuz3",fontsize=16,color="magenta"];60 -> 65[label="",style="dashed", color="magenta", weight=3]; 61 -> 59[label="",style="dashed", color="red", weight=0]; 61[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="magenta"];62[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) vuz500) vuz3",fontsize=16,color="burlywood",shape="box"];4752[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];62 -> 4752[label="",style="solid", color="burlywood", weight=9]; 4752 -> 66[label="",style="solid", color="burlywood", weight=3]; 4753[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 4753[label="",style="solid", color="burlywood", weight=9]; 4753 -> 67[label="",style="solid", color="burlywood", weight=3]; 63[label="pr2F3 (primEqInt (primMinusNat Zero vuz500) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero vuz500) vuz3",fontsize=16,color="burlywood",shape="box"];4754[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];63 -> 4754[label="",style="solid", color="burlywood", weight=9]; 4754 -> 68[label="",style="solid", color="burlywood", weight=3]; 4755[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 4755[label="",style="solid", color="burlywood", weight=9]; 4755 -> 69[label="",style="solid", color="burlywood", weight=3]; 64[label="pr2F3 (primEqInt (Pos (Succ vuz400)) (Pos Zero)) vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];64 -> 70[label="",style="solid", color="black", weight=3]; 65[label="Succ (primPlusNat vuz400 vuz500)",fontsize=16,color="green",shape="box"];65 -> 71[label="",style="dashed", color="green", weight=3]; 66[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3]; 67[label="pr2F3 (primEqInt (primMinusNat (Succ vuz4000) Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat (Succ vuz4000) Zero) vuz3",fontsize=16,color="black",shape="box"];67 -> 73[label="",style="solid", color="black", weight=3]; 68[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];68 -> 74[label="",style="solid", color="black", weight=3]; 69[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) vuz3 (primMinusNat Zero Zero) vuz3",fontsize=16,color="black",shape="box"];69 -> 75[label="",style="solid", color="black", weight=3]; 70[label="pr2F3 False vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];70 -> 76[label="",style="solid", color="black", weight=3]; 71[label="primPlusNat vuz400 vuz500",fontsize=16,color="burlywood",shape="triangle"];4756[label="vuz400/Succ vuz4000",fontsize=10,color="white",style="solid",shape="box"];71 -> 4756[label="",style="solid", color="burlywood", weight=9]; 4756 -> 77[label="",style="solid", color="burlywood", weight=3]; 4757[label="vuz400/Zero",fontsize=10,color="white",style="solid",shape="box"];71 -> 4757[label="",style="solid", color="burlywood", weight=9]; 4757 -> 78[label="",style="solid", color="burlywood", weight=3]; 72 -> 58[label="",style="dashed", color="red", weight=0]; 72[label="pr2F3 (primEqInt (primMinusNat vuz4000 vuz5000) (fromInt (Pos Zero))) vuz3 (primMinusNat vuz4000 vuz5000) vuz3",fontsize=16,color="magenta"];72 -> 79[label="",style="dashed", color="magenta", weight=3]; 72 -> 80[label="",style="dashed", color="magenta", weight=3]; 73 -> 59[label="",style="dashed", color="red", weight=0]; 73[label="pr2F3 (primEqInt (Pos (Succ vuz4000)) (fromInt (Pos Zero))) vuz3 (Pos (Succ vuz4000)) vuz3",fontsize=16,color="magenta"];73 -> 81[label="",style="dashed", color="magenta", weight=3]; 74[label="pr2F3 (primEqInt (Neg (Succ vuz5000)) (fromInt (Pos Zero))) vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];74 -> 82[label="",style="solid", color="black", weight=3]; 75[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];75 -> 83[label="",style="solid", color="black", weight=3]; 76[label="pr2F0 vuz3 (Pos (Succ vuz400)) vuz3",fontsize=16,color="black",shape="box"];76 -> 84[label="",style="solid", color="black", weight=3]; 77[label="primPlusNat (Succ vuz4000) vuz500",fontsize=16,color="burlywood",shape="box"];4758[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];77 -> 4758[label="",style="solid", color="burlywood", weight=9]; 4758 -> 85[label="",style="solid", color="burlywood", weight=3]; 4759[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];77 -> 4759[label="",style="solid", color="burlywood", weight=9]; 4759 -> 86[label="",style="solid", color="burlywood", weight=3]; 78[label="primPlusNat Zero vuz500",fontsize=16,color="burlywood",shape="box"];4760[label="vuz500/Succ vuz5000",fontsize=10,color="white",style="solid",shape="box"];78 -> 4760[label="",style="solid", color="burlywood", weight=9]; 4760 -> 87[label="",style="solid", color="burlywood", weight=3]; 4761[label="vuz500/Zero",fontsize=10,color="white",style="solid",shape="box"];78 -> 4761[label="",style="solid", color="burlywood", weight=9]; 4761 -> 88[label="",style="solid", color="burlywood", weight=3]; 79[label="vuz4000",fontsize=16,color="green",shape="box"];80[label="vuz5000",fontsize=16,color="green",shape="box"];81[label="vuz4000",fontsize=16,color="green",shape="box"];82[label="pr2F3 (primEqInt (Neg (Succ vuz5000)) (Pos Zero)) vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];82 -> 89[label="",style="solid", color="black", weight=3]; 83[label="pr2F3 (primEqInt (Pos Zero) (Pos Zero)) vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];83 -> 90[label="",style="solid", color="black", weight=3]; 84[label="pr2F0G vuz3 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];84 -> 91[label="",style="solid", color="black", weight=3]; 85[label="primPlusNat (Succ vuz4000) (Succ vuz5000)",fontsize=16,color="black",shape="box"];85 -> 92[label="",style="solid", color="black", weight=3]; 86[label="primPlusNat (Succ vuz4000) Zero",fontsize=16,color="black",shape="box"];86 -> 93[label="",style="solid", color="black", weight=3]; 87[label="primPlusNat Zero (Succ vuz5000)",fontsize=16,color="black",shape="box"];87 -> 94[label="",style="solid", color="black", weight=3]; 88[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];88 -> 95[label="",style="solid", color="black", weight=3]; 89[label="pr2F3 False vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];89 -> 96[label="",style="solid", color="black", weight=3]; 90[label="pr2F3 True vuz3 (Pos Zero) vuz3",fontsize=16,color="black",shape="box"];90 -> 97[label="",style="solid", color="black", weight=3]; 91[label="pr2F0G2 vuz3 vuz3 (Pos (Succ vuz400))",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3]; 92[label="Succ (Succ (primPlusNat vuz4000 vuz5000))",fontsize=16,color="green",shape="box"];92 -> 99[label="",style="dashed", color="green", weight=3]; 93[label="Succ vuz4000",fontsize=16,color="green",shape="box"];94[label="Succ vuz5000",fontsize=16,color="green",shape="box"];95[label="Zero",fontsize=16,color="green",shape="box"];96[label="pr2F0 vuz3 (Neg (Succ vuz5000)) vuz3",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3]; 97[label="vuz3",fontsize=16,color="green",shape="box"];98[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (even (Pos (Succ vuz400)))",fontsize=16,color="black",shape="box"];98 -> 101[label="",style="solid", color="black", weight=3]; 99 -> 71[label="",style="dashed", color="red", weight=0]; 99[label="primPlusNat vuz4000 vuz5000",fontsize=16,color="magenta"];99 -> 102[label="",style="dashed", color="magenta", weight=3]; 99 -> 103[label="",style="dashed", color="magenta", weight=3]; 100[label="pr2F0G vuz3 vuz3 (Neg (Succ vuz5000))",fontsize=16,color="black",shape="box"];100 -> 104[label="",style="solid", color="black", weight=3]; 101[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (primEvenInt (Pos (Succ vuz400)))",fontsize=16,color="black",shape="box"];101 -> 105[label="",style="solid", color="black", weight=3]; 102[label="vuz4000",fontsize=16,color="green",shape="box"];103[label="vuz5000",fontsize=16,color="green",shape="box"];104[label="pr2F0G2 vuz3 vuz3 (Neg (Succ vuz5000))",fontsize=16,color="black",shape="box"];104 -> 106[label="",style="solid", color="black", weight=3]; 105 -> 256[label="",style="dashed", color="red", weight=0]; 105[label="pr2F0G1 vuz3 vuz3 (Pos (Succ vuz400)) (primEvenNat (Succ vuz400))",fontsize=16,color="magenta"];105 -> 257[label="",style="dashed", color="magenta", weight=3]; 105 -> 258[label="",style="dashed", color="magenta", weight=3]; 105 -> 259[label="",style="dashed", color="magenta", weight=3]; 106[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (even (Neg (Succ vuz5000)))",fontsize=16,color="black",shape="box"];106 -> 109[label="",style="solid", color="black", weight=3]; 257[label="vuz3",fontsize=16,color="green",shape="box"];258[label="vuz400",fontsize=16,color="green",shape="box"];259[label="Succ vuz400",fontsize=16,color="green",shape="box"];256[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat vuz22)",fontsize=16,color="burlywood",shape="triangle"];4762[label="vuz22/Succ vuz220",fontsize=10,color="white",style="solid",shape="box"];256 -> 4762[label="",style="solid", color="burlywood", weight=9]; 4762 -> 275[label="",style="solid", color="burlywood", weight=3]; 4763[label="vuz22/Zero",fontsize=10,color="white",style="solid",shape="box"];256 -> 4763[label="",style="solid", color="burlywood", weight=9]; 4763 -> 276[label="",style="solid", color="burlywood", weight=3]; 109[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (primEvenInt (Neg (Succ vuz5000)))",fontsize=16,color="black",shape="box"];109 -> 112[label="",style="solid", color="black", weight=3]; 275[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ vuz220))",fontsize=16,color="burlywood",shape="box"];4764[label="vuz220/Succ vuz2200",fontsize=10,color="white",style="solid",shape="box"];275 -> 4764[label="",style="solid", color="burlywood", weight=9]; 4764 -> 279[label="",style="solid", color="burlywood", weight=3]; 4765[label="vuz220/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 4765[label="",style="solid", color="burlywood", weight=9]; 4765 -> 280[label="",style="solid", color="burlywood", weight=3]; 276[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];276 -> 281[label="",style="solid", color="black", weight=3]; 112 -> 199[label="",style="dashed", color="red", weight=0]; 112[label="pr2F0G1 vuz3 vuz3 (Neg (Succ vuz5000)) (primEvenNat (Succ vuz5000))",fontsize=16,color="magenta"];112 -> 200[label="",style="dashed", color="magenta", weight=3]; 112 -> 201[label="",style="dashed", color="magenta", weight=3]; 112 -> 202[label="",style="dashed", color="magenta", weight=3]; 279[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ (Succ vuz2200)))",fontsize=16,color="black",shape="box"];279 -> 284[label="",style="solid", color="black", weight=3]; 280[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];280 -> 285[label="",style="solid", color="black", weight=3]; 281[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) True",fontsize=16,color="black",shape="box"];281 -> 286[label="",style="solid", color="black", weight=3]; 200[label="Succ vuz5000",fontsize=16,color="green",shape="box"];201[label="vuz3",fontsize=16,color="green",shape="box"];202[label="vuz5000",fontsize=16,color="green",shape="box"];199[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat vuz14)",fontsize=16,color="burlywood",shape="triangle"];4766[label="vuz14/Succ vuz140",fontsize=10,color="white",style="solid",shape="box"];199 -> 4766[label="",style="solid", color="burlywood", weight=9]; 4766 -> 212[label="",style="solid", color="burlywood", weight=3]; 4767[label="vuz14/Zero",fontsize=10,color="white",style="solid",shape="box"];199 -> 4767[label="",style="solid", color="burlywood", weight=9]; 4767 -> 213[label="",style="solid", color="burlywood", weight=3]; 284 -> 256[label="",style="dashed", color="red", weight=0]; 284[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) (primEvenNat vuz2200)",fontsize=16,color="magenta"];284 -> 289[label="",style="dashed", color="magenta", weight=3]; 285[label="pr2F0G1 vuz20 vuz20 (Pos (Succ vuz21)) False",fontsize=16,color="black",shape="box"];285 -> 290[label="",style="solid", color="black", weight=3]; 286[label="pr2F0G vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];286 -> 291[label="",style="solid", color="black", weight=3]; 212[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ vuz140))",fontsize=16,color="burlywood",shape="box"];4768[label="vuz140/Succ vuz1400",fontsize=10,color="white",style="solid",shape="box"];212 -> 4768[label="",style="solid", color="burlywood", weight=9]; 4768 -> 216[label="",style="solid", color="burlywood", weight=3]; 4769[label="vuz140/Zero",fontsize=10,color="white",style="solid",shape="box"];212 -> 4769[label="",style="solid", color="burlywood", weight=9]; 4769 -> 217[label="",style="solid", color="burlywood", weight=3]; 213[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];213 -> 218[label="",style="solid", color="black", weight=3]; 289[label="vuz2200",fontsize=16,color="green",shape="box"];290[label="pr2F0G0 vuz20 vuz20 (Pos (Succ vuz21)) otherwise",fontsize=16,color="black",shape="box"];290 -> 294[label="",style="solid", color="black", weight=3]; 291[label="pr2F0G2 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];291 -> 295[label="",style="solid", color="black", weight=3]; 216[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ (Succ vuz1400)))",fontsize=16,color="black",shape="box"];216 -> 227[label="",style="solid", color="black", weight=3]; 217[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];217 -> 228[label="",style="solid", color="black", weight=3]; 218[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) True",fontsize=16,color="black",shape="box"];218 -> 229[label="",style="solid", color="black", weight=3]; 294[label="pr2F0G0 vuz20 vuz20 (Pos (Succ vuz21)) True",fontsize=16,color="black",shape="box"];294 -> 298[label="",style="solid", color="black", weight=3]; 295[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];295 -> 299[label="",style="solid", color="black", weight=3]; 227 -> 199[label="",style="dashed", color="red", weight=0]; 227[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) (primEvenNat vuz1400)",fontsize=16,color="magenta"];227 -> 237[label="",style="dashed", color="magenta", weight=3]; 228[label="pr2F0G1 vuz12 vuz12 (Neg (Succ vuz13)) False",fontsize=16,color="black",shape="box"];228 -> 238[label="",style="solid", color="black", weight=3]; 229[label="pr2F0G vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];229 -> 239[label="",style="solid", color="black", weight=3]; 298 -> 302[label="",style="dashed", color="red", weight=0]; 298[label="pr2F vuz20 (Pos (Succ vuz21) - fromInt (Pos (Succ Zero))) (vuz20 * vuz20)",fontsize=16,color="magenta"];298 -> 303[label="",style="dashed", color="magenta", weight=3]; 299[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos (Succ vuz21) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];299 -> 304[label="",style="solid", color="black", weight=3]; 237[label="vuz1400",fontsize=16,color="green",shape="box"];238[label="pr2F0G0 vuz12 vuz12 (Neg (Succ vuz13)) otherwise",fontsize=16,color="black",shape="box"];238 -> 253[label="",style="solid", color="black", weight=3]; 239[label="pr2F0G2 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];239 -> 254[label="",style="solid", color="black", weight=3]; 303 -> 23[label="",style="dashed", color="red", weight=0]; 303[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];302[label="pr2F vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="triangle"];302 -> 305[label="",style="solid", color="black", weight=3]; 304[label="pr2F0G1 vuz20 (vuz20 * vuz20) (primQuotInt (Pos (Succ vuz21)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos (Succ vuz21)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];304 -> 309[label="",style="solid", color="black", weight=3]; 253[label="pr2F0G0 vuz12 vuz12 (Neg (Succ vuz13)) True",fontsize=16,color="black",shape="box"];253 -> 277[label="",style="solid", color="black", weight=3]; 254[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];254 -> 278[label="",style="solid", color="black", weight=3]; 305[label="pr2F4 vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="box"];305 -> 310[label="",style="solid", color="black", weight=3]; 309[label="pr2F0G1 vuz20 (vuz20 * vuz20) (primQuotInt (Pos (Succ vuz21)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos (Succ vuz21)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];309 -> 315[label="",style="solid", color="black", weight=3]; 277 -> 282[label="",style="dashed", color="red", weight=0]; 277[label="pr2F vuz12 (Neg (Succ vuz13) - fromInt (Pos (Succ Zero))) (vuz12 * vuz12)",fontsize=16,color="magenta"];277 -> 283[label="",style="dashed", color="magenta", weight=3]; 278[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg (Succ vuz13) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];278 -> 287[label="",style="solid", color="black", weight=3]; 310[label="pr2F3 (Pos (Succ vuz21) - vuz24 == fromInt (Pos Zero)) vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="black",shape="box"];310 -> 316[label="",style="solid", color="black", weight=3]; 315 -> 1605[label="",style="dashed", color="red", weight=0]; 315[label="pr2F0G1 vuz20 (vuz20 * vuz20) (Pos (primDivNatS (Succ vuz21) (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS (Succ vuz21) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];315 -> 1606[label="",style="dashed", color="magenta", weight=3]; 315 -> 1607[label="",style="dashed", color="magenta", weight=3]; 315 -> 1608[label="",style="dashed", color="magenta", weight=3]; 315 -> 1609[label="",style="dashed", color="magenta", weight=3]; 283 -> 23[label="",style="dashed", color="red", weight=0]; 283[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];282[label="pr2F vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="triangle"];282 -> 288[label="",style="solid", color="black", weight=3]; 287[label="pr2F0G1 vuz12 (vuz12 * vuz12) (primQuotInt (Neg (Succ vuz13)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg (Succ vuz13)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];287 -> 292[label="",style="solid", color="black", weight=3]; 316 -> 3789[label="",style="dashed", color="red", weight=0]; 316[label="pr2F3 (primEqInt (Pos (Succ vuz21) - vuz24) (fromInt (Pos Zero))) vuz20 (Pos (Succ vuz21) - vuz24) (vuz20 * vuz20)",fontsize=16,color="magenta"];316 -> 3790[label="",style="dashed", color="magenta", weight=3]; 316 -> 3791[label="",style="dashed", color="magenta", weight=3]; 316 -> 3792[label="",style="dashed", color="magenta", weight=3]; 316 -> 3793[label="",style="dashed", color="magenta", weight=3]; 1606 -> 1226[label="",style="dashed", color="red", weight=0]; 1606[label="primDivNatS (Succ vuz21) (Succ (Succ Zero))",fontsize=16,color="magenta"];1606 -> 1624[label="",style="dashed", color="magenta", weight=3]; 1607[label="vuz20",fontsize=16,color="green",shape="box"];1608[label="vuz20",fontsize=16,color="green",shape="box"];1609 -> 1226[label="",style="dashed", color="red", weight=0]; 1609[label="primDivNatS (Succ vuz21) (Succ (Succ Zero))",fontsize=16,color="magenta"];1609 -> 1625[label="",style="dashed", color="magenta", weight=3]; 1605[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenInt (Pos vuz106))",fontsize=16,color="black",shape="triangle"];1605 -> 1626[label="",style="solid", color="black", weight=3]; 288[label="pr2F4 vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="box"];288 -> 293[label="",style="solid", color="black", weight=3]; 292[label="pr2F0G1 vuz12 (vuz12 * vuz12) (primQuotInt (Neg (Succ vuz13)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg (Succ vuz13)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];292 -> 296[label="",style="solid", color="black", weight=3]; 3790[label="vuz24",fontsize=16,color="green",shape="box"];3791[label="vuz21",fontsize=16,color="green",shape="box"];3792[label="vuz20",fontsize=16,color="green",shape="box"];3793[label="vuz20",fontsize=16,color="green",shape="box"];3789[label="pr2F3 (primEqInt (Pos (Succ vuz202) - vuz203) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz202) - vuz203) (vuz204 * vuz205)",fontsize=16,color="black",shape="triangle"];3789 -> 3814[label="",style="solid", color="black", weight=3]; 1624[label="Succ vuz21",fontsize=16,color="green",shape="box"];1226[label="primDivNatS vuz55 (Succ (Succ Zero))",fontsize=16,color="burlywood",shape="triangle"];4770[label="vuz55/Succ vuz550",fontsize=10,color="white",style="solid",shape="box"];1226 -> 4770[label="",style="solid", color="burlywood", weight=9]; 4770 -> 1241[label="",style="solid", color="burlywood", weight=3]; 4771[label="vuz55/Zero",fontsize=10,color="white",style="solid",shape="box"];1226 -> 4771[label="",style="solid", color="burlywood", weight=9]; 4771 -> 1242[label="",style="solid", color="burlywood", weight=3]; 1625[label="Succ vuz21",fontsize=16,color="green",shape="box"];1626[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat vuz106)",fontsize=16,color="burlywood",shape="triangle"];4772[label="vuz106/Succ vuz1060",fontsize=10,color="white",style="solid",shape="box"];1626 -> 4772[label="",style="solid", color="burlywood", weight=9]; 4772 -> 1659[label="",style="solid", color="burlywood", weight=3]; 4773[label="vuz106/Zero",fontsize=10,color="white",style="solid",shape="box"];1626 -> 4773[label="",style="solid", color="burlywood", weight=9]; 4773 -> 1660[label="",style="solid", color="burlywood", weight=3]; 293[label="pr2F3 (Neg (Succ vuz13) - vuz23 == fromInt (Pos Zero)) vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="black",shape="box"];293 -> 297[label="",style="solid", color="black", weight=3]; 296 -> 1755[label="",style="dashed", color="red", weight=0]; 296[label="pr2F0G1 vuz12 (vuz12 * vuz12) (Neg (primDivNatS (Succ vuz13) (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS (Succ vuz13) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];296 -> 1756[label="",style="dashed", color="magenta", weight=3]; 296 -> 1757[label="",style="dashed", color="magenta", weight=3]; 296 -> 1758[label="",style="dashed", color="magenta", weight=3]; 296 -> 1759[label="",style="dashed", color="magenta", weight=3]; 3814[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) vuz203) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) vuz203) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4774[label="vuz203/Pos vuz2030",fontsize=10,color="white",style="solid",shape="box"];3814 -> 4774[label="",style="solid", color="burlywood", weight=9]; 4774 -> 3843[label="",style="solid", color="burlywood", weight=3]; 4775[label="vuz203/Neg vuz2030",fontsize=10,color="white",style="solid",shape="box"];3814 -> 4775[label="",style="solid", color="burlywood", weight=9]; 4775 -> 3844[label="",style="solid", color="burlywood", weight=3]; 1241[label="primDivNatS (Succ vuz550) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];1241 -> 1287[label="",style="solid", color="black", weight=3]; 1242[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];1242 -> 1288[label="",style="solid", color="black", weight=3]; 1659[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ vuz1060))",fontsize=16,color="burlywood",shape="box"];4776[label="vuz1060/Succ vuz10600",fontsize=10,color="white",style="solid",shape="box"];1659 -> 4776[label="",style="solid", color="burlywood", weight=9]; 4776 -> 1713[label="",style="solid", color="burlywood", weight=3]; 4777[label="vuz1060/Zero",fontsize=10,color="white",style="solid",shape="box"];1659 -> 4777[label="",style="solid", color="burlywood", weight=9]; 4777 -> 1714[label="",style="solid", color="burlywood", weight=3]; 1660[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];1660 -> 1715[label="",style="solid", color="black", weight=3]; 297 -> 4256[label="",style="dashed", color="red", weight=0]; 297[label="pr2F3 (primEqInt (Neg (Succ vuz13) - vuz23) (fromInt (Pos Zero))) vuz12 (Neg (Succ vuz13) - vuz23) (vuz12 * vuz12)",fontsize=16,color="magenta"];297 -> 4257[label="",style="dashed", color="magenta", weight=3]; 297 -> 4258[label="",style="dashed", color="magenta", weight=3]; 297 -> 4259[label="",style="dashed", color="magenta", weight=3]; 297 -> 4260[label="",style="dashed", color="magenta", weight=3]; 1756[label="vuz12",fontsize=16,color="green",shape="box"];1757 -> 1226[label="",style="dashed", color="red", weight=0]; 1757[label="primDivNatS (Succ vuz13) (Succ (Succ Zero))",fontsize=16,color="magenta"];1757 -> 1770[label="",style="dashed", color="magenta", weight=3]; 1758[label="vuz12",fontsize=16,color="green",shape="box"];1759 -> 1226[label="",style="dashed", color="red", weight=0]; 1759[label="primDivNatS (Succ vuz13) (Succ (Succ Zero))",fontsize=16,color="magenta"];1759 -> 1771[label="",style="dashed", color="magenta", weight=3]; 1755[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenInt (Neg vuz114))",fontsize=16,color="black",shape="triangle"];1755 -> 1772[label="",style="solid", color="black", weight=3]; 3843[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) (Pos vuz2030)) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) (Pos vuz2030)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];3843 -> 3919[label="",style="solid", color="black", weight=3]; 3844[label="pr2F3 (primEqInt (primMinusInt (Pos (Succ vuz202)) (Neg vuz2030)) (fromInt (Pos Zero))) vuz204 (primMinusInt (Pos (Succ vuz202)) (Neg vuz2030)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];3844 -> 3920[label="",style="solid", color="black", weight=3]; 1287 -> 562[label="",style="dashed", color="red", weight=0]; 1287[label="primDivNatS0 vuz550 (Succ Zero) (primGEqNatS vuz550 (Succ Zero))",fontsize=16,color="magenta"];1287 -> 1313[label="",style="dashed", color="magenta", weight=3]; 1288[label="Zero",fontsize=16,color="green",shape="box"];1713[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ (Succ vuz10600)))",fontsize=16,color="black",shape="box"];1713 -> 1773[label="",style="solid", color="black", weight=3]; 1714[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];1714 -> 1774[label="",style="solid", color="black", weight=3]; 1715[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) True",fontsize=16,color="black",shape="box"];1715 -> 1775[label="",style="solid", color="black", weight=3]; 4257[label="vuz13",fontsize=16,color="green",shape="box"];4258[label="vuz12",fontsize=16,color="green",shape="box"];4259[label="vuz23",fontsize=16,color="green",shape="box"];4260[label="vuz12",fontsize=16,color="green",shape="box"];4256[label="pr2F3 (primEqInt (Neg (Succ vuz214) - vuz215) (fromInt (Pos Zero))) vuz216 (Neg (Succ vuz214) - vuz215) (vuz216 * vuz217)",fontsize=16,color="black",shape="triangle"];4256 -> 4281[label="",style="solid", color="black", weight=3]; 1770[label="Succ vuz13",fontsize=16,color="green",shape="box"];1771[label="Succ vuz13",fontsize=16,color="green",shape="box"];1772[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat vuz114)",fontsize=16,color="burlywood",shape="triangle"];4778[label="vuz114/Succ vuz1140",fontsize=10,color="white",style="solid",shape="box"];1772 -> 4778[label="",style="solid", color="burlywood", weight=9]; 4778 -> 1793[label="",style="solid", color="burlywood", weight=3]; 4779[label="vuz114/Zero",fontsize=10,color="white",style="solid",shape="box"];1772 -> 4779[label="",style="solid", color="burlywood", weight=9]; 4779 -> 1794[label="",style="solid", color="burlywood", weight=3]; 3919[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) vuz2030) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) vuz2030) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4780[label="vuz2030/Succ vuz20300",fontsize=10,color="white",style="solid",shape="box"];3919 -> 4780[label="",style="solid", color="burlywood", weight=9]; 4780 -> 4005[label="",style="solid", color="burlywood", weight=3]; 4781[label="vuz2030/Zero",fontsize=10,color="white",style="solid",shape="box"];3919 -> 4781[label="",style="solid", color="burlywood", weight=9]; 4781 -> 4006[label="",style="solid", color="burlywood", weight=3]; 3920 -> 4007[label="",style="dashed", color="red", weight=0]; 3920[label="pr2F3 (primEqInt (Pos (primPlusNat (Succ vuz202) vuz2030)) (fromInt (Pos Zero))) vuz204 (Pos (primPlusNat (Succ vuz202) vuz2030)) (vuz204 * vuz205)",fontsize=16,color="magenta"];3920 -> 4008[label="",style="dashed", color="magenta", weight=3]; 3920 -> 4009[label="",style="dashed", color="magenta", weight=3]; 1313[label="vuz550",fontsize=16,color="green",shape="box"];562[label="primDivNatS0 vuz1300 (Succ Zero) (primGEqNatS vuz1300 (Succ Zero))",fontsize=16,color="burlywood",shape="triangle"];4782[label="vuz1300/Succ vuz13000",fontsize=10,color="white",style="solid",shape="box"];562 -> 4782[label="",style="solid", color="burlywood", weight=9]; 4782 -> 570[label="",style="solid", color="burlywood", weight=3]; 4783[label="vuz1300/Zero",fontsize=10,color="white",style="solid",shape="box"];562 -> 4783[label="",style="solid", color="burlywood", weight=9]; 4783 -> 571[label="",style="solid", color="burlywood", weight=3]; 1773 -> 1626[label="",style="dashed", color="red", weight=0]; 1773[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) (primEvenNat vuz10600)",fontsize=16,color="magenta"];1773 -> 1795[label="",style="dashed", color="magenta", weight=3]; 1774[label="pr2F0G1 vuz102 (vuz103 * vuz103) (Pos vuz105) False",fontsize=16,color="black",shape="box"];1774 -> 1796[label="",style="solid", color="black", weight=3]; 1775[label="pr2F0G vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1775 -> 1797[label="",style="solid", color="black", weight=3]; 4281[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) vuz215) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) vuz215) (vuz216 * vuz217)",fontsize=16,color="burlywood",shape="box"];4784[label="vuz215/Pos vuz2150",fontsize=10,color="white",style="solid",shape="box"];4281 -> 4784[label="",style="solid", color="burlywood", weight=9]; 4784 -> 4288[label="",style="solid", color="burlywood", weight=3]; 4785[label="vuz215/Neg vuz2150",fontsize=10,color="white",style="solid",shape="box"];4281 -> 4785[label="",style="solid", color="burlywood", weight=9]; 4785 -> 4289[label="",style="solid", color="burlywood", weight=3]; 1793[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ vuz1140))",fontsize=16,color="burlywood",shape="box"];4786[label="vuz1140/Succ vuz11400",fontsize=10,color="white",style="solid",shape="box"];1793 -> 4786[label="",style="solid", color="burlywood", weight=9]; 4786 -> 1807[label="",style="solid", color="burlywood", weight=3]; 4787[label="vuz1140/Zero",fontsize=10,color="white",style="solid",shape="box"];1793 -> 4787[label="",style="solid", color="burlywood", weight=9]; 4787 -> 1808[label="",style="solid", color="burlywood", weight=3]; 1794[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];1794 -> 1809[label="",style="solid", color="black", weight=3]; 4005[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) (Succ vuz20300)) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) (Succ vuz20300)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4005 -> 4086[label="",style="solid", color="black", weight=3]; 4006[label="pr2F3 (primEqInt (primMinusNat (Succ vuz202) Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz202) Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4006 -> 4087[label="",style="solid", color="black", weight=3]; 4008 -> 71[label="",style="dashed", color="red", weight=0]; 4008[label="primPlusNat (Succ vuz202) vuz2030",fontsize=16,color="magenta"];4008 -> 4088[label="",style="dashed", color="magenta", weight=3]; 4008 -> 4089[label="",style="dashed", color="magenta", weight=3]; 4009 -> 71[label="",style="dashed", color="red", weight=0]; 4009[label="primPlusNat (Succ vuz202) vuz2030",fontsize=16,color="magenta"];4009 -> 4090[label="",style="dashed", color="magenta", weight=3]; 4009 -> 4091[label="",style="dashed", color="magenta", weight=3]; 4007[label="pr2F3 (primEqInt (Pos vuz212) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="triangle"];4788[label="vuz212/Succ vuz2120",fontsize=10,color="white",style="solid",shape="box"];4007 -> 4788[label="",style="solid", color="burlywood", weight=9]; 4788 -> 4092[label="",style="solid", color="burlywood", weight=3]; 4789[label="vuz212/Zero",fontsize=10,color="white",style="solid",shape="box"];4007 -> 4789[label="",style="solid", color="burlywood", weight=9]; 4789 -> 4093[label="",style="solid", color="burlywood", weight=3]; 570[label="primDivNatS0 (Succ vuz13000) (Succ Zero) (primGEqNatS (Succ vuz13000) (Succ Zero))",fontsize=16,color="black",shape="box"];570 -> 584[label="",style="solid", color="black", weight=3]; 571[label="primDivNatS0 Zero (Succ Zero) (primGEqNatS Zero (Succ Zero))",fontsize=16,color="black",shape="box"];571 -> 585[label="",style="solid", color="black", weight=3]; 1795[label="vuz10600",fontsize=16,color="green",shape="box"];1796[label="pr2F0G0 vuz102 (vuz103 * vuz103) (Pos vuz105) otherwise",fontsize=16,color="black",shape="box"];1796 -> 1810[label="",style="solid", color="black", weight=3]; 1797[label="pr2F0G2 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1797 -> 1811[label="",style="solid", color="black", weight=3]; 4288[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) (Pos vuz2150)) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) (Pos vuz2150)) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4288 -> 4296[label="",style="solid", color="black", weight=3]; 4289[label="pr2F3 (primEqInt (primMinusInt (Neg (Succ vuz214)) (Neg vuz2150)) (fromInt (Pos Zero))) vuz216 (primMinusInt (Neg (Succ vuz214)) (Neg vuz2150)) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4289 -> 4297[label="",style="solid", color="black", weight=3]; 1807[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ (Succ vuz11400)))",fontsize=16,color="black",shape="box"];1807 -> 1817[label="",style="solid", color="black", weight=3]; 1808[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];1808 -> 1818[label="",style="solid", color="black", weight=3]; 1809[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) True",fontsize=16,color="black",shape="box"];1809 -> 1819[label="",style="solid", color="black", weight=3]; 4086[label="pr2F3 (primEqInt (primMinusNat vuz202 vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat vuz202 vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="triangle"];4790[label="vuz202/Succ vuz2020",fontsize=10,color="white",style="solid",shape="box"];4086 -> 4790[label="",style="solid", color="burlywood", weight=9]; 4790 -> 4165[label="",style="solid", color="burlywood", weight=3]; 4791[label="vuz202/Zero",fontsize=10,color="white",style="solid",shape="box"];4086 -> 4791[label="",style="solid", color="burlywood", weight=9]; 4791 -> 4166[label="",style="solid", color="burlywood", weight=3]; 4087 -> 4007[label="",style="dashed", color="red", weight=0]; 4087[label="pr2F3 (primEqInt (Pos (Succ vuz202)) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz202)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4087 -> 4167[label="",style="dashed", color="magenta", weight=3]; 4087 -> 4168[label="",style="dashed", color="magenta", weight=3]; 4088[label="Succ vuz202",fontsize=16,color="green",shape="box"];4089[label="vuz2030",fontsize=16,color="green",shape="box"];4090[label="Succ vuz202",fontsize=16,color="green",shape="box"];4091[label="vuz2030",fontsize=16,color="green",shape="box"];4092[label="pr2F3 (primEqInt (Pos (Succ vuz2120)) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4092 -> 4169[label="",style="solid", color="black", weight=3]; 4093[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4093 -> 4170[label="",style="solid", color="black", weight=3]; 584[label="primDivNatS0 (Succ vuz13000) (Succ Zero) (primGEqNatS vuz13000 Zero)",fontsize=16,color="burlywood",shape="box"];4792[label="vuz13000/Succ vuz130000",fontsize=10,color="white",style="solid",shape="box"];584 -> 4792[label="",style="solid", color="burlywood", weight=9]; 4792 -> 606[label="",style="solid", color="burlywood", weight=3]; 4793[label="vuz13000/Zero",fontsize=10,color="white",style="solid",shape="box"];584 -> 4793[label="",style="solid", color="burlywood", weight=9]; 4793 -> 607[label="",style="solid", color="burlywood", weight=3]; 585[label="primDivNatS0 Zero (Succ Zero) False",fontsize=16,color="black",shape="box"];585 -> 608[label="",style="solid", color="black", weight=3]; 1810[label="pr2F0G0 vuz102 (vuz103 * vuz103) (Pos vuz105) True",fontsize=16,color="black",shape="box"];1810 -> 1820[label="",style="solid", color="black", weight=3]; 1811[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1811 -> 1821[label="",style="solid", color="black", weight=3]; 4296 -> 4311[label="",style="dashed", color="red", weight=0]; 4296[label="pr2F3 (primEqInt (Neg (primPlusNat (Succ vuz214) vuz2150)) (fromInt (Pos Zero))) vuz216 (Neg (primPlusNat (Succ vuz214) vuz2150)) (vuz216 * vuz217)",fontsize=16,color="magenta"];4296 -> 4312[label="",style="dashed", color="magenta", weight=3]; 4296 -> 4313[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4086[label="",style="dashed", color="red", weight=0]; 4297[label="pr2F3 (primEqInt (primMinusNat vuz2150 (Succ vuz214)) (fromInt (Pos Zero))) vuz216 (primMinusNat vuz2150 (Succ vuz214)) (vuz216 * vuz217)",fontsize=16,color="magenta"];4297 -> 4330[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4331[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4332[label="",style="dashed", color="magenta", weight=3]; 4297 -> 4333[label="",style="dashed", color="magenta", weight=3]; 1817 -> 1772[label="",style="dashed", color="red", weight=0]; 1817[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) (primEvenNat vuz11400)",fontsize=16,color="magenta"];1817 -> 1829[label="",style="dashed", color="magenta", weight=3]; 1818[label="pr2F0G1 vuz110 (vuz111 * vuz111) (Neg vuz113) False",fontsize=16,color="black",shape="box"];1818 -> 1830[label="",style="solid", color="black", weight=3]; 1819[label="pr2F0G vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1819 -> 1831[label="",style="solid", color="black", weight=3]; 4165[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4794[label="vuz20300/Succ vuz203000",fontsize=10,color="white",style="solid",shape="box"];4165 -> 4794[label="",style="solid", color="burlywood", weight=9]; 4794 -> 4282[label="",style="solid", color="burlywood", weight=3]; 4795[label="vuz20300/Zero",fontsize=10,color="white",style="solid",shape="box"];4165 -> 4795[label="",style="solid", color="burlywood", weight=9]; 4795 -> 4283[label="",style="solid", color="burlywood", weight=3]; 4166[label="pr2F3 (primEqInt (primMinusNat Zero vuz20300) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero vuz20300) (vuz204 * vuz205)",fontsize=16,color="burlywood",shape="box"];4796[label="vuz20300/Succ vuz203000",fontsize=10,color="white",style="solid",shape="box"];4166 -> 4796[label="",style="solid", color="burlywood", weight=9]; 4796 -> 4284[label="",style="solid", color="burlywood", weight=3]; 4797[label="vuz20300/Zero",fontsize=10,color="white",style="solid",shape="box"];4166 -> 4797[label="",style="solid", color="burlywood", weight=9]; 4797 -> 4285[label="",style="solid", color="burlywood", weight=3]; 4167[label="Succ vuz202",fontsize=16,color="green",shape="box"];4168[label="Succ vuz202",fontsize=16,color="green",shape="box"];4169[label="pr2F3 (primEqInt (Pos (Succ vuz2120)) (Pos Zero)) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4169 -> 4286[label="",style="solid", color="black", weight=3]; 4170[label="pr2F3 (primEqInt (Pos Zero) (Pos Zero)) vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4170 -> 4287[label="",style="solid", color="black", weight=3]; 606[label="primDivNatS0 (Succ (Succ vuz130000)) (Succ Zero) (primGEqNatS (Succ vuz130000) Zero)",fontsize=16,color="black",shape="box"];606 -> 619[label="",style="solid", color="black", weight=3]; 607[label="primDivNatS0 (Succ Zero) (Succ Zero) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];607 -> 620[label="",style="solid", color="black", weight=3]; 608[label="Zero",fontsize=16,color="green",shape="box"];1820 -> 1832[label="",style="dashed", color="red", weight=0]; 1820[label="pr2F (vuz103 * vuz103) (Pos vuz105 - fromInt (Pos (Succ Zero))) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1820 -> 1833[label="",style="dashed", color="magenta", weight=3]; 1821[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos vuz105 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1821 -> 1834[label="",style="solid", color="black", weight=3]; 4312 -> 71[label="",style="dashed", color="red", weight=0]; 4312[label="primPlusNat (Succ vuz214) vuz2150",fontsize=16,color="magenta"];4312 -> 4334[label="",style="dashed", color="magenta", weight=3]; 4312 -> 4335[label="",style="dashed", color="magenta", weight=3]; 4313 -> 71[label="",style="dashed", color="red", weight=0]; 4313[label="primPlusNat (Succ vuz214) vuz2150",fontsize=16,color="magenta"];4313 -> 4336[label="",style="dashed", color="magenta", weight=3]; 4313 -> 4337[label="",style="dashed", color="magenta", weight=3]; 4311[label="pr2F3 (primEqInt (Neg vuz219) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="burlywood",shape="triangle"];4798[label="vuz219/Succ vuz2190",fontsize=10,color="white",style="solid",shape="box"];4311 -> 4798[label="",style="solid", color="burlywood", weight=9]; 4798 -> 4338[label="",style="solid", color="burlywood", weight=3]; 4799[label="vuz219/Zero",fontsize=10,color="white",style="solid",shape="box"];4311 -> 4799[label="",style="solid", color="burlywood", weight=9]; 4799 -> 4339[label="",style="solid", color="burlywood", weight=3]; 4330[label="Succ vuz214",fontsize=16,color="green",shape="box"];4331[label="vuz2150",fontsize=16,color="green",shape="box"];4332[label="vuz217",fontsize=16,color="green",shape="box"];4333[label="vuz216",fontsize=16,color="green",shape="box"];1829[label="vuz11400",fontsize=16,color="green",shape="box"];1830[label="pr2F0G0 vuz110 (vuz111 * vuz111) (Neg vuz113) otherwise",fontsize=16,color="black",shape="box"];1830 -> 1835[label="",style="solid", color="black", weight=3]; 1831[label="pr2F0G2 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];1831 -> 1836[label="",style="solid", color="black", weight=3]; 4282[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4282 -> 4290[label="",style="solid", color="black", weight=3]; 4283[label="pr2F3 (primEqInt (primMinusNat (Succ vuz2020) Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat (Succ vuz2020) Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4283 -> 4291[label="",style="solid", color="black", weight=3]; 4284[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4284 -> 4292[label="",style="solid", color="black", weight=3]; 4285[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) vuz204 (primMinusNat Zero Zero) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4285 -> 4293[label="",style="solid", color="black", weight=3]; 4286[label="pr2F3 False vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4286 -> 4294[label="",style="solid", color="black", weight=3]; 4287[label="pr2F3 True vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4287 -> 4295[label="",style="solid", color="black", weight=3]; 619[label="primDivNatS0 (Succ (Succ vuz130000)) (Succ Zero) True",fontsize=16,color="black",shape="box"];619 -> 645[label="",style="solid", color="black", weight=3]; 620[label="primDivNatS0 (Succ Zero) (Succ Zero) True",fontsize=16,color="black",shape="box"];620 -> 646[label="",style="solid", color="black", weight=3]; 1833 -> 23[label="",style="dashed", color="red", weight=0]; 1833[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];1832[label="pr2F (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="triangle"];1832 -> 1837[label="",style="solid", color="black", weight=3]; 1834[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (primQuotInt (Pos vuz105) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos vuz105) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1834 -> 1846[label="",style="solid", color="black", weight=3]; 4334[label="Succ vuz214",fontsize=16,color="green",shape="box"];4335[label="vuz2150",fontsize=16,color="green",shape="box"];4336[label="Succ vuz214",fontsize=16,color="green",shape="box"];4337[label="vuz2150",fontsize=16,color="green",shape="box"];4338[label="pr2F3 (primEqInt (Neg (Succ vuz2190)) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4338 -> 4365[label="",style="solid", color="black", weight=3]; 4339[label="pr2F3 (primEqInt (Neg Zero) (fromInt (Pos Zero))) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4339 -> 4366[label="",style="solid", color="black", weight=3]; 1835[label="pr2F0G0 vuz110 (vuz111 * vuz111) (Neg vuz113) True",fontsize=16,color="black",shape="box"];1835 -> 1847[label="",style="solid", color="black", weight=3]; 1836[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1836 -> 1848[label="",style="solid", color="black", weight=3]; 4290 -> 4086[label="",style="dashed", color="red", weight=0]; 4290[label="pr2F3 (primEqInt (primMinusNat vuz2020 vuz203000) (fromInt (Pos Zero))) vuz204 (primMinusNat vuz2020 vuz203000) (vuz204 * vuz205)",fontsize=16,color="magenta"];4290 -> 4298[label="",style="dashed", color="magenta", weight=3]; 4290 -> 4299[label="",style="dashed", color="magenta", weight=3]; 4291 -> 4007[label="",style="dashed", color="red", weight=0]; 4291[label="pr2F3 (primEqInt (Pos (Succ vuz2020)) (fromInt (Pos Zero))) vuz204 (Pos (Succ vuz2020)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4291 -> 4300[label="",style="dashed", color="magenta", weight=3]; 4291 -> 4301[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4311[label="",style="dashed", color="red", weight=0]; 4292[label="pr2F3 (primEqInt (Neg (Succ vuz203000)) (fromInt (Pos Zero))) vuz204 (Neg (Succ vuz203000)) (vuz204 * vuz205)",fontsize=16,color="magenta"];4292 -> 4314[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4315[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4316[label="",style="dashed", color="magenta", weight=3]; 4292 -> 4317[label="",style="dashed", color="magenta", weight=3]; 4293 -> 4007[label="",style="dashed", color="red", weight=0]; 4293[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) vuz204 (Pos Zero) (vuz204 * vuz205)",fontsize=16,color="magenta"];4293 -> 4303[label="",style="dashed", color="magenta", weight=3]; 4293 -> 4304[label="",style="dashed", color="magenta", weight=3]; 4294[label="pr2F0 vuz204 (Pos vuz211) (vuz204 * vuz205)",fontsize=16,color="black",shape="box"];4294 -> 4305[label="",style="solid", color="black", weight=3]; 4295[label="vuz204 * vuz205",fontsize=16,color="blue",shape="box"];4800[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4800[label="",style="solid", color="blue", weight=9]; 4800 -> 4306[label="",style="solid", color="blue", weight=3]; 4801[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4801[label="",style="solid", color="blue", weight=9]; 4801 -> 4307[label="",style="solid", color="blue", weight=3]; 4802[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4802[label="",style="solid", color="blue", weight=9]; 4802 -> 4308[label="",style="solid", color="blue", weight=3]; 4803[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4803[label="",style="solid", color="blue", weight=9]; 4803 -> 4309[label="",style="solid", color="blue", weight=3]; 4804[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4295 -> 4804[label="",style="solid", color="blue", weight=9]; 4804 -> 4310[label="",style="solid", color="blue", weight=3]; 645[label="Succ (primDivNatS (primMinusNatS (Succ (Succ vuz130000)) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];645 -> 673[label="",style="dashed", color="green", weight=3]; 646[label="Succ (primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero)))",fontsize=16,color="green",shape="box"];646 -> 674[label="",style="dashed", color="green", weight=3]; 1837[label="pr2F4 (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1837 -> 1849[label="",style="solid", color="black", weight=3]; 1846[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (primQuotInt (Pos vuz105) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos vuz105) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1846 -> 1859[label="",style="solid", color="black", weight=3]; 4365[label="pr2F3 (primEqInt (Neg (Succ vuz2190)) (Pos Zero)) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4365 -> 4378[label="",style="solid", color="black", weight=3]; 4366[label="pr2F3 (primEqInt (Neg Zero) (Pos Zero)) vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4366 -> 4379[label="",style="solid", color="black", weight=3]; 1847 -> 1860[label="",style="dashed", color="red", weight=0]; 1847[label="pr2F (vuz111 * vuz111) (Neg vuz113 - fromInt (Pos (Succ Zero))) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1847 -> 1861[label="",style="dashed", color="magenta", weight=3]; 1848[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg vuz113 `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1848 -> 1862[label="",style="solid", color="black", weight=3]; 4298[label="vuz203000",fontsize=16,color="green",shape="box"];4299[label="vuz2020",fontsize=16,color="green",shape="box"];4300[label="Succ vuz2020",fontsize=16,color="green",shape="box"];4301[label="Succ vuz2020",fontsize=16,color="green",shape="box"];4314[label="Succ vuz203000",fontsize=16,color="green",shape="box"];4315[label="vuz204",fontsize=16,color="green",shape="box"];4316[label="Succ vuz203000",fontsize=16,color="green",shape="box"];4317[label="vuz205",fontsize=16,color="green",shape="box"];4303[label="Zero",fontsize=16,color="green",shape="box"];4304[label="Zero",fontsize=16,color="green",shape="box"];4305[label="pr2F0G (vuz204 * vuz205) vuz204 (Pos vuz211)",fontsize=16,color="black",shape="box"];4305 -> 4340[label="",style="solid", color="black", weight=3]; 4306 -> 1024[label="",style="dashed", color="red", weight=0]; 4306[label="vuz204 * vuz205",fontsize=16,color="magenta"];4306 -> 4341[label="",style="dashed", color="magenta", weight=3]; 4306 -> 4342[label="",style="dashed", color="magenta", weight=3]; 4307 -> 1041[label="",style="dashed", color="red", weight=0]; 4307[label="vuz204 * vuz205",fontsize=16,color="magenta"];4307 -> 4343[label="",style="dashed", color="magenta", weight=3]; 4307 -> 4344[label="",style="dashed", color="magenta", weight=3]; 4308 -> 1051[label="",style="dashed", color="red", weight=0]; 4308[label="vuz204 * vuz205",fontsize=16,color="magenta"];4308 -> 4345[label="",style="dashed", color="magenta", weight=3]; 4308 -> 4346[label="",style="dashed", color="magenta", weight=3]; 4309 -> 1061[label="",style="dashed", color="red", weight=0]; 4309[label="vuz204 * vuz205",fontsize=16,color="magenta"];4309 -> 4347[label="",style="dashed", color="magenta", weight=3]; 4309 -> 4348[label="",style="dashed", color="magenta", weight=3]; 4310 -> 1073[label="",style="dashed", color="red", weight=0]; 4310[label="vuz204 * vuz205",fontsize=16,color="magenta"];4310 -> 4349[label="",style="dashed", color="magenta", weight=3]; 4310 -> 4350[label="",style="dashed", color="magenta", weight=3]; 673 -> 508[label="",style="dashed", color="red", weight=0]; 673[label="primDivNatS (primMinusNatS (Succ (Succ vuz130000)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];673 -> 739[label="",style="dashed", color="magenta", weight=3]; 674 -> 510[label="",style="dashed", color="red", weight=0]; 674[label="primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="magenta"];1849[label="pr2F3 (Pos vuz105 - vuz115 == fromInt (Pos Zero)) (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1849 -> 1863[label="",style="solid", color="black", weight=3]; 1859 -> 1605[label="",style="dashed", color="red", weight=0]; 1859[label="pr2F0G1 vuz102 (vuz103 * vuz103 * (vuz103 * vuz103)) (Pos (primDivNatS vuz105 (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS vuz105 (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1859 -> 1864[label="",style="dashed", color="magenta", weight=3]; 1859 -> 1865[label="",style="dashed", color="magenta", weight=3]; 1859 -> 1866[label="",style="dashed", color="magenta", weight=3]; 4378[label="pr2F3 False vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4378 -> 4381[label="",style="solid", color="black", weight=3]; 4379[label="pr2F3 True vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4379 -> 4382[label="",style="solid", color="black", weight=3]; 1861 -> 23[label="",style="dashed", color="red", weight=0]; 1861[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];1860[label="pr2F (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="triangle"];1860 -> 1867[label="",style="solid", color="black", weight=3]; 1862[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (primQuotInt (Neg vuz113) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg vuz113) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];1862 -> 1874[label="",style="solid", color="black", weight=3]; 4340[label="pr2F0G2 (vuz204 * vuz205) vuz204 (Pos vuz211)",fontsize=16,color="black",shape="box"];4340 -> 4367[label="",style="solid", color="black", weight=3]; 4341[label="vuz204",fontsize=16,color="green",shape="box"];4342[label="vuz205",fontsize=16,color="green",shape="box"];1024[label="vuz69 * vuz20",fontsize=16,color="black",shape="triangle"];1024 -> 1029[label="",style="solid", color="black", weight=3]; 4343[label="vuz204",fontsize=16,color="green",shape="box"];4344[label="vuz205",fontsize=16,color="green",shape="box"];1041[label="vuz70 * vuz20",fontsize=16,color="black",shape="triangle"];1041 -> 1046[label="",style="solid", color="black", weight=3]; 4345[label="vuz205",fontsize=16,color="green",shape="box"];4346[label="vuz204",fontsize=16,color="green",shape="box"];1051[label="vuz71 * vuz20",fontsize=16,color="black",shape="triangle"];1051 -> 1056[label="",style="solid", color="black", weight=3]; 4347[label="vuz204",fontsize=16,color="green",shape="box"];4348[label="vuz205",fontsize=16,color="green",shape="box"];1061[label="vuz72 * vuz20",fontsize=16,color="black",shape="triangle"];1061 -> 1066[label="",style="solid", color="black", weight=3]; 4349[label="vuz205",fontsize=16,color="green",shape="box"];4350[label="vuz204",fontsize=16,color="green",shape="box"];1073[label="vuz73 * vuz20",fontsize=16,color="black",shape="triangle"];1073 -> 1078[label="",style="solid", color="black", weight=3]; 739[label="vuz130000",fontsize=16,color="green",shape="box"];508[label="primDivNatS (primMinusNatS (Succ (Succ vuz1300)) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];508 -> 538[label="",style="solid", color="black", weight=3]; 510[label="primDivNatS (primMinusNatS (Succ Zero) (Succ Zero)) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];510 -> 541[label="",style="solid", color="black", weight=3]; 1863[label="pr2F3 (primEqInt (Pos vuz105 - vuz115) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos vuz105 - vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1863 -> 1875[label="",style="solid", color="black", weight=3]; 1864 -> 1226[label="",style="dashed", color="red", weight=0]; 1864[label="primDivNatS vuz105 (Succ (Succ Zero))",fontsize=16,color="magenta"];1864 -> 1876[label="",style="dashed", color="magenta", weight=3]; 1865[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4805[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4805[label="",style="solid", color="blue", weight=9]; 4805 -> 1877[label="",style="solid", color="blue", weight=3]; 4806[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4806[label="",style="solid", color="blue", weight=9]; 4806 -> 1878[label="",style="solid", color="blue", weight=3]; 4807[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4807[label="",style="solid", color="blue", weight=9]; 4807 -> 1879[label="",style="solid", color="blue", weight=3]; 4808[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4808[label="",style="solid", color="blue", weight=9]; 4808 -> 1880[label="",style="solid", color="blue", weight=3]; 4809[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];1865 -> 4809[label="",style="solid", color="blue", weight=9]; 4809 -> 1881[label="",style="solid", color="blue", weight=3]; 1866 -> 1226[label="",style="dashed", color="red", weight=0]; 1866[label="primDivNatS vuz105 (Succ (Succ Zero))",fontsize=16,color="magenta"];1866 -> 1882[label="",style="dashed", color="magenta", weight=3]; 4381[label="pr2F0 vuz216 (Neg vuz218) (vuz216 * vuz217)",fontsize=16,color="black",shape="box"];4381 -> 4384[label="",style="solid", color="black", weight=3]; 4382[label="vuz216 * vuz217",fontsize=16,color="blue",shape="box"];4810[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4810[label="",style="solid", color="blue", weight=9]; 4810 -> 4385[label="",style="solid", color="blue", weight=3]; 4811[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4811[label="",style="solid", color="blue", weight=9]; 4811 -> 4386[label="",style="solid", color="blue", weight=3]; 4812[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4812[label="",style="solid", color="blue", weight=9]; 4812 -> 4387[label="",style="solid", color="blue", weight=3]; 4813[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4813[label="",style="solid", color="blue", weight=9]; 4813 -> 4388[label="",style="solid", color="blue", weight=3]; 4814[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4382 -> 4814[label="",style="solid", color="blue", weight=9]; 4814 -> 4389[label="",style="solid", color="blue", weight=3]; 1867[label="pr2F4 (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1867 -> 1883[label="",style="solid", color="black", weight=3]; 1874[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (primQuotInt (Neg vuz113) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg vuz113) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];1874 -> 1896[label="",style="solid", color="black", weight=3]; 4367[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (even (Pos vuz211))",fontsize=16,color="black",shape="box"];4367 -> 4380[label="",style="solid", color="black", weight=3]; 1029[label="error []",fontsize=16,color="red",shape="box"];1046[label="primMulInt vuz70 vuz20",fontsize=16,color="burlywood",shape="box"];4815[label="vuz70/Pos vuz700",fontsize=10,color="white",style="solid",shape="box"];1046 -> 4815[label="",style="solid", color="burlywood", weight=9]; 4815 -> 1057[label="",style="solid", color="burlywood", weight=3]; 4816[label="vuz70/Neg vuz700",fontsize=10,color="white",style="solid",shape="box"];1046 -> 4816[label="",style="solid", color="burlywood", weight=9]; 4816 -> 1058[label="",style="solid", color="burlywood", weight=3]; 1056[label="error []",fontsize=16,color="red",shape="box"];1066[label="error []",fontsize=16,color="red",shape="box"];1078[label="error []",fontsize=16,color="red",shape="box"];538[label="primDivNatS (primMinusNatS (Succ vuz1300) Zero) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];538 -> 554[label="",style="solid", color="black", weight=3]; 541[label="primDivNatS (primMinusNatS Zero Zero) (Succ (Succ Zero))",fontsize=16,color="black",shape="box"];541 -> 557[label="",style="solid", color="black", weight=3]; 1875[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) vuz115) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) vuz115) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4817[label="vuz115/Pos vuz1150",fontsize=10,color="white",style="solid",shape="box"];1875 -> 4817[label="",style="solid", color="burlywood", weight=9]; 4817 -> 1897[label="",style="solid", color="burlywood", weight=3]; 4818[label="vuz115/Neg vuz1150",fontsize=10,color="white",style="solid",shape="box"];1875 -> 4818[label="",style="solid", color="burlywood", weight=9]; 4818 -> 1898[label="",style="solid", color="burlywood", weight=3]; 1876[label="vuz105",fontsize=16,color="green",shape="box"];1877 -> 397[label="",style="dashed", color="red", weight=0]; 1877[label="vuz103 * vuz103",fontsize=16,color="magenta"];1877 -> 1899[label="",style="dashed", color="magenta", weight=3]; 1878 -> 398[label="",style="dashed", color="red", weight=0]; 1878[label="vuz103 * vuz103",fontsize=16,color="magenta"];1878 -> 1900[label="",style="dashed", color="magenta", weight=3]; 1879 -> 399[label="",style="dashed", color="red", weight=0]; 1879[label="vuz103 * vuz103",fontsize=16,color="magenta"];1879 -> 1901[label="",style="dashed", color="magenta", weight=3]; 1880 -> 400[label="",style="dashed", color="red", weight=0]; 1880[label="vuz103 * vuz103",fontsize=16,color="magenta"];1880 -> 1902[label="",style="dashed", color="magenta", weight=3]; 1881 -> 401[label="",style="dashed", color="red", weight=0]; 1881[label="vuz103 * vuz103",fontsize=16,color="magenta"];1881 -> 1903[label="",style="dashed", color="magenta", weight=3]; 1882[label="vuz105",fontsize=16,color="green",shape="box"];4384[label="pr2F0G (vuz216 * vuz217) vuz216 (Neg vuz218)",fontsize=16,color="black",shape="box"];4384 -> 4392[label="",style="solid", color="black", weight=3]; 4385 -> 1024[label="",style="dashed", color="red", weight=0]; 4385[label="vuz216 * vuz217",fontsize=16,color="magenta"];4385 -> 4393[label="",style="dashed", color="magenta", weight=3]; 4385 -> 4394[label="",style="dashed", color="magenta", weight=3]; 4386 -> 1041[label="",style="dashed", color="red", weight=0]; 4386[label="vuz216 * vuz217",fontsize=16,color="magenta"];4386 -> 4395[label="",style="dashed", color="magenta", weight=3]; 4386 -> 4396[label="",style="dashed", color="magenta", weight=3]; 4387 -> 1051[label="",style="dashed", color="red", weight=0]; 4387[label="vuz216 * vuz217",fontsize=16,color="magenta"];4387 -> 4397[label="",style="dashed", color="magenta", weight=3]; 4387 -> 4398[label="",style="dashed", color="magenta", weight=3]; 4388 -> 1061[label="",style="dashed", color="red", weight=0]; 4388[label="vuz216 * vuz217",fontsize=16,color="magenta"];4388 -> 4399[label="",style="dashed", color="magenta", weight=3]; 4388 -> 4400[label="",style="dashed", color="magenta", weight=3]; 4389 -> 1073[label="",style="dashed", color="red", weight=0]; 4389[label="vuz216 * vuz217",fontsize=16,color="magenta"];4389 -> 4401[label="",style="dashed", color="magenta", weight=3]; 4389 -> 4402[label="",style="dashed", color="magenta", weight=3]; 1883[label="pr2F3 (Neg vuz113 - vuz116 == fromInt (Pos Zero)) (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1883 -> 1904[label="",style="solid", color="black", weight=3]; 1896 -> 1755[label="",style="dashed", color="red", weight=0]; 1896[label="pr2F0G1 vuz110 (vuz111 * vuz111 * (vuz111 * vuz111)) (Neg (primDivNatS vuz113 (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS vuz113 (Succ (Succ Zero)))))",fontsize=16,color="magenta"];1896 -> 1917[label="",style="dashed", color="magenta", weight=3]; 1896 -> 1918[label="",style="dashed", color="magenta", weight=3]; 1896 -> 1919[label="",style="dashed", color="magenta", weight=3]; 4380[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (primEvenInt (Pos vuz211))",fontsize=16,color="black",shape="box"];4380 -> 4383[label="",style="solid", color="black", weight=3]; 1057[label="primMulInt (Pos vuz700) vuz20",fontsize=16,color="burlywood",shape="box"];4819[label="vuz20/Pos vuz200",fontsize=10,color="white",style="solid",shape="box"];1057 -> 4819[label="",style="solid", color="burlywood", weight=9]; 4819 -> 1067[label="",style="solid", color="burlywood", weight=3]; 4820[label="vuz20/Neg vuz200",fontsize=10,color="white",style="solid",shape="box"];1057 -> 4820[label="",style="solid", color="burlywood", weight=9]; 4820 -> 1068[label="",style="solid", color="burlywood", weight=3]; 1058[label="primMulInt (Neg vuz700) vuz20",fontsize=16,color="burlywood",shape="box"];4821[label="vuz20/Pos vuz200",fontsize=10,color="white",style="solid",shape="box"];1058 -> 4821[label="",style="solid", color="burlywood", weight=9]; 4821 -> 1069[label="",style="solid", color="burlywood", weight=3]; 4822[label="vuz20/Neg vuz200",fontsize=10,color="white",style="solid",shape="box"];1058 -> 4822[label="",style="solid", color="burlywood", weight=9]; 4822 -> 1070[label="",style="solid", color="burlywood", weight=3]; 554[label="primDivNatS (Succ vuz1300) (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];554 -> 562[label="",style="solid", color="black", weight=3]; 557[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="black",shape="triangle"];557 -> 566[label="",style="solid", color="black", weight=3]; 1897[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) (Pos vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) (Pos vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1897 -> 1920[label="",style="solid", color="black", weight=3]; 1898[label="pr2F3 (primEqInt (primMinusInt (Pos vuz105) (Neg vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusInt (Pos vuz105) (Neg vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1898 -> 1921[label="",style="solid", color="black", weight=3]; 1899[label="vuz103",fontsize=16,color="green",shape="box"];397 -> 1024[label="",style="dashed", color="red", weight=0]; 397[label="vuz12 * vuz12",fontsize=16,color="magenta"];397 -> 1026[label="",style="dashed", color="magenta", weight=3]; 397 -> 1027[label="",style="dashed", color="magenta", weight=3]; 1900[label="vuz103",fontsize=16,color="green",shape="box"];398 -> 1041[label="",style="dashed", color="red", weight=0]; 398[label="vuz12 * vuz12",fontsize=16,color="magenta"];398 -> 1043[label="",style="dashed", color="magenta", weight=3]; 398 -> 1044[label="",style="dashed", color="magenta", weight=3]; 1901[label="vuz103",fontsize=16,color="green",shape="box"];399 -> 1051[label="",style="dashed", color="red", weight=0]; 399[label="vuz12 * vuz12",fontsize=16,color="magenta"];399 -> 1053[label="",style="dashed", color="magenta", weight=3]; 399 -> 1054[label="",style="dashed", color="magenta", weight=3]; 1902[label="vuz103",fontsize=16,color="green",shape="box"];400 -> 1061[label="",style="dashed", color="red", weight=0]; 400[label="vuz12 * vuz12",fontsize=16,color="magenta"];400 -> 1063[label="",style="dashed", color="magenta", weight=3]; 400 -> 1064[label="",style="dashed", color="magenta", weight=3]; 1903[label="vuz103",fontsize=16,color="green",shape="box"];401 -> 1073[label="",style="dashed", color="red", weight=0]; 401[label="vuz12 * vuz12",fontsize=16,color="magenta"];401 -> 1075[label="",style="dashed", color="magenta", weight=3]; 401 -> 1076[label="",style="dashed", color="magenta", weight=3]; 4392[label="pr2F0G2 (vuz216 * vuz217) vuz216 (Neg vuz218)",fontsize=16,color="black",shape="box"];4392 -> 4406[label="",style="solid", color="black", weight=3]; 4393[label="vuz216",fontsize=16,color="green",shape="box"];4394[label="vuz217",fontsize=16,color="green",shape="box"];4395[label="vuz216",fontsize=16,color="green",shape="box"];4396[label="vuz217",fontsize=16,color="green",shape="box"];4397[label="vuz217",fontsize=16,color="green",shape="box"];4398[label="vuz216",fontsize=16,color="green",shape="box"];4399[label="vuz216",fontsize=16,color="green",shape="box"];4400[label="vuz217",fontsize=16,color="green",shape="box"];4401[label="vuz217",fontsize=16,color="green",shape="box"];4402[label="vuz216",fontsize=16,color="green",shape="box"];1904[label="pr2F3 (primEqInt (Neg vuz113 - vuz116) (fromInt (Pos Zero))) (vuz111 * vuz111) (Neg vuz113 - vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1904 -> 1922[label="",style="solid", color="black", weight=3]; 1917[label="vuz111 * vuz111",fontsize=16,color="blue",shape="box"];4823[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4823[label="",style="solid", color="blue", weight=9]; 4823 -> 1930[label="",style="solid", color="blue", weight=3]; 4824[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4824[label="",style="solid", color="blue", weight=9]; 4824 -> 1931[label="",style="solid", color="blue", weight=3]; 4825[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4825[label="",style="solid", color="blue", weight=9]; 4825 -> 1932[label="",style="solid", color="blue", weight=3]; 4826[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4826[label="",style="solid", color="blue", weight=9]; 4826 -> 1933[label="",style="solid", color="blue", weight=3]; 4827[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];1917 -> 4827[label="",style="solid", color="blue", weight=9]; 4827 -> 1934[label="",style="solid", color="blue", weight=3]; 1918 -> 1226[label="",style="dashed", color="red", weight=0]; 1918[label="primDivNatS vuz113 (Succ (Succ Zero))",fontsize=16,color="magenta"];1918 -> 1935[label="",style="dashed", color="magenta", weight=3]; 1919 -> 1226[label="",style="dashed", color="red", weight=0]; 1919[label="primDivNatS vuz113 (Succ (Succ Zero))",fontsize=16,color="magenta"];1919 -> 1936[label="",style="dashed", color="magenta", weight=3]; 4383[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos vuz211) (primEvenNat vuz211)",fontsize=16,color="burlywood",shape="box"];4828[label="vuz211/Succ vuz2110",fontsize=10,color="white",style="solid",shape="box"];4383 -> 4828[label="",style="solid", color="burlywood", weight=9]; 4828 -> 4390[label="",style="solid", color="burlywood", weight=3]; 4829[label="vuz211/Zero",fontsize=10,color="white",style="solid",shape="box"];4383 -> 4829[label="",style="solid", color="burlywood", weight=9]; 4829 -> 4391[label="",style="solid", color="burlywood", weight=3]; 1067[label="primMulInt (Pos vuz700) (Pos vuz200)",fontsize=16,color="black",shape="box"];1067 -> 1079[label="",style="solid", color="black", weight=3]; 1068[label="primMulInt (Pos vuz700) (Neg vuz200)",fontsize=16,color="black",shape="box"];1068 -> 1080[label="",style="solid", color="black", weight=3]; 1069[label="primMulInt (Neg vuz700) (Pos vuz200)",fontsize=16,color="black",shape="box"];1069 -> 1081[label="",style="solid", color="black", weight=3]; 1070[label="primMulInt (Neg vuz700) (Neg vuz200)",fontsize=16,color="black",shape="box"];1070 -> 1082[label="",style="solid", color="black", weight=3]; 566[label="Zero",fontsize=16,color="green",shape="box"];1920[label="pr2F3 (primEqInt (primMinusNat vuz105 vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat vuz105 vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="triangle"];4830[label="vuz105/Succ vuz1050",fontsize=10,color="white",style="solid",shape="box"];1920 -> 4830[label="",style="solid", color="burlywood", weight=9]; 4830 -> 1937[label="",style="solid", color="burlywood", weight=3]; 4831[label="vuz105/Zero",fontsize=10,color="white",style="solid",shape="box"];1920 -> 4831[label="",style="solid", color="burlywood", weight=9]; 4831 -> 1938[label="",style="solid", color="burlywood", weight=3]; 1921 -> 4007[label="",style="dashed", color="red", weight=0]; 1921[label="pr2F3 (primEqInt (Pos (primPlusNat vuz105 vuz1150)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos (primPlusNat vuz105 vuz1150)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1921 -> 4022[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4023[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4024[label="",style="dashed", color="magenta", weight=3]; 1921 -> 4025[label="",style="dashed", color="magenta", weight=3]; 1026[label="vuz12",fontsize=16,color="green",shape="box"];1027[label="vuz12",fontsize=16,color="green",shape="box"];1043[label="vuz12",fontsize=16,color="green",shape="box"];1044[label="vuz12",fontsize=16,color="green",shape="box"];1053[label="vuz12",fontsize=16,color="green",shape="box"];1054[label="vuz12",fontsize=16,color="green",shape="box"];1063[label="vuz12",fontsize=16,color="green",shape="box"];1064[label="vuz12",fontsize=16,color="green",shape="box"];1075[label="vuz12",fontsize=16,color="green",shape="box"];1076[label="vuz12",fontsize=16,color="green",shape="box"];4406[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (even (Neg vuz218))",fontsize=16,color="black",shape="box"];4406 -> 4410[label="",style="solid", color="black", weight=3]; 1922[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) vuz116) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) vuz116) (vuz111 * vuz111 * vuz110)",fontsize=16,color="burlywood",shape="box"];4832[label="vuz116/Pos vuz1160",fontsize=10,color="white",style="solid",shape="box"];1922 -> 4832[label="",style="solid", color="burlywood", weight=9]; 4832 -> 1942[label="",style="solid", color="burlywood", weight=3]; 4833[label="vuz116/Neg vuz1160",fontsize=10,color="white",style="solid",shape="box"];1922 -> 4833[label="",style="solid", color="burlywood", weight=9]; 4833 -> 1943[label="",style="solid", color="burlywood", weight=3]; 1930 -> 397[label="",style="dashed", color="red", weight=0]; 1930[label="vuz111 * vuz111",fontsize=16,color="magenta"];1930 -> 1944[label="",style="dashed", color="magenta", weight=3]; 1931 -> 398[label="",style="dashed", color="red", weight=0]; 1931[label="vuz111 * vuz111",fontsize=16,color="magenta"];1931 -> 1945[label="",style="dashed", color="magenta", weight=3]; 1932 -> 399[label="",style="dashed", color="red", weight=0]; 1932[label="vuz111 * vuz111",fontsize=16,color="magenta"];1932 -> 1946[label="",style="dashed", color="magenta", weight=3]; 1933 -> 400[label="",style="dashed", color="red", weight=0]; 1933[label="vuz111 * vuz111",fontsize=16,color="magenta"];1933 -> 1947[label="",style="dashed", color="magenta", weight=3]; 1934 -> 401[label="",style="dashed", color="red", weight=0]; 1934[label="vuz111 * vuz111",fontsize=16,color="magenta"];1934 -> 1948[label="",style="dashed", color="magenta", weight=3]; 1935[label="vuz113",fontsize=16,color="green",shape="box"];1936[label="vuz113",fontsize=16,color="green",shape="box"];4390[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ vuz2110)) (primEvenNat (Succ vuz2110))",fontsize=16,color="burlywood",shape="box"];4834[label="vuz2110/Succ vuz21100",fontsize=10,color="white",style="solid",shape="box"];4390 -> 4834[label="",style="solid", color="burlywood", weight=9]; 4834 -> 4403[label="",style="solid", color="burlywood", weight=3]; 4835[label="vuz2110/Zero",fontsize=10,color="white",style="solid",shape="box"];4390 -> 4835[label="",style="solid", color="burlywood", weight=9]; 4835 -> 4404[label="",style="solid", color="burlywood", weight=3]; 4391[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos Zero) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4391 -> 4405[label="",style="solid", color="black", weight=3]; 1079[label="Pos (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1079 -> 1085[label="",style="dashed", color="green", weight=3]; 1080[label="Neg (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1080 -> 1086[label="",style="dashed", color="green", weight=3]; 1081[label="Neg (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1081 -> 1087[label="",style="dashed", color="green", weight=3]; 1082[label="Pos (primMulNat vuz700 vuz200)",fontsize=16,color="green",shape="box"];1082 -> 1088[label="",style="dashed", color="green", weight=3]; 1937[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4836[label="vuz1150/Succ vuz11500",fontsize=10,color="white",style="solid",shape="box"];1937 -> 4836[label="",style="solid", color="burlywood", weight=9]; 4836 -> 1949[label="",style="solid", color="burlywood", weight=3]; 4837[label="vuz1150/Zero",fontsize=10,color="white",style="solid",shape="box"];1937 -> 4837[label="",style="solid", color="burlywood", weight=9]; 4837 -> 1950[label="",style="solid", color="burlywood", weight=3]; 1938[label="pr2F3 (primEqInt (primMinusNat Zero vuz1150) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero vuz1150) (vuz103 * vuz103 * vuz102)",fontsize=16,color="burlywood",shape="box"];4838[label="vuz1150/Succ vuz11500",fontsize=10,color="white",style="solid",shape="box"];1938 -> 4838[label="",style="solid", color="burlywood", weight=9]; 4838 -> 1951[label="",style="solid", color="burlywood", weight=3]; 4839[label="vuz1150/Zero",fontsize=10,color="white",style="solid",shape="box"];1938 -> 4839[label="",style="solid", color="burlywood", weight=9]; 4839 -> 1952[label="",style="solid", color="burlywood", weight=3]; 4022[label="vuz102",fontsize=16,color="green",shape="box"];4023 -> 71[label="",style="dashed", color="red", weight=0]; 4023[label="primPlusNat vuz105 vuz1150",fontsize=16,color="magenta"];4023 -> 4094[label="",style="dashed", color="magenta", weight=3]; 4023 -> 4095[label="",style="dashed", color="magenta", weight=3]; 4024[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4840[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4840[label="",style="solid", color="blue", weight=9]; 4840 -> 4096[label="",style="solid", color="blue", weight=3]; 4841[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4841[label="",style="solid", color="blue", weight=9]; 4841 -> 4097[label="",style="solid", color="blue", weight=3]; 4842[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4842[label="",style="solid", color="blue", weight=9]; 4842 -> 4098[label="",style="solid", color="blue", weight=3]; 4843[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4843[label="",style="solid", color="blue", weight=9]; 4843 -> 4099[label="",style="solid", color="blue", weight=3]; 4844[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4024 -> 4844[label="",style="solid", color="blue", weight=9]; 4844 -> 4100[label="",style="solid", color="blue", weight=3]; 4025 -> 71[label="",style="dashed", color="red", weight=0]; 4025[label="primPlusNat vuz105 vuz1150",fontsize=16,color="magenta"];4025 -> 4101[label="",style="dashed", color="magenta", weight=3]; 4025 -> 4102[label="",style="dashed", color="magenta", weight=3]; 4410[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (primEvenInt (Neg vuz218))",fontsize=16,color="black",shape="box"];4410 -> 4415[label="",style="solid", color="black", weight=3]; 1942[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) (Pos vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) (Pos vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1942 -> 1969[label="",style="solid", color="black", weight=3]; 1943[label="pr2F3 (primEqInt (primMinusInt (Neg vuz113) (Neg vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusInt (Neg vuz113) (Neg vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="black",shape="box"];1943 -> 1970[label="",style="solid", color="black", weight=3]; 1944[label="vuz111",fontsize=16,color="green",shape="box"];1945[label="vuz111",fontsize=16,color="green",shape="box"];1946[label="vuz111",fontsize=16,color="green",shape="box"];1947[label="vuz111",fontsize=16,color="green",shape="box"];1948[label="vuz111",fontsize=16,color="green",shape="box"];4403[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ (Succ vuz21100))) (primEvenNat (Succ (Succ vuz21100)))",fontsize=16,color="black",shape="box"];4403 -> 4407[label="",style="solid", color="black", weight=3]; 4404[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4404 -> 4408[label="",style="solid", color="black", weight=3]; 4405[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos Zero) True",fontsize=16,color="black",shape="box"];4405 -> 4409[label="",style="solid", color="black", weight=3]; 1085[label="primMulNat vuz700 vuz200",fontsize=16,color="burlywood",shape="triangle"];4845[label="vuz700/Succ vuz7000",fontsize=10,color="white",style="solid",shape="box"];1085 -> 4845[label="",style="solid", color="burlywood", weight=9]; 4845 -> 1111[label="",style="solid", color="burlywood", weight=3]; 4846[label="vuz700/Zero",fontsize=10,color="white",style="solid",shape="box"];1085 -> 4846[label="",style="solid", color="burlywood", weight=9]; 4846 -> 1112[label="",style="solid", color="burlywood", weight=3]; 1086 -> 1085[label="",style="dashed", color="red", weight=0]; 1086[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1086 -> 1113[label="",style="dashed", color="magenta", weight=3]; 1087 -> 1085[label="",style="dashed", color="red", weight=0]; 1087[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1087 -> 1114[label="",style="dashed", color="magenta", weight=3]; 1088 -> 1085[label="",style="dashed", color="red", weight=0]; 1088[label="primMulNat vuz700 vuz200",fontsize=16,color="magenta"];1088 -> 1115[label="",style="dashed", color="magenta", weight=3]; 1088 -> 1116[label="",style="dashed", color="magenta", weight=3]; 1949[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1949 -> 1971[label="",style="solid", color="black", weight=3]; 1950[label="pr2F3 (primEqInt (primMinusNat (Succ vuz1050) Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat (Succ vuz1050) Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1950 -> 1972[label="",style="solid", color="black", weight=3]; 1951[label="pr2F3 (primEqInt (primMinusNat Zero (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1951 -> 1973[label="",style="solid", color="black", weight=3]; 1952[label="pr2F3 (primEqInt (primMinusNat Zero Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat Zero Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="black",shape="box"];1952 -> 1974[label="",style="solid", color="black", weight=3]; 4094[label="vuz105",fontsize=16,color="green",shape="box"];4095[label="vuz1150",fontsize=16,color="green",shape="box"];4096 -> 397[label="",style="dashed", color="red", weight=0]; 4096[label="vuz103 * vuz103",fontsize=16,color="magenta"];4096 -> 4171[label="",style="dashed", color="magenta", weight=3]; 4097 -> 398[label="",style="dashed", color="red", weight=0]; 4097[label="vuz103 * vuz103",fontsize=16,color="magenta"];4097 -> 4172[label="",style="dashed", color="magenta", weight=3]; 4098 -> 399[label="",style="dashed", color="red", weight=0]; 4098[label="vuz103 * vuz103",fontsize=16,color="magenta"];4098 -> 4173[label="",style="dashed", color="magenta", weight=3]; 4099 -> 400[label="",style="dashed", color="red", weight=0]; 4099[label="vuz103 * vuz103",fontsize=16,color="magenta"];4099 -> 4174[label="",style="dashed", color="magenta", weight=3]; 4100 -> 401[label="",style="dashed", color="red", weight=0]; 4100[label="vuz103 * vuz103",fontsize=16,color="magenta"];4100 -> 4175[label="",style="dashed", color="magenta", weight=3]; 4101[label="vuz105",fontsize=16,color="green",shape="box"];4102[label="vuz1150",fontsize=16,color="green",shape="box"];4415[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg vuz218) (primEvenNat vuz218)",fontsize=16,color="burlywood",shape="box"];4847[label="vuz218/Succ vuz2180",fontsize=10,color="white",style="solid",shape="box"];4415 -> 4847[label="",style="solid", color="burlywood", weight=9]; 4847 -> 4421[label="",style="solid", color="burlywood", weight=3]; 4848[label="vuz218/Zero",fontsize=10,color="white",style="solid",shape="box"];4415 -> 4848[label="",style="solid", color="burlywood", weight=9]; 4848 -> 4422[label="",style="solid", color="burlywood", weight=3]; 1969 -> 4311[label="",style="dashed", color="red", weight=0]; 1969[label="pr2F3 (primEqInt (Neg (primPlusNat vuz113 vuz1160)) (fromInt (Pos Zero))) (vuz111 * vuz111) (Neg (primPlusNat vuz113 vuz1160)) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1969 -> 4318[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4319[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4320[label="",style="dashed", color="magenta", weight=3]; 1969 -> 4321[label="",style="dashed", color="magenta", weight=3]; 1970 -> 1920[label="",style="dashed", color="red", weight=0]; 1970[label="pr2F3 (primEqInt (primMinusNat vuz1160 vuz113) (fromInt (Pos Zero))) (vuz111 * vuz111) (primMinusNat vuz1160 vuz113) (vuz111 * vuz111 * vuz110)",fontsize=16,color="magenta"];1970 -> 2001[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2002[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2003[label="",style="dashed", color="magenta", weight=3]; 1970 -> 2004[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4464[label="",style="dashed", color="red", weight=0]; 4407[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ (Succ vuz21100))) (primEvenNat vuz21100)",fontsize=16,color="magenta"];4407 -> 4465[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4466[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4467[label="",style="dashed", color="magenta", weight=3]; 4407 -> 4468[label="",style="dashed", color="magenta", weight=3]; 4408[label="pr2F0G1 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) False",fontsize=16,color="black",shape="box"];4408 -> 4413[label="",style="solid", color="black", weight=3]; 4409[label="pr2F0G (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4409 -> 4414[label="",style="solid", color="black", weight=3]; 1111[label="primMulNat (Succ vuz7000) vuz200",fontsize=16,color="burlywood",shape="box"];4849[label="vuz200/Succ vuz2000",fontsize=10,color="white",style="solid",shape="box"];1111 -> 4849[label="",style="solid", color="burlywood", weight=9]; 4849 -> 1179[label="",style="solid", color="burlywood", weight=3]; 4850[label="vuz200/Zero",fontsize=10,color="white",style="solid",shape="box"];1111 -> 4850[label="",style="solid", color="burlywood", weight=9]; 4850 -> 1180[label="",style="solid", color="burlywood", weight=3]; 1112[label="primMulNat Zero vuz200",fontsize=16,color="burlywood",shape="box"];4851[label="vuz200/Succ vuz2000",fontsize=10,color="white",style="solid",shape="box"];1112 -> 4851[label="",style="solid", color="burlywood", weight=9]; 4851 -> 1181[label="",style="solid", color="burlywood", weight=3]; 4852[label="vuz200/Zero",fontsize=10,color="white",style="solid",shape="box"];1112 -> 4852[label="",style="solid", color="burlywood", weight=9]; 4852 -> 1182[label="",style="solid", color="burlywood", weight=3]; 1113[label="vuz200",fontsize=16,color="green",shape="box"];1114[label="vuz700",fontsize=16,color="green",shape="box"];1115[label="vuz200",fontsize=16,color="green",shape="box"];1116[label="vuz700",fontsize=16,color="green",shape="box"];1971 -> 1920[label="",style="dashed", color="red", weight=0]; 1971[label="pr2F3 (primEqInt (primMinusNat vuz1050 vuz11500) (fromInt (Pos Zero))) (vuz103 * vuz103) (primMinusNat vuz1050 vuz11500) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1971 -> 2005[label="",style="dashed", color="magenta", weight=3]; 1971 -> 2006[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4007[label="",style="dashed", color="red", weight=0]; 1972[label="pr2F3 (primEqInt (Pos (Succ vuz1050)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos (Succ vuz1050)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1972 -> 4030[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4031[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4032[label="",style="dashed", color="magenta", weight=3]; 1972 -> 4033[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4311[label="",style="dashed", color="red", weight=0]; 1973[label="pr2F3 (primEqInt (Neg (Succ vuz11500)) (fromInt (Pos Zero))) (vuz103 * vuz103) (Neg (Succ vuz11500)) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1973 -> 4322[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4323[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4324[label="",style="dashed", color="magenta", weight=3]; 1973 -> 4325[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4007[label="",style="dashed", color="red", weight=0]; 1974[label="pr2F3 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (vuz103 * vuz103) (Pos Zero) (vuz103 * vuz103 * vuz102)",fontsize=16,color="magenta"];1974 -> 4034[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4035[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4036[label="",style="dashed", color="magenta", weight=3]; 1974 -> 4037[label="",style="dashed", color="magenta", weight=3]; 4171[label="vuz103",fontsize=16,color="green",shape="box"];4172[label="vuz103",fontsize=16,color="green",shape="box"];4173[label="vuz103",fontsize=16,color="green",shape="box"];4174[label="vuz103",fontsize=16,color="green",shape="box"];4175[label="vuz103",fontsize=16,color="green",shape="box"];4421[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ vuz2180)) (primEvenNat (Succ vuz2180))",fontsize=16,color="burlywood",shape="box"];4853[label="vuz2180/Succ vuz21800",fontsize=10,color="white",style="solid",shape="box"];4421 -> 4853[label="",style="solid", color="burlywood", weight=9]; 4853 -> 4428[label="",style="solid", color="burlywood", weight=3]; 4854[label="vuz2180/Zero",fontsize=10,color="white",style="solid",shape="box"];4421 -> 4854[label="",style="solid", color="burlywood", weight=9]; 4854 -> 4429[label="",style="solid", color="burlywood", weight=3]; 4422[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg Zero) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4422 -> 4430[label="",style="solid", color="black", weight=3]; 4318 -> 71[label="",style="dashed", color="red", weight=0]; 4318[label="primPlusNat vuz113 vuz1160",fontsize=16,color="magenta"];4318 -> 4351[label="",style="dashed", color="magenta", weight=3]; 4318 -> 4352[label="",style="dashed", color="magenta", weight=3]; 4319[label="vuz111 * vuz111",fontsize=16,color="blue",shape="box"];4855[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4855[label="",style="solid", color="blue", weight=9]; 4855 -> 4353[label="",style="solid", color="blue", weight=3]; 4856[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4856[label="",style="solid", color="blue", weight=9]; 4856 -> 4354[label="",style="solid", color="blue", weight=3]; 4857[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4857[label="",style="solid", color="blue", weight=9]; 4857 -> 4355[label="",style="solid", color="blue", weight=3]; 4858[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4858[label="",style="solid", color="blue", weight=9]; 4858 -> 4356[label="",style="solid", color="blue", weight=3]; 4859[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4319 -> 4859[label="",style="solid", color="blue", weight=9]; 4859 -> 4357[label="",style="solid", color="blue", weight=3]; 4320 -> 71[label="",style="dashed", color="red", weight=0]; 4320[label="primPlusNat vuz113 vuz1160",fontsize=16,color="magenta"];4320 -> 4358[label="",style="dashed", color="magenta", weight=3]; 4320 -> 4359[label="",style="dashed", color="magenta", weight=3]; 4321[label="vuz110",fontsize=16,color="green",shape="box"];2001[label="vuz1160",fontsize=16,color="green",shape="box"];2002[label="vuz113",fontsize=16,color="green",shape="box"];2003[label="vuz111",fontsize=16,color="green",shape="box"];2004[label="vuz110",fontsize=16,color="green",shape="box"];4465[label="vuz205",fontsize=16,color="green",shape="box"];4466[label="vuz21100",fontsize=16,color="green",shape="box"];4467[label="vuz204",fontsize=16,color="green",shape="box"];4468[label="Succ vuz21100",fontsize=16,color="green",shape="box"];4464[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat vuz225)",fontsize=16,color="burlywood",shape="triangle"];4860[label="vuz225/Succ vuz2250",fontsize=10,color="white",style="solid",shape="box"];4464 -> 4860[label="",style="solid", color="burlywood", weight=9]; 4860 -> 4477[label="",style="solid", color="burlywood", weight=3]; 4861[label="vuz225/Zero",fontsize=10,color="white",style="solid",shape="box"];4464 -> 4861[label="",style="solid", color="burlywood", weight=9]; 4861 -> 4478[label="",style="solid", color="burlywood", weight=3]; 4413[label="pr2F0G0 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) otherwise",fontsize=16,color="black",shape="box"];4413 -> 4419[label="",style="solid", color="black", weight=3]; 4414[label="pr2F0G2 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4414 -> 4420[label="",style="solid", color="black", weight=3]; 1179[label="primMulNat (Succ vuz7000) (Succ vuz2000)",fontsize=16,color="black",shape="box"];1179 -> 1232[label="",style="solid", color="black", weight=3]; 1180[label="primMulNat (Succ vuz7000) Zero",fontsize=16,color="black",shape="box"];1180 -> 1233[label="",style="solid", color="black", weight=3]; 1181[label="primMulNat Zero (Succ vuz2000)",fontsize=16,color="black",shape="box"];1181 -> 1234[label="",style="solid", color="black", weight=3]; 1182[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];1182 -> 1235[label="",style="solid", color="black", weight=3]; 2005[label="vuz1050",fontsize=16,color="green",shape="box"];2006[label="vuz11500",fontsize=16,color="green",shape="box"];4030[label="vuz102",fontsize=16,color="green",shape="box"];4031[label="Succ vuz1050",fontsize=16,color="green",shape="box"];4032[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4862[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4862[label="",style="solid", color="blue", weight=9]; 4862 -> 4103[label="",style="solid", color="blue", weight=3]; 4863[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4863[label="",style="solid", color="blue", weight=9]; 4863 -> 4104[label="",style="solid", color="blue", weight=3]; 4864[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4864[label="",style="solid", color="blue", weight=9]; 4864 -> 4105[label="",style="solid", color="blue", weight=3]; 4865[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4865[label="",style="solid", color="blue", weight=9]; 4865 -> 4106[label="",style="solid", color="blue", weight=3]; 4866[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4032 -> 4866[label="",style="solid", color="blue", weight=9]; 4866 -> 4107[label="",style="solid", color="blue", weight=3]; 4033[label="Succ vuz1050",fontsize=16,color="green",shape="box"];4322[label="Succ vuz11500",fontsize=16,color="green",shape="box"];4323[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4867[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4867[label="",style="solid", color="blue", weight=9]; 4867 -> 4360[label="",style="solid", color="blue", weight=3]; 4868[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4868[label="",style="solid", color="blue", weight=9]; 4868 -> 4361[label="",style="solid", color="blue", weight=3]; 4869[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4869[label="",style="solid", color="blue", weight=9]; 4869 -> 4362[label="",style="solid", color="blue", weight=3]; 4870[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4870[label="",style="solid", color="blue", weight=9]; 4870 -> 4363[label="",style="solid", color="blue", weight=3]; 4871[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4323 -> 4871[label="",style="solid", color="blue", weight=9]; 4871 -> 4364[label="",style="solid", color="blue", weight=3]; 4324[label="Succ vuz11500",fontsize=16,color="green",shape="box"];4325[label="vuz102",fontsize=16,color="green",shape="box"];4034[label="vuz102",fontsize=16,color="green",shape="box"];4035[label="Zero",fontsize=16,color="green",shape="box"];4036[label="vuz103 * vuz103",fontsize=16,color="blue",shape="box"];4872[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4872[label="",style="solid", color="blue", weight=9]; 4872 -> 4108[label="",style="solid", color="blue", weight=3]; 4873[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4873[label="",style="solid", color="blue", weight=9]; 4873 -> 4109[label="",style="solid", color="blue", weight=3]; 4874[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4874[label="",style="solid", color="blue", weight=9]; 4874 -> 4110[label="",style="solid", color="blue", weight=3]; 4875[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4875[label="",style="solid", color="blue", weight=9]; 4875 -> 4111[label="",style="solid", color="blue", weight=3]; 4876[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4036 -> 4876[label="",style="solid", color="blue", weight=9]; 4876 -> 4112[label="",style="solid", color="blue", weight=3]; 4037[label="Zero",fontsize=16,color="green",shape="box"];4428[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ (Succ vuz21800))) (primEvenNat (Succ (Succ vuz21800)))",fontsize=16,color="black",shape="box"];4428 -> 4437[label="",style="solid", color="black", weight=3]; 4429[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4429 -> 4438[label="",style="solid", color="black", weight=3]; 4430[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg Zero) True",fontsize=16,color="black",shape="box"];4430 -> 4439[label="",style="solid", color="black", weight=3]; 4351[label="vuz113",fontsize=16,color="green",shape="box"];4352[label="vuz1160",fontsize=16,color="green",shape="box"];4353 -> 397[label="",style="dashed", color="red", weight=0]; 4353[label="vuz111 * vuz111",fontsize=16,color="magenta"];4353 -> 4368[label="",style="dashed", color="magenta", weight=3]; 4354 -> 398[label="",style="dashed", color="red", weight=0]; 4354[label="vuz111 * vuz111",fontsize=16,color="magenta"];4354 -> 4369[label="",style="dashed", color="magenta", weight=3]; 4355 -> 399[label="",style="dashed", color="red", weight=0]; 4355[label="vuz111 * vuz111",fontsize=16,color="magenta"];4355 -> 4370[label="",style="dashed", color="magenta", weight=3]; 4356 -> 400[label="",style="dashed", color="red", weight=0]; 4356[label="vuz111 * vuz111",fontsize=16,color="magenta"];4356 -> 4371[label="",style="dashed", color="magenta", weight=3]; 4357 -> 401[label="",style="dashed", color="red", weight=0]; 4357[label="vuz111 * vuz111",fontsize=16,color="magenta"];4357 -> 4372[label="",style="dashed", color="magenta", weight=3]; 4358[label="vuz113",fontsize=16,color="green",shape="box"];4359[label="vuz1160",fontsize=16,color="green",shape="box"];4477[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ vuz2250))",fontsize=16,color="burlywood",shape="box"];4877[label="vuz2250/Succ vuz22500",fontsize=10,color="white",style="solid",shape="box"];4477 -> 4877[label="",style="solid", color="burlywood", weight=9]; 4877 -> 4489[label="",style="solid", color="burlywood", weight=3]; 4878[label="vuz2250/Zero",fontsize=10,color="white",style="solid",shape="box"];4477 -> 4878[label="",style="solid", color="burlywood", weight=9]; 4878 -> 4490[label="",style="solid", color="burlywood", weight=3]; 4478[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4478 -> 4491[label="",style="solid", color="black", weight=3]; 4419[label="pr2F0G0 (vuz204 * vuz205) vuz204 (Pos (Succ Zero)) True",fontsize=16,color="black",shape="box"];4419 -> 4426[label="",style="solid", color="black", weight=3]; 4420[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4420 -> 4427[label="",style="solid", color="black", weight=3]; 1232 -> 71[label="",style="dashed", color="red", weight=0]; 1232[label="primPlusNat (primMulNat vuz7000 (Succ vuz2000)) (Succ vuz2000)",fontsize=16,color="magenta"];1232 -> 1267[label="",style="dashed", color="magenta", weight=3]; 1232 -> 1268[label="",style="dashed", color="magenta", weight=3]; 1233[label="Zero",fontsize=16,color="green",shape="box"];1234[label="Zero",fontsize=16,color="green",shape="box"];1235[label="Zero",fontsize=16,color="green",shape="box"];4103 -> 397[label="",style="dashed", color="red", weight=0]; 4103[label="vuz103 * vuz103",fontsize=16,color="magenta"];4103 -> 4176[label="",style="dashed", color="magenta", weight=3]; 4104 -> 398[label="",style="dashed", color="red", weight=0]; 4104[label="vuz103 * vuz103",fontsize=16,color="magenta"];4104 -> 4177[label="",style="dashed", color="magenta", weight=3]; 4105 -> 399[label="",style="dashed", color="red", weight=0]; 4105[label="vuz103 * vuz103",fontsize=16,color="magenta"];4105 -> 4178[label="",style="dashed", color="magenta", weight=3]; 4106 -> 400[label="",style="dashed", color="red", weight=0]; 4106[label="vuz103 * vuz103",fontsize=16,color="magenta"];4106 -> 4179[label="",style="dashed", color="magenta", weight=3]; 4107 -> 401[label="",style="dashed", color="red", weight=0]; 4107[label="vuz103 * vuz103",fontsize=16,color="magenta"];4107 -> 4180[label="",style="dashed", color="magenta", weight=3]; 4360 -> 397[label="",style="dashed", color="red", weight=0]; 4360[label="vuz103 * vuz103",fontsize=16,color="magenta"];4360 -> 4373[label="",style="dashed", color="magenta", weight=3]; 4361 -> 398[label="",style="dashed", color="red", weight=0]; 4361[label="vuz103 * vuz103",fontsize=16,color="magenta"];4361 -> 4374[label="",style="dashed", color="magenta", weight=3]; 4362 -> 399[label="",style="dashed", color="red", weight=0]; 4362[label="vuz103 * vuz103",fontsize=16,color="magenta"];4362 -> 4375[label="",style="dashed", color="magenta", weight=3]; 4363 -> 400[label="",style="dashed", color="red", weight=0]; 4363[label="vuz103 * vuz103",fontsize=16,color="magenta"];4363 -> 4376[label="",style="dashed", color="magenta", weight=3]; 4364 -> 401[label="",style="dashed", color="red", weight=0]; 4364[label="vuz103 * vuz103",fontsize=16,color="magenta"];4364 -> 4377[label="",style="dashed", color="magenta", weight=3]; 4108 -> 397[label="",style="dashed", color="red", weight=0]; 4108[label="vuz103 * vuz103",fontsize=16,color="magenta"];4108 -> 4181[label="",style="dashed", color="magenta", weight=3]; 4109 -> 398[label="",style="dashed", color="red", weight=0]; 4109[label="vuz103 * vuz103",fontsize=16,color="magenta"];4109 -> 4182[label="",style="dashed", color="magenta", weight=3]; 4110 -> 399[label="",style="dashed", color="red", weight=0]; 4110[label="vuz103 * vuz103",fontsize=16,color="magenta"];4110 -> 4183[label="",style="dashed", color="magenta", weight=3]; 4111 -> 400[label="",style="dashed", color="red", weight=0]; 4111[label="vuz103 * vuz103",fontsize=16,color="magenta"];4111 -> 4184[label="",style="dashed", color="magenta", weight=3]; 4112 -> 401[label="",style="dashed", color="red", weight=0]; 4112[label="vuz103 * vuz103",fontsize=16,color="magenta"];4112 -> 4185[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4546[label="",style="dashed", color="red", weight=0]; 4437[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ (Succ vuz21800))) (primEvenNat vuz21800)",fontsize=16,color="magenta"];4437 -> 4547[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4548[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4549[label="",style="dashed", color="magenta", weight=3]; 4437 -> 4550[label="",style="dashed", color="magenta", weight=3]; 4438[label="pr2F0G1 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) False",fontsize=16,color="black",shape="box"];4438 -> 4449[label="",style="solid", color="black", weight=3]; 4439[label="pr2F0G (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4439 -> 4450[label="",style="solid", color="black", weight=3]; 4368[label="vuz111",fontsize=16,color="green",shape="box"];4369[label="vuz111",fontsize=16,color="green",shape="box"];4370[label="vuz111",fontsize=16,color="green",shape="box"];4371[label="vuz111",fontsize=16,color="green",shape="box"];4372[label="vuz111",fontsize=16,color="green",shape="box"];4489[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ (Succ vuz22500)))",fontsize=16,color="black",shape="box"];4489 -> 4498[label="",style="solid", color="black", weight=3]; 4490[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4490 -> 4499[label="",style="solid", color="black", weight=3]; 4491[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) True",fontsize=16,color="black",shape="box"];4491 -> 4500[label="",style="solid", color="black", weight=3]; 4426 -> 4581[label="",style="dashed", color="red", weight=0]; 4426[label="pr2F vuz204 (Pos (Succ Zero) - fromInt (Pos (Succ Zero))) (vuz204 * (vuz204 * vuz205))",fontsize=16,color="magenta"];4426 -> 4582[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4583[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4584[label="",style="dashed", color="magenta", weight=3]; 4426 -> 4585[label="",style="dashed", color="magenta", weight=3]; 4427[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4427 -> 4440[label="",style="solid", color="black", weight=3]; 1267 -> 1085[label="",style="dashed", color="red", weight=0]; 1267[label="primMulNat vuz7000 (Succ vuz2000)",fontsize=16,color="magenta"];1267 -> 1275[label="",style="dashed", color="magenta", weight=3]; 1267 -> 1276[label="",style="dashed", color="magenta", weight=3]; 1268[label="Succ vuz2000",fontsize=16,color="green",shape="box"];4176[label="vuz103",fontsize=16,color="green",shape="box"];4177[label="vuz103",fontsize=16,color="green",shape="box"];4178[label="vuz103",fontsize=16,color="green",shape="box"];4179[label="vuz103",fontsize=16,color="green",shape="box"];4180[label="vuz103",fontsize=16,color="green",shape="box"];4373[label="vuz103",fontsize=16,color="green",shape="box"];4374[label="vuz103",fontsize=16,color="green",shape="box"];4375[label="vuz103",fontsize=16,color="green",shape="box"];4376[label="vuz103",fontsize=16,color="green",shape="box"];4377[label="vuz103",fontsize=16,color="green",shape="box"];4181[label="vuz103",fontsize=16,color="green",shape="box"];4182[label="vuz103",fontsize=16,color="green",shape="box"];4183[label="vuz103",fontsize=16,color="green",shape="box"];4184[label="vuz103",fontsize=16,color="green",shape="box"];4185[label="vuz103",fontsize=16,color="green",shape="box"];4547[label="vuz216",fontsize=16,color="green",shape="box"];4548[label="vuz21800",fontsize=16,color="green",shape="box"];4549[label="Succ vuz21800",fontsize=16,color="green",shape="box"];4550[label="vuz217",fontsize=16,color="green",shape="box"];4546[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat vuz231)",fontsize=16,color="burlywood",shape="triangle"];4879[label="vuz231/Succ vuz2310",fontsize=10,color="white",style="solid",shape="box"];4546 -> 4879[label="",style="solid", color="burlywood", weight=9]; 4879 -> 4559[label="",style="solid", color="burlywood", weight=3]; 4880[label="vuz231/Zero",fontsize=10,color="white",style="solid",shape="box"];4546 -> 4880[label="",style="solid", color="burlywood", weight=9]; 4880 -> 4560[label="",style="solid", color="burlywood", weight=3]; 4449[label="pr2F0G0 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) otherwise",fontsize=16,color="black",shape="box"];4449 -> 4461[label="",style="solid", color="black", weight=3]; 4450[label="pr2F0G2 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4450 -> 4462[label="",style="solid", color="black", weight=3]; 4498 -> 4464[label="",style="dashed", color="red", weight=0]; 4498[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) (primEvenNat vuz22500)",fontsize=16,color="magenta"];4498 -> 4518[label="",style="dashed", color="magenta", weight=3]; 4499[label="pr2F0G1 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) False",fontsize=16,color="black",shape="box"];4499 -> 4519[label="",style="solid", color="black", weight=3]; 4500[label="pr2F0G (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4500 -> 4520[label="",style="solid", color="black", weight=3]; 4582[label="vuz205",fontsize=16,color="green",shape="box"];4583 -> 23[label="",style="dashed", color="red", weight=0]; 4583[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4584[label="vuz204",fontsize=16,color="green",shape="box"];4585[label="Zero",fontsize=16,color="green",shape="box"];4581[label="pr2F vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="triangle"];4581 -> 4591[label="",style="solid", color="black", weight=3]; 4440[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (primQuotInt (Pos Zero) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos Zero) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4440 -> 4451[label="",style="solid", color="black", weight=3]; 1275[label="Succ vuz2000",fontsize=16,color="green",shape="box"];1276[label="vuz7000",fontsize=16,color="green",shape="box"];4559[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ vuz2310))",fontsize=16,color="burlywood",shape="box"];4881[label="vuz2310/Succ vuz23100",fontsize=10,color="white",style="solid",shape="box"];4559 -> 4881[label="",style="solid", color="burlywood", weight=9]; 4881 -> 4578[label="",style="solid", color="burlywood", weight=3]; 4882[label="vuz2310/Zero",fontsize=10,color="white",style="solid",shape="box"];4559 -> 4882[label="",style="solid", color="burlywood", weight=9]; 4882 -> 4579[label="",style="solid", color="burlywood", weight=3]; 4560[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat Zero)",fontsize=16,color="black",shape="box"];4560 -> 4580[label="",style="solid", color="black", weight=3]; 4461[label="pr2F0G0 (vuz216 * vuz217) vuz216 (Neg (Succ Zero)) True",fontsize=16,color="black",shape="box"];4461 -> 4482[label="",style="solid", color="black", weight=3]; 4462[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4462 -> 4483[label="",style="solid", color="black", weight=3]; 4518[label="vuz22500",fontsize=16,color="green",shape="box"];4519[label="pr2F0G0 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) otherwise",fontsize=16,color="black",shape="box"];4519 -> 4543[label="",style="solid", color="black", weight=3]; 4520[label="pr2F0G2 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4520 -> 4544[label="",style="solid", color="black", weight=3]; 4591[label="pr2F4 vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="box"];4591 -> 4606[label="",style="solid", color="black", weight=3]; 4451[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (primQuotInt (Pos Zero) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos Zero) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4451 -> 4463[label="",style="solid", color="black", weight=3]; 4578[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ (Succ vuz23100)))",fontsize=16,color="black",shape="box"];4578 -> 4592[label="",style="solid", color="black", weight=3]; 4579[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];4579 -> 4593[label="",style="solid", color="black", weight=3]; 4580[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) True",fontsize=16,color="black",shape="box"];4580 -> 4594[label="",style="solid", color="black", weight=3]; 4482 -> 4655[label="",style="dashed", color="red", weight=0]; 4482[label="pr2F vuz216 (Neg (Succ Zero) - fromInt (Pos (Succ Zero))) (vuz216 * (vuz216 * vuz217))",fontsize=16,color="magenta"];4482 -> 4656[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4657[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4658[label="",style="dashed", color="magenta", weight=3]; 4482 -> 4659[label="",style="dashed", color="magenta", weight=3]; 4483[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg Zero `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4483 -> 4501[label="",style="solid", color="black", weight=3]; 4543[label="pr2F0G0 (vuz222 * vuz223) vuz222 (Pos (Succ vuz224)) True",fontsize=16,color="black",shape="box"];4543 -> 4561[label="",style="solid", color="black", weight=3]; 4544[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4544 -> 4562[label="",style="solid", color="black", weight=3]; 4606[label="pr2F3 (Pos (Succ vuz224) - vuz232 == fromInt (Pos Zero)) vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="black",shape="box"];4606 -> 4626[label="",style="solid", color="black", weight=3]; 4463 -> 1605[label="",style="dashed", color="red", weight=0]; 4463[label="pr2F0G1 (vuz204 * vuz205) (vuz204 * vuz204) (Pos (primDivNatS Zero (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS Zero (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4463 -> 4485[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4486[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4487[label="",style="dashed", color="magenta", weight=3]; 4463 -> 4488[label="",style="dashed", color="magenta", weight=3]; 4592 -> 4546[label="",style="dashed", color="red", weight=0]; 4592[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) (primEvenNat vuz23100)",fontsize=16,color="magenta"];4592 -> 4607[label="",style="dashed", color="magenta", weight=3]; 4593[label="pr2F0G1 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) False",fontsize=16,color="black",shape="box"];4593 -> 4608[label="",style="solid", color="black", weight=3]; 4594[label="pr2F0G (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4594 -> 4609[label="",style="solid", color="black", weight=3]; 4656[label="vuz216",fontsize=16,color="green",shape="box"];4657[label="Zero",fontsize=16,color="green",shape="box"];4658 -> 23[label="",style="dashed", color="red", weight=0]; 4658[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4659[label="vuz217",fontsize=16,color="green",shape="box"];4655[label="pr2F vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="triangle"];4655 -> 4665[label="",style="solid", color="black", weight=3]; 4501[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (primQuotInt (Neg Zero) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg Zero) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4501 -> 4521[label="",style="solid", color="black", weight=3]; 4561 -> 4581[label="",style="dashed", color="red", weight=0]; 4561[label="pr2F vuz222 (Pos (Succ vuz224) - fromInt (Pos (Succ Zero))) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="magenta"];4561 -> 4590[label="",style="dashed", color="magenta", weight=3]; 4562[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Pos (Succ vuz224) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4562 -> 4595[label="",style="solid", color="black", weight=3]; 4626 -> 3789[label="",style="dashed", color="red", weight=0]; 4626[label="pr2F3 (primEqInt (Pos (Succ vuz224) - vuz232) (fromInt (Pos Zero))) vuz222 (Pos (Succ vuz224) - vuz232) (vuz222 * (vuz222 * vuz223))",fontsize=16,color="magenta"];4626 -> 4640[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4641[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4642[label="",style="dashed", color="magenta", weight=3]; 4626 -> 4643[label="",style="dashed", color="magenta", weight=3]; 4485 -> 1226[label="",style="dashed", color="red", weight=0]; 4485[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4485 -> 4511[label="",style="dashed", color="magenta", weight=3]; 4486[label="vuz204",fontsize=16,color="green",shape="box"];4487[label="vuz204 * vuz205",fontsize=16,color="blue",shape="box"];4883[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4883[label="",style="solid", color="blue", weight=9]; 4883 -> 4512[label="",style="solid", color="blue", weight=3]; 4884[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4884[label="",style="solid", color="blue", weight=9]; 4884 -> 4513[label="",style="solid", color="blue", weight=3]; 4885[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4885[label="",style="solid", color="blue", weight=9]; 4885 -> 4514[label="",style="solid", color="blue", weight=3]; 4886[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4886[label="",style="solid", color="blue", weight=9]; 4886 -> 4515[label="",style="solid", color="blue", weight=3]; 4887[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4487 -> 4887[label="",style="solid", color="blue", weight=9]; 4887 -> 4516[label="",style="solid", color="blue", weight=3]; 4488 -> 1226[label="",style="dashed", color="red", weight=0]; 4488[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4488 -> 4517[label="",style="dashed", color="magenta", weight=3]; 4607[label="vuz23100",fontsize=16,color="green",shape="box"];4608[label="pr2F0G0 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) otherwise",fontsize=16,color="black",shape="box"];4608 -> 4627[label="",style="solid", color="black", weight=3]; 4609[label="pr2F0G2 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero))))",fontsize=16,color="black",shape="box"];4609 -> 4628[label="",style="solid", color="black", weight=3]; 4665[label="pr2F4 vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="box"];4665 -> 4684[label="",style="solid", color="black", weight=3]; 4521[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (primQuotInt (Neg Zero) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg Zero) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4521 -> 4545[label="",style="solid", color="black", weight=3]; 4590 -> 23[label="",style="dashed", color="red", weight=0]; 4590[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4595[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (primQuotInt (Pos (Succ vuz224)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Pos (Succ vuz224)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4595 -> 4610[label="",style="solid", color="black", weight=3]; 4640[label="vuz232",fontsize=16,color="green",shape="box"];4641[label="vuz224",fontsize=16,color="green",shape="box"];4642[label="vuz222 * vuz223",fontsize=16,color="blue",shape="box"];4888[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4888[label="",style="solid", color="blue", weight=9]; 4888 -> 4650[label="",style="solid", color="blue", weight=3]; 4889[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4889[label="",style="solid", color="blue", weight=9]; 4889 -> 4651[label="",style="solid", color="blue", weight=3]; 4890[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4890[label="",style="solid", color="blue", weight=9]; 4890 -> 4652[label="",style="solid", color="blue", weight=3]; 4891[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4891[label="",style="solid", color="blue", weight=9]; 4891 -> 4653[label="",style="solid", color="blue", weight=3]; 4892[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4642 -> 4892[label="",style="solid", color="blue", weight=9]; 4892 -> 4654[label="",style="solid", color="blue", weight=3]; 4643[label="vuz222",fontsize=16,color="green",shape="box"];4511[label="Zero",fontsize=16,color="green",shape="box"];4512 -> 1024[label="",style="dashed", color="red", weight=0]; 4512[label="vuz204 * vuz205",fontsize=16,color="magenta"];4512 -> 4533[label="",style="dashed", color="magenta", weight=3]; 4512 -> 4534[label="",style="dashed", color="magenta", weight=3]; 4513 -> 1041[label="",style="dashed", color="red", weight=0]; 4513[label="vuz204 * vuz205",fontsize=16,color="magenta"];4513 -> 4535[label="",style="dashed", color="magenta", weight=3]; 4513 -> 4536[label="",style="dashed", color="magenta", weight=3]; 4514 -> 1051[label="",style="dashed", color="red", weight=0]; 4514[label="vuz204 * vuz205",fontsize=16,color="magenta"];4514 -> 4537[label="",style="dashed", color="magenta", weight=3]; 4514 -> 4538[label="",style="dashed", color="magenta", weight=3]; 4515 -> 1061[label="",style="dashed", color="red", weight=0]; 4515[label="vuz204 * vuz205",fontsize=16,color="magenta"];4515 -> 4539[label="",style="dashed", color="magenta", weight=3]; 4515 -> 4540[label="",style="dashed", color="magenta", weight=3]; 4516 -> 1073[label="",style="dashed", color="red", weight=0]; 4516[label="vuz204 * vuz205",fontsize=16,color="magenta"];4516 -> 4541[label="",style="dashed", color="magenta", weight=3]; 4516 -> 4542[label="",style="dashed", color="magenta", weight=3]; 4517[label="Zero",fontsize=16,color="green",shape="box"];4627[label="pr2F0G0 (vuz228 * vuz229) vuz228 (Neg (Succ vuz230)) True",fontsize=16,color="black",shape="box"];4627 -> 4644[label="",style="solid", color="black", weight=3]; 4628[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))) (even (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4628 -> 4645[label="",style="solid", color="black", weight=3]; 4684[label="pr2F3 (Neg (Succ vuz230) - vuz233 == fromInt (Pos Zero)) vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="black",shape="box"];4684 -> 4696[label="",style="solid", color="black", weight=3]; 4545 -> 1755[label="",style="dashed", color="red", weight=0]; 4545[label="pr2F0G1 (vuz216 * vuz217) (vuz216 * vuz216) (Neg (primDivNatS Zero (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS Zero (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4545 -> 4564[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4565[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4566[label="",style="dashed", color="magenta", weight=3]; 4545 -> 4567[label="",style="dashed", color="magenta", weight=3]; 4610[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (primQuotInt (Pos (Succ vuz224)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Pos (Succ vuz224)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4610 -> 4629[label="",style="solid", color="black", weight=3]; 4650 -> 1024[label="",style="dashed", color="red", weight=0]; 4650[label="vuz222 * vuz223",fontsize=16,color="magenta"];4650 -> 4666[label="",style="dashed", color="magenta", weight=3]; 4650 -> 4667[label="",style="dashed", color="magenta", weight=3]; 4651 -> 1041[label="",style="dashed", color="red", weight=0]; 4651[label="vuz222 * vuz223",fontsize=16,color="magenta"];4651 -> 4668[label="",style="dashed", color="magenta", weight=3]; 4651 -> 4669[label="",style="dashed", color="magenta", weight=3]; 4652 -> 1051[label="",style="dashed", color="red", weight=0]; 4652[label="vuz222 * vuz223",fontsize=16,color="magenta"];4652 -> 4670[label="",style="dashed", color="magenta", weight=3]; 4652 -> 4671[label="",style="dashed", color="magenta", weight=3]; 4653 -> 1061[label="",style="dashed", color="red", weight=0]; 4653[label="vuz222 * vuz223",fontsize=16,color="magenta"];4653 -> 4672[label="",style="dashed", color="magenta", weight=3]; 4653 -> 4673[label="",style="dashed", color="magenta", weight=3]; 4654 -> 1073[label="",style="dashed", color="red", weight=0]; 4654[label="vuz222 * vuz223",fontsize=16,color="magenta"];4654 -> 4674[label="",style="dashed", color="magenta", weight=3]; 4654 -> 4675[label="",style="dashed", color="magenta", weight=3]; 4533[label="vuz204",fontsize=16,color="green",shape="box"];4534[label="vuz205",fontsize=16,color="green",shape="box"];4535[label="vuz204",fontsize=16,color="green",shape="box"];4536[label="vuz205",fontsize=16,color="green",shape="box"];4537[label="vuz205",fontsize=16,color="green",shape="box"];4538[label="vuz204",fontsize=16,color="green",shape="box"];4539[label="vuz204",fontsize=16,color="green",shape="box"];4540[label="vuz205",fontsize=16,color="green",shape="box"];4541[label="vuz205",fontsize=16,color="green",shape="box"];4542[label="vuz204",fontsize=16,color="green",shape="box"];4644 -> 4655[label="",style="dashed", color="red", weight=0]; 4644[label="pr2F vuz228 (Neg (Succ vuz230) - fromInt (Pos (Succ Zero))) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="magenta"];4644 -> 4664[label="",style="dashed", color="magenta", weight=3]; 4645[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))) (primEvenInt (Neg (Succ vuz230) `quot` fromInt (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4645 -> 4676[label="",style="solid", color="black", weight=3]; 4696 -> 4256[label="",style="dashed", color="red", weight=0]; 4696[label="pr2F3 (primEqInt (Neg (Succ vuz230) - vuz233) (fromInt (Pos Zero))) vuz228 (Neg (Succ vuz230) - vuz233) (vuz228 * (vuz228 * vuz229))",fontsize=16,color="magenta"];4696 -> 4698[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4699[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4700[label="",style="dashed", color="magenta", weight=3]; 4696 -> 4701[label="",style="dashed", color="magenta", weight=3]; 4564[label="vuz216",fontsize=16,color="green",shape="box"];4565 -> 1226[label="",style="dashed", color="red", weight=0]; 4565[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4565 -> 4599[label="",style="dashed", color="magenta", weight=3]; 4566[label="vuz216 * vuz217",fontsize=16,color="blue",shape="box"];4893[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4893[label="",style="solid", color="blue", weight=9]; 4893 -> 4600[label="",style="solid", color="blue", weight=3]; 4894[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4894[label="",style="solid", color="blue", weight=9]; 4894 -> 4601[label="",style="solid", color="blue", weight=3]; 4895[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4895[label="",style="solid", color="blue", weight=9]; 4895 -> 4602[label="",style="solid", color="blue", weight=3]; 4896[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4896[label="",style="solid", color="blue", weight=9]; 4896 -> 4603[label="",style="solid", color="blue", weight=3]; 4897[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4566 -> 4897[label="",style="solid", color="blue", weight=9]; 4897 -> 4604[label="",style="solid", color="blue", weight=3]; 4567 -> 1226[label="",style="dashed", color="red", weight=0]; 4567[label="primDivNatS Zero (Succ (Succ Zero))",fontsize=16,color="magenta"];4567 -> 4605[label="",style="dashed", color="magenta", weight=3]; 4629 -> 1605[label="",style="dashed", color="red", weight=0]; 4629[label="pr2F0G1 (vuz222 * vuz223) (vuz222 * vuz222) (Pos (primDivNatS (Succ vuz224) (Succ (Succ Zero)))) (primEvenInt (Pos (primDivNatS (Succ vuz224) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4629 -> 4646[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4647[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4648[label="",style="dashed", color="magenta", weight=3]; 4629 -> 4649[label="",style="dashed", color="magenta", weight=3]; 4666[label="vuz222",fontsize=16,color="green",shape="box"];4667[label="vuz223",fontsize=16,color="green",shape="box"];4668[label="vuz222",fontsize=16,color="green",shape="box"];4669[label="vuz223",fontsize=16,color="green",shape="box"];4670[label="vuz223",fontsize=16,color="green",shape="box"];4671[label="vuz222",fontsize=16,color="green",shape="box"];4672[label="vuz222",fontsize=16,color="green",shape="box"];4673[label="vuz223",fontsize=16,color="green",shape="box"];4674[label="vuz223",fontsize=16,color="green",shape="box"];4675[label="vuz222",fontsize=16,color="green",shape="box"];4664 -> 23[label="",style="dashed", color="red", weight=0]; 4664[label="fromInt (Pos (Succ Zero))",fontsize=16,color="magenta"];4676[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (primQuotInt (Neg (Succ vuz230)) (fromInt (Pos (Succ (Succ Zero))))) (primEvenInt (primQuotInt (Neg (Succ vuz230)) (fromInt (Pos (Succ (Succ Zero))))))",fontsize=16,color="black",shape="box"];4676 -> 4685[label="",style="solid", color="black", weight=3]; 4698[label="vuz230",fontsize=16,color="green",shape="box"];4699[label="vuz228",fontsize=16,color="green",shape="box"];4700[label="vuz233",fontsize=16,color="green",shape="box"];4701[label="vuz228 * vuz229",fontsize=16,color="blue",shape="box"];4898[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4898[label="",style="solid", color="blue", weight=9]; 4898 -> 4706[label="",style="solid", color="blue", weight=3]; 4899[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4899[label="",style="solid", color="blue", weight=9]; 4899 -> 4707[label="",style="solid", color="blue", weight=3]; 4900[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4900[label="",style="solid", color="blue", weight=9]; 4900 -> 4708[label="",style="solid", color="blue", weight=3]; 4901[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4901[label="",style="solid", color="blue", weight=9]; 4901 -> 4709[label="",style="solid", color="blue", weight=3]; 4902[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4701 -> 4902[label="",style="solid", color="blue", weight=9]; 4902 -> 4710[label="",style="solid", color="blue", weight=3]; 4599[label="Zero",fontsize=16,color="green",shape="box"];4600 -> 1024[label="",style="dashed", color="red", weight=0]; 4600[label="vuz216 * vuz217",fontsize=16,color="magenta"];4600 -> 4616[label="",style="dashed", color="magenta", weight=3]; 4600 -> 4617[label="",style="dashed", color="magenta", weight=3]; 4601 -> 1041[label="",style="dashed", color="red", weight=0]; 4601[label="vuz216 * vuz217",fontsize=16,color="magenta"];4601 -> 4618[label="",style="dashed", color="magenta", weight=3]; 4601 -> 4619[label="",style="dashed", color="magenta", weight=3]; 4602 -> 1051[label="",style="dashed", color="red", weight=0]; 4602[label="vuz216 * vuz217",fontsize=16,color="magenta"];4602 -> 4620[label="",style="dashed", color="magenta", weight=3]; 4602 -> 4621[label="",style="dashed", color="magenta", weight=3]; 4603 -> 1061[label="",style="dashed", color="red", weight=0]; 4603[label="vuz216 * vuz217",fontsize=16,color="magenta"];4603 -> 4622[label="",style="dashed", color="magenta", weight=3]; 4603 -> 4623[label="",style="dashed", color="magenta", weight=3]; 4604 -> 1073[label="",style="dashed", color="red", weight=0]; 4604[label="vuz216 * vuz217",fontsize=16,color="magenta"];4604 -> 4624[label="",style="dashed", color="magenta", weight=3]; 4604 -> 4625[label="",style="dashed", color="magenta", weight=3]; 4605[label="Zero",fontsize=16,color="green",shape="box"];4646 -> 1226[label="",style="dashed", color="red", weight=0]; 4646[label="primDivNatS (Succ vuz224) (Succ (Succ Zero))",fontsize=16,color="magenta"];4646 -> 4677[label="",style="dashed", color="magenta", weight=3]; 4647[label="vuz222",fontsize=16,color="green",shape="box"];4648[label="vuz222 * vuz223",fontsize=16,color="blue",shape="box"];4903[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4903[label="",style="solid", color="blue", weight=9]; 4903 -> 4678[label="",style="solid", color="blue", weight=3]; 4904[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4904[label="",style="solid", color="blue", weight=9]; 4904 -> 4679[label="",style="solid", color="blue", weight=3]; 4905[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4905[label="",style="solid", color="blue", weight=9]; 4905 -> 4680[label="",style="solid", color="blue", weight=3]; 4906[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4906[label="",style="solid", color="blue", weight=9]; 4906 -> 4681[label="",style="solid", color="blue", weight=3]; 4907[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4648 -> 4907[label="",style="solid", color="blue", weight=9]; 4907 -> 4682[label="",style="solid", color="blue", weight=3]; 4649 -> 1226[label="",style="dashed", color="red", weight=0]; 4649[label="primDivNatS (Succ vuz224) (Succ (Succ Zero))",fontsize=16,color="magenta"];4649 -> 4683[label="",style="dashed", color="magenta", weight=3]; 4685[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (primQuotInt (Neg (Succ vuz230)) (Pos (Succ (Succ Zero)))) (primEvenInt (primQuotInt (Neg (Succ vuz230)) (Pos (Succ (Succ Zero)))))",fontsize=16,color="black",shape="box"];4685 -> 4697[label="",style="solid", color="black", weight=3]; 4706 -> 1024[label="",style="dashed", color="red", weight=0]; 4706[label="vuz228 * vuz229",fontsize=16,color="magenta"];4706 -> 4718[label="",style="dashed", color="magenta", weight=3]; 4706 -> 4719[label="",style="dashed", color="magenta", weight=3]; 4707 -> 1041[label="",style="dashed", color="red", weight=0]; 4707[label="vuz228 * vuz229",fontsize=16,color="magenta"];4707 -> 4720[label="",style="dashed", color="magenta", weight=3]; 4707 -> 4721[label="",style="dashed", color="magenta", weight=3]; 4708 -> 1051[label="",style="dashed", color="red", weight=0]; 4708[label="vuz228 * vuz229",fontsize=16,color="magenta"];4708 -> 4722[label="",style="dashed", color="magenta", weight=3]; 4708 -> 4723[label="",style="dashed", color="magenta", weight=3]; 4709 -> 1061[label="",style="dashed", color="red", weight=0]; 4709[label="vuz228 * vuz229",fontsize=16,color="magenta"];4709 -> 4724[label="",style="dashed", color="magenta", weight=3]; 4709 -> 4725[label="",style="dashed", color="magenta", weight=3]; 4710 -> 1073[label="",style="dashed", color="red", weight=0]; 4710[label="vuz228 * vuz229",fontsize=16,color="magenta"];4710 -> 4726[label="",style="dashed", color="magenta", weight=3]; 4710 -> 4727[label="",style="dashed", color="magenta", weight=3]; 4616[label="vuz216",fontsize=16,color="green",shape="box"];4617[label="vuz217",fontsize=16,color="green",shape="box"];4618[label="vuz216",fontsize=16,color="green",shape="box"];4619[label="vuz217",fontsize=16,color="green",shape="box"];4620[label="vuz217",fontsize=16,color="green",shape="box"];4621[label="vuz216",fontsize=16,color="green",shape="box"];4622[label="vuz216",fontsize=16,color="green",shape="box"];4623[label="vuz217",fontsize=16,color="green",shape="box"];4624[label="vuz217",fontsize=16,color="green",shape="box"];4625[label="vuz216",fontsize=16,color="green",shape="box"];4677[label="Succ vuz224",fontsize=16,color="green",shape="box"];4678 -> 1024[label="",style="dashed", color="red", weight=0]; 4678[label="vuz222 * vuz223",fontsize=16,color="magenta"];4678 -> 4686[label="",style="dashed", color="magenta", weight=3]; 4678 -> 4687[label="",style="dashed", color="magenta", weight=3]; 4679 -> 1041[label="",style="dashed", color="red", weight=0]; 4679[label="vuz222 * vuz223",fontsize=16,color="magenta"];4679 -> 4688[label="",style="dashed", color="magenta", weight=3]; 4679 -> 4689[label="",style="dashed", color="magenta", weight=3]; 4680 -> 1051[label="",style="dashed", color="red", weight=0]; 4680[label="vuz222 * vuz223",fontsize=16,color="magenta"];4680 -> 4690[label="",style="dashed", color="magenta", weight=3]; 4680 -> 4691[label="",style="dashed", color="magenta", weight=3]; 4681 -> 1061[label="",style="dashed", color="red", weight=0]; 4681[label="vuz222 * vuz223",fontsize=16,color="magenta"];4681 -> 4692[label="",style="dashed", color="magenta", weight=3]; 4681 -> 4693[label="",style="dashed", color="magenta", weight=3]; 4682 -> 1073[label="",style="dashed", color="red", weight=0]; 4682[label="vuz222 * vuz223",fontsize=16,color="magenta"];4682 -> 4694[label="",style="dashed", color="magenta", weight=3]; 4682 -> 4695[label="",style="dashed", color="magenta", weight=3]; 4683[label="Succ vuz224",fontsize=16,color="green",shape="box"];4697 -> 1755[label="",style="dashed", color="red", weight=0]; 4697[label="pr2F0G1 (vuz228 * vuz229) (vuz228 * vuz228) (Neg (primDivNatS (Succ vuz230) (Succ (Succ Zero)))) (primEvenInt (Neg (primDivNatS (Succ vuz230) (Succ (Succ Zero)))))",fontsize=16,color="magenta"];4697 -> 4702[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4703[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4704[label="",style="dashed", color="magenta", weight=3]; 4697 -> 4705[label="",style="dashed", color="magenta", weight=3]; 4718[label="vuz228",fontsize=16,color="green",shape="box"];4719[label="vuz229",fontsize=16,color="green",shape="box"];4720[label="vuz228",fontsize=16,color="green",shape="box"];4721[label="vuz229",fontsize=16,color="green",shape="box"];4722[label="vuz229",fontsize=16,color="green",shape="box"];4723[label="vuz228",fontsize=16,color="green",shape="box"];4724[label="vuz228",fontsize=16,color="green",shape="box"];4725[label="vuz229",fontsize=16,color="green",shape="box"];4726[label="vuz229",fontsize=16,color="green",shape="box"];4727[label="vuz228",fontsize=16,color="green",shape="box"];4686[label="vuz222",fontsize=16,color="green",shape="box"];4687[label="vuz223",fontsize=16,color="green",shape="box"];4688[label="vuz222",fontsize=16,color="green",shape="box"];4689[label="vuz223",fontsize=16,color="green",shape="box"];4690[label="vuz223",fontsize=16,color="green",shape="box"];4691[label="vuz222",fontsize=16,color="green",shape="box"];4692[label="vuz222",fontsize=16,color="green",shape="box"];4693[label="vuz223",fontsize=16,color="green",shape="box"];4694[label="vuz223",fontsize=16,color="green",shape="box"];4695[label="vuz222",fontsize=16,color="green",shape="box"];4702[label="vuz228",fontsize=16,color="green",shape="box"];4703 -> 1226[label="",style="dashed", color="red", weight=0]; 4703[label="primDivNatS (Succ vuz230) (Succ (Succ Zero))",fontsize=16,color="magenta"];4703 -> 4711[label="",style="dashed", color="magenta", weight=3]; 4704[label="vuz228 * vuz229",fontsize=16,color="blue",shape="box"];4908[label="* :: (Ratio a) -> (Ratio a) -> Ratio a",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4908[label="",style="solid", color="blue", weight=9]; 4908 -> 4712[label="",style="solid", color="blue", weight=3]; 4909[label="* :: Int -> Int -> Int",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4909[label="",style="solid", color="blue", weight=9]; 4909 -> 4713[label="",style="solid", color="blue", weight=3]; 4910[label="* :: Integer -> Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4910[label="",style="solid", color="blue", weight=9]; 4910 -> 4714[label="",style="solid", color="blue", weight=3]; 4911[label="* :: Double -> Double -> Double",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4911[label="",style="solid", color="blue", weight=9]; 4911 -> 4715[label="",style="solid", color="blue", weight=3]; 4912[label="* :: Float -> Float -> Float",fontsize=10,color="white",style="solid",shape="box"];4704 -> 4912[label="",style="solid", color="blue", weight=9]; 4912 -> 4716[label="",style="solid", color="blue", weight=3]; 4705 -> 1226[label="",style="dashed", color="red", weight=0]; 4705[label="primDivNatS (Succ vuz230) (Succ (Succ Zero))",fontsize=16,color="magenta"];4705 -> 4717[label="",style="dashed", color="magenta", weight=3]; 4711[label="Succ vuz230",fontsize=16,color="green",shape="box"];4712 -> 1024[label="",style="dashed", color="red", weight=0]; 4712[label="vuz228 * vuz229",fontsize=16,color="magenta"];4712 -> 4728[label="",style="dashed", color="magenta", weight=3]; 4712 -> 4729[label="",style="dashed", color="magenta", weight=3]; 4713 -> 1041[label="",style="dashed", color="red", weight=0]; 4713[label="vuz228 * vuz229",fontsize=16,color="magenta"];4713 -> 4730[label="",style="dashed", color="magenta", weight=3]; 4713 -> 4731[label="",style="dashed", color="magenta", weight=3]; 4714 -> 1051[label="",style="dashed", color="red", weight=0]; 4714[label="vuz228 * vuz229",fontsize=16,color="magenta"];4714 -> 4732[label="",style="dashed", color="magenta", weight=3]; 4714 -> 4733[label="",style="dashed", color="magenta", weight=3]; 4715 -> 1061[label="",style="dashed", color="red", weight=0]; 4715[label="vuz228 * vuz229",fontsize=16,color="magenta"];4715 -> 4734[label="",style="dashed", color="magenta", weight=3]; 4715 -> 4735[label="",style="dashed", color="magenta", weight=3]; 4716 -> 1073[label="",style="dashed", color="red", weight=0]; 4716[label="vuz228 * vuz229",fontsize=16,color="magenta"];4716 -> 4736[label="",style="dashed", color="magenta", weight=3]; 4716 -> 4737[label="",style="dashed", color="magenta", weight=3]; 4717[label="Succ vuz230",fontsize=16,color="green",shape="box"];4728[label="vuz228",fontsize=16,color="green",shape="box"];4729[label="vuz229",fontsize=16,color="green",shape="box"];4730[label="vuz228",fontsize=16,color="green",shape="box"];4731[label="vuz229",fontsize=16,color="green",shape="box"];4732[label="vuz229",fontsize=16,color="green",shape="box"];4733[label="vuz228",fontsize=16,color="green",shape="box"];4734[label="vuz228",fontsize=16,color="green",shape="box"];4735[label="vuz229",fontsize=16,color="green",shape="box"];4736[label="vuz229",fontsize=16,color="green",shape="box"];4737[label="vuz228",fontsize=16,color="green",shape="box"];} ---------------------------------------- (139) TRUE