/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) IFR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 24 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) DependencyGraphProof [EQUIVALENT, 0 ms] (15) QDP (16) QDPOrderProof [EQUIVALENT, 93 ms] (17) QDP (18) DependencyGraphProof [EQUIVALENT, 0 ms] (19) QDP (20) TransformationProof [EQUIVALENT, 0 ms] (21) QDP (22) DependencyGraphProof [EQUIVALENT, 0 ms] (23) AND (24) QDP (25) UsableRulesProof [EQUIVALENT, 0 ms] (26) QDP (27) QReductionProof [EQUIVALENT, 0 ms] (28) QDP (29) QDPOrderProof [EQUIVALENT, 0 ms] (30) QDP (31) DependencyGraphProof [EQUIVALENT, 0 ms] (32) QDP (33) TransformationProof [EQUIVALENT, 0 ms] (34) QDP (35) DependencyGraphProof [EQUIVALENT, 0 ms] (36) QDP (37) UsableRulesProof [EQUIVALENT, 0 ms] (38) QDP (39) QReductionProof [EQUIVALENT, 0 ms] (40) QDP (41) TransformationProof [EQUIVALENT, 0 ms] (42) QDP (43) InductionCalculusProof [EQUIVALENT, 0 ms] (44) QDP (45) NonInfProof [EQUIVALENT, 54 ms] (46) AND (47) QDP (48) DependencyGraphProof [EQUIVALENT, 0 ms] (49) AND (50) QDP (51) QDPSizeChangeProof [EQUIVALENT, 0 ms] (52) YES (53) QDP (54) QDPSizeChangeProof [EQUIVALENT, 0 ms] (55) YES (56) QDP (57) DependencyGraphProof [EQUIVALENT, 0 ms] (58) AND (59) QDP (60) QDPSizeChangeProof [EQUIVALENT, 0 ms] (61) YES (62) QDP (63) QDPSizeChangeProof [EQUIVALENT, 0 ms] (64) YES (65) QDP (66) UsableRulesProof [EQUIVALENT, 0 ms] (67) QDP (68) QReductionProof [EQUIVALENT, 0 ms] (69) QDP (70) TransformationProof [EQUIVALENT, 0 ms] (71) QDP (72) DependencyGraphProof [EQUIVALENT, 0 ms] (73) AND (74) QDP (75) UsableRulesProof [EQUIVALENT, 0 ms] (76) QDP (77) QReductionProof [EQUIVALENT, 0 ms] (78) QDP (79) MRRProof [EQUIVALENT, 0 ms] (80) QDP (81) DependencyGraphProof [EQUIVALENT, 0 ms] (82) TRUE (83) QDP (84) UsableRulesProof [EQUIVALENT, 0 ms] (85) QDP (86) QReductionProof [EQUIVALENT, 0 ms] (87) QDP (88) QDPOrderProof [EQUIVALENT, 0 ms] (89) QDP (90) DependencyGraphProof [EQUIVALENT, 0 ms] (91) QDP (92) InductionCalculusProof [EQUIVALENT, 0 ms] (93) QDP (94) NonInfProof [EQUIVALENT, 0 ms] (95) QDP (96) DependencyGraphProof [EQUIVALENT, 0 ms] (97) QDP (98) QDPSizeChangeProof [EQUIVALENT, 0 ms] (99) YES (100) QDP (101) QDPSizeChangeProof [EQUIVALENT, 0 ms] (102) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) IFR (EQUIVALENT) If Reductions: The following If expression "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" is transformed to "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); primModNatS0 x y False = Succ x; " ---------------------------------------- (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 "absReal x|x >= 0x|otherwise`negate` x; " is transformed to "absReal x = absReal2 x; " "absReal0 x True = `negate` x; " "absReal1 x True = x; absReal1 x False = absReal0 x otherwise; " "absReal2 x = absReal1 x (x >= 0); " The following Function with conditions "gcd' x 0 = x; gcd' x y = gcd' y (x `rem` y); " is transformed to "gcd' x xx = gcd'2 x xx; gcd' x y = gcd'0 x y; " "gcd'0 x y = gcd' y (x `rem` y); " "gcd'1 True x xx = x; gcd'1 xy xz yu = gcd'0 xz yu; " "gcd'2 x xx = gcd'1 (xx == 0) x xx; gcd'2 yv yw = gcd'0 yv yw; " The following Function with conditions "gcd 0 0 = error []; gcd x y = gcd' (abs x) (abs y) where { gcd' x 0 = x; gcd' x y = gcd' y (x `rem` y); } ; " is transformed to "gcd yx yy = gcd3 yx yy; gcd x y = gcd0 x y; " "gcd0 x y = gcd' (abs x) (abs y) where { gcd' x xx = gcd'2 x xx; gcd' x y = gcd'0 x y; ; gcd'0 x y = gcd' y (x `rem` y); ; gcd'1 True x xx = x; gcd'1 xy xz yu = gcd'0 xz yu; ; gcd'2 x xx = gcd'1 (xx == 0) x xx; gcd'2 yv yw = gcd'0 yv yw; } ; " "gcd1 True yx yy = error []; gcd1 yz zu zv = gcd0 zu zv; " "gcd2 True yx yy = gcd1 (yy == 0) yx yy; gcd2 zw zx zy = gcd0 zx zy; " "gcd3 yx yy = gcd2 (yx == 0) yx yy; gcd3 zz vuu = gcd0 zz vuu; " The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "gcd' (abs x) (abs y) where { gcd' x xx = gcd'2 x xx; gcd' x y = gcd'0 x y; ; gcd'0 x y = gcd' y (x `rem` y); ; gcd'1 True x xx = x; gcd'1 xy xz yu = gcd'0 xz yu; ; gcd'2 x xx = gcd'1 (xx == 0) x xx; gcd'2 yv yw = gcd'0 yv yw; } " are unpacked to the following functions on top level "gcd0Gcd'1 True x xx = x; gcd0Gcd'1 xy xz yu = gcd0Gcd'0 xz yu; " "gcd0Gcd'2 x xx = gcd0Gcd'1 (xx == 0) x xx; gcd0Gcd'2 yv yw = gcd0Gcd'0 yv yw; " "gcd0Gcd'0 x y = gcd0Gcd' y (x `rem` y); " "gcd0Gcd' x xx = gcd0Gcd'2 x xx; gcd0Gcd' x y = gcd0Gcd'0 x y; " ---------------------------------------- (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="gcd",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="gcd vuv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="gcd vuv3 vuv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="gcd3 vuv3 vuv4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="gcd2 (vuv3 == fromInt (Pos Zero)) vuv3 vuv4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="gcd2 (primEqInt vuv3 (fromInt (Pos Zero))) vuv3 vuv4",fontsize=16,color="burlywood",shape="box"];1509[label="vuv3/Pos vuv30",fontsize=10,color="white",style="solid",shape="box"];7 -> 1509[label="",style="solid", color="burlywood", weight=9]; 1509 -> 8[label="",style="solid", color="burlywood", weight=3]; 1510[label="vuv3/Neg vuv30",fontsize=10,color="white",style="solid",shape="box"];7 -> 1510[label="",style="solid", color="burlywood", weight=9]; 1510 -> 9[label="",style="solid", color="burlywood", weight=3]; 8[label="gcd2 (primEqInt (Pos vuv30) (fromInt (Pos Zero))) (Pos vuv30) vuv4",fontsize=16,color="burlywood",shape="box"];1511[label="vuv30/Succ vuv300",fontsize=10,color="white",style="solid",shape="box"];8 -> 1511[label="",style="solid", color="burlywood", weight=9]; 1511 -> 10[label="",style="solid", color="burlywood", weight=3]; 1512[label="vuv30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 1512[label="",style="solid", color="burlywood", weight=9]; 1512 -> 11[label="",style="solid", color="burlywood", weight=3]; 9[label="gcd2 (primEqInt (Neg vuv30) (fromInt (Pos Zero))) (Neg vuv30) vuv4",fontsize=16,color="burlywood",shape="box"];1513[label="vuv30/Succ vuv300",fontsize=10,color="white",style="solid",shape="box"];9 -> 1513[label="",style="solid", color="burlywood", weight=9]; 1513 -> 12[label="",style="solid", color="burlywood", weight=3]; 1514[label="vuv30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 1514[label="",style="solid", color="burlywood", weight=9]; 1514 -> 13[label="",style="solid", color="burlywood", weight=3]; 10[label="gcd2 (primEqInt (Pos (Succ vuv300)) (fromInt (Pos Zero))) (Pos (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3]; 11[label="gcd2 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) vuv4",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3]; 12[label="gcd2 (primEqInt (Neg (Succ vuv300)) (fromInt (Pos Zero))) (Neg (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 13[label="gcd2 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) vuv4",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 14[label="gcd2 (primEqInt (Pos (Succ vuv300)) (Pos Zero)) (Pos (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 15[label="gcd2 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) vuv4",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 16[label="gcd2 (primEqInt (Neg (Succ vuv300)) (Pos Zero)) (Neg (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 17[label="gcd2 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) vuv4",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 18[label="gcd2 False (Pos (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 19[label="gcd2 True (Pos Zero) vuv4",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 20[label="gcd2 False (Neg (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 21[label="gcd2 True (Neg Zero) vuv4",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 22[label="gcd0 (Pos (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 23[label="gcd1 (vuv4 == fromInt (Pos Zero)) (Pos Zero) vuv4",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 24[label="gcd0 (Neg (Succ vuv300)) vuv4",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 25[label="gcd1 (vuv4 == fromInt (Pos Zero)) (Neg Zero) vuv4",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 26[label="gcd0Gcd' (abs (Pos (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 27[label="gcd1 (primEqInt vuv4 (fromInt (Pos Zero))) (Pos Zero) vuv4",fontsize=16,color="burlywood",shape="box"];1515[label="vuv4/Pos vuv40",fontsize=10,color="white",style="solid",shape="box"];27 -> 1515[label="",style="solid", color="burlywood", weight=9]; 1515 -> 31[label="",style="solid", color="burlywood", weight=3]; 1516[label="vuv4/Neg vuv40",fontsize=10,color="white",style="solid",shape="box"];27 -> 1516[label="",style="solid", color="burlywood", weight=9]; 1516 -> 32[label="",style="solid", color="burlywood", weight=3]; 28[label="gcd0Gcd' (abs (Neg (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];28 -> 33[label="",style="solid", color="black", weight=3]; 29[label="gcd1 (primEqInt vuv4 (fromInt (Pos Zero))) (Neg Zero) vuv4",fontsize=16,color="burlywood",shape="box"];1517[label="vuv4/Pos vuv40",fontsize=10,color="white",style="solid",shape="box"];29 -> 1517[label="",style="solid", color="burlywood", weight=9]; 1517 -> 34[label="",style="solid", color="burlywood", weight=3]; 1518[label="vuv4/Neg vuv40",fontsize=10,color="white",style="solid",shape="box"];29 -> 1518[label="",style="solid", color="burlywood", weight=9]; 1518 -> 35[label="",style="solid", color="burlywood", weight=3]; 30[label="gcd0Gcd'2 (abs (Pos (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];30 -> 36[label="",style="solid", color="black", weight=3]; 31[label="gcd1 (primEqInt (Pos vuv40) (fromInt (Pos Zero))) (Pos Zero) (Pos vuv40)",fontsize=16,color="burlywood",shape="box"];1519[label="vuv40/Succ vuv400",fontsize=10,color="white",style="solid",shape="box"];31 -> 1519[label="",style="solid", color="burlywood", weight=9]; 1519 -> 37[label="",style="solid", color="burlywood", weight=3]; 1520[label="vuv40/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 1520[label="",style="solid", color="burlywood", weight=9]; 1520 -> 38[label="",style="solid", color="burlywood", weight=3]; 32[label="gcd1 (primEqInt (Neg vuv40) (fromInt (Pos Zero))) (Pos Zero) (Neg vuv40)",fontsize=16,color="burlywood",shape="box"];1521[label="vuv40/Succ vuv400",fontsize=10,color="white",style="solid",shape="box"];32 -> 1521[label="",style="solid", color="burlywood", weight=9]; 1521 -> 39[label="",style="solid", color="burlywood", weight=3]; 1522[label="vuv40/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 1522[label="",style="solid", color="burlywood", weight=9]; 1522 -> 40[label="",style="solid", color="burlywood", weight=3]; 33[label="gcd0Gcd'2 (abs (Neg (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];33 -> 41[label="",style="solid", color="black", weight=3]; 34[label="gcd1 (primEqInt (Pos vuv40) (fromInt (Pos Zero))) (Neg Zero) (Pos vuv40)",fontsize=16,color="burlywood",shape="box"];1523[label="vuv40/Succ vuv400",fontsize=10,color="white",style="solid",shape="box"];34 -> 1523[label="",style="solid", color="burlywood", weight=9]; 1523 -> 42[label="",style="solid", color="burlywood", weight=3]; 1524[label="vuv40/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 1524[label="",style="solid", color="burlywood", weight=9]; 1524 -> 43[label="",style="solid", color="burlywood", weight=3]; 35[label="gcd1 (primEqInt (Neg vuv40) (fromInt (Pos Zero))) (Neg Zero) (Neg vuv40)",fontsize=16,color="burlywood",shape="box"];1525[label="vuv40/Succ vuv400",fontsize=10,color="white",style="solid",shape="box"];35 -> 1525[label="",style="solid", color="burlywood", weight=9]; 1525 -> 44[label="",style="solid", color="burlywood", weight=3]; 1526[label="vuv40/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 1526[label="",style="solid", color="burlywood", weight=9]; 1526 -> 45[label="",style="solid", color="burlywood", weight=3]; 36[label="gcd0Gcd'1 (abs vuv4 == fromInt (Pos Zero)) (abs (Pos (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];36 -> 46[label="",style="solid", color="black", weight=3]; 37[label="gcd1 (primEqInt (Pos (Succ vuv400)) (fromInt (Pos Zero))) (Pos Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3]; 38[label="gcd1 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3]; 39[label="gcd1 (primEqInt (Neg (Succ vuv400)) (fromInt (Pos Zero))) (Pos Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3]; 40[label="gcd1 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];40 -> 50[label="",style="solid", color="black", weight=3]; 41[label="gcd0Gcd'1 (abs vuv4 == fromInt (Pos Zero)) (abs (Neg (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];41 -> 51[label="",style="solid", color="black", weight=3]; 42[label="gcd1 (primEqInt (Pos (Succ vuv400)) (fromInt (Pos Zero))) (Neg Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];42 -> 52[label="",style="solid", color="black", weight=3]; 43[label="gcd1 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];43 -> 53[label="",style="solid", color="black", weight=3]; 44[label="gcd1 (primEqInt (Neg (Succ vuv400)) (fromInt (Pos Zero))) (Neg Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];44 -> 54[label="",style="solid", color="black", weight=3]; 45[label="gcd1 (primEqInt (Neg Zero) (fromInt (Pos Zero))) (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];45 -> 55[label="",style="solid", color="black", weight=3]; 46[label="gcd0Gcd'1 (primEqInt (abs vuv4) (fromInt (Pos Zero))) (abs (Pos (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];46 -> 56[label="",style="solid", color="black", weight=3]; 47[label="gcd1 (primEqInt (Pos (Succ vuv400)) (Pos Zero)) (Pos Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];47 -> 57[label="",style="solid", color="black", weight=3]; 48[label="gcd1 (primEqInt (Pos Zero) (Pos Zero)) (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];48 -> 58[label="",style="solid", color="black", weight=3]; 49[label="gcd1 (primEqInt (Neg (Succ vuv400)) (Pos Zero)) (Pos Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];49 -> 59[label="",style="solid", color="black", weight=3]; 50[label="gcd1 (primEqInt (Neg Zero) (Pos Zero)) (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];50 -> 60[label="",style="solid", color="black", weight=3]; 51[label="gcd0Gcd'1 (primEqInt (abs vuv4) (fromInt (Pos Zero))) (abs (Neg (Succ vuv300))) (abs vuv4)",fontsize=16,color="black",shape="box"];51 -> 61[label="",style="solid", color="black", weight=3]; 52[label="gcd1 (primEqInt (Pos (Succ vuv400)) (Pos Zero)) (Neg Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];52 -> 62[label="",style="solid", color="black", weight=3]; 53[label="gcd1 (primEqInt (Pos Zero) (Pos Zero)) (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];53 -> 63[label="",style="solid", color="black", weight=3]; 54[label="gcd1 (primEqInt (Neg (Succ vuv400)) (Pos Zero)) (Neg Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];54 -> 64[label="",style="solid", color="black", weight=3]; 55[label="gcd1 (primEqInt (Neg Zero) (Pos Zero)) (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];55 -> 65[label="",style="solid", color="black", weight=3]; 56[label="gcd0Gcd'1 (primEqInt (absReal vuv4) (fromInt (Pos Zero))) (abs (Pos (Succ vuv300))) (absReal vuv4)",fontsize=16,color="black",shape="box"];56 -> 66[label="",style="solid", color="black", weight=3]; 57[label="gcd1 False (Pos Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];57 -> 67[label="",style="solid", color="black", weight=3]; 58[label="gcd1 True (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];58 -> 68[label="",style="solid", color="black", weight=3]; 59[label="gcd1 False (Pos Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];59 -> 69[label="",style="solid", color="black", weight=3]; 60[label="gcd1 True (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];60 -> 70[label="",style="solid", color="black", weight=3]; 61[label="gcd0Gcd'1 (primEqInt (absReal vuv4) (fromInt (Pos Zero))) (abs (Neg (Succ vuv300))) (absReal vuv4)",fontsize=16,color="black",shape="box"];61 -> 71[label="",style="solid", color="black", weight=3]; 62[label="gcd1 False (Neg Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];62 -> 72[label="",style="solid", color="black", weight=3]; 63[label="gcd1 True (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];63 -> 73[label="",style="solid", color="black", weight=3]; 64[label="gcd1 False (Neg Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];64 -> 74[label="",style="solid", color="black", weight=3]; 65[label="gcd1 True (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];65 -> 75[label="",style="solid", color="black", weight=3]; 66[label="gcd0Gcd'1 (primEqInt (absReal2 vuv4) (fromInt (Pos Zero))) (abs (Pos (Succ vuv300))) (absReal2 vuv4)",fontsize=16,color="black",shape="box"];66 -> 76[label="",style="solid", color="black", weight=3]; 67[label="gcd0 (Pos Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];67 -> 77[label="",style="solid", color="black", weight=3]; 68[label="error []",fontsize=16,color="black",shape="triangle"];68 -> 78[label="",style="solid", color="black", weight=3]; 69[label="gcd0 (Pos Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];69 -> 79[label="",style="solid", color="black", weight=3]; 70 -> 68[label="",style="dashed", color="red", weight=0]; 70[label="error []",fontsize=16,color="magenta"];71[label="gcd0Gcd'1 (primEqInt (absReal2 vuv4) (fromInt (Pos Zero))) (abs (Neg (Succ vuv300))) (absReal2 vuv4)",fontsize=16,color="black",shape="box"];71 -> 80[label="",style="solid", color="black", weight=3]; 72[label="gcd0 (Neg Zero) (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];72 -> 81[label="",style="solid", color="black", weight=3]; 73 -> 68[label="",style="dashed", color="red", weight=0]; 73[label="error []",fontsize=16,color="magenta"];74[label="gcd0 (Neg Zero) (Neg (Succ vuv400))",fontsize=16,color="black",shape="box"];74 -> 82[label="",style="solid", color="black", weight=3]; 75 -> 68[label="",style="dashed", 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234[label="gcd0Gcd'1 (primEqInt (absReal0 (Neg (Succ vuv400)) True) (fromInt (Pos Zero))) (abs (Pos Zero)) (absReal0 (Neg (Succ vuv400)) True)",fontsize=16,color="black",shape="box"];234 -> 243[label="",style="solid", color="black", weight=3]; 230 -> 212[label="",style="dashed", color="red", weight=0]; 230[label="abs (Neg (Succ vuv300))",fontsize=16,color="magenta"];235[label="absReal2 (Neg (Succ vuv300))",fontsize=16,color="black",shape="box"];235 -> 244[label="",style="solid", color="black", weight=3]; 236 -> 232[label="",style="dashed", color="red", weight=0]; 236[label="gcd0Gcd'1 (primEqInt (Pos (Succ vuv400)) (fromInt (Pos Zero))) (abs (Neg Zero)) (Pos (Succ vuv400))",fontsize=16,color="magenta"];236 -> 245[label="",style="dashed", color="magenta", weight=3]; 236 -> 246[label="",style="dashed", color="magenta", weight=3]; 237[label="gcd0Gcd'1 (primEqInt (absReal0 (Neg (Succ vuv400)) True) (fromInt (Pos Zero))) (abs (Neg Zero)) (absReal0 (Neg (Succ vuv400)) 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260[label="",style="solid", color="black", weight=3]; 249[label="absReal1 (Pos (Succ vuv300)) (compare (Pos (Succ vuv300)) (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];249 -> 261[label="",style="solid", color="black", weight=3]; 240[label="gcd0Gcd'1 (primEqInt (Pos (Succ vuv400)) (Pos Zero)) vuv5 (Pos (Succ vuv400))",fontsize=16,color="black",shape="box"];240 -> 250[label="",style="solid", color="black", weight=3]; 251[label="absReal (Pos Zero)",fontsize=16,color="black",shape="box"];251 -> 262[label="",style="solid", color="black", weight=3]; 255 -> 241[label="",style="dashed", color="red", weight=0]; 255[label="abs (Pos Zero)",fontsize=16,color="magenta"];258[label="absReal1 (Neg (Succ vuv300)) (compare (Neg (Succ vuv300)) (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];258 -> 265[label="",style="solid", color="black", weight=3]; 259[label="absReal (Neg Zero)",fontsize=16,color="black",shape="box"];259 -> 266[label="",style="solid", 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1540[label="",style="solid", color="burlywood", weight=9]; 1540 -> 281[label="",style="solid", color="burlywood", weight=3]; 276[label="absReal1 (Pos (Succ vuv300)) (not (primCmpInt (Pos (Succ vuv300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];276 -> 282[label="",style="solid", color="black", weight=3]; 270[label="vuv400",fontsize=16,color="green",shape="box"];271[label="vuv5",fontsize=16,color="green",shape="box"];277[label="absReal1 (Pos Zero) (compare (Pos Zero) (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];277 -> 283[label="",style="solid", color="black", weight=3]; 278[label="absReal1 (Neg (Succ vuv300)) (not (primCmpInt (Neg (Succ vuv300)) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];278 -> 284[label="",style="solid", color="black", weight=3]; 279[label="absReal1 (Neg Zero) (compare (Neg Zero) (fromInt (Pos Zero)) /= LT)",fontsize=16,color="black",shape="box"];279 -> 285[label="",style="solid", color="black", weight=3]; 280[label="gcd0Gcd'1 (primEqInt (primRemInt (Pos vuv60) (Pos (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (primRemInt (Pos vuv60) (Pos (Succ vuv400)))",fontsize=16,color="black",shape="box"];280 -> 286[label="",style="solid", color="black", weight=3]; 281[label="gcd0Gcd'1 (primEqInt (primRemInt (Neg vuv60) (Pos (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (primRemInt (Neg vuv60) (Pos (Succ vuv400)))",fontsize=16,color="black",shape="box"];281 -> 287[label="",style="solid", color="black", weight=3]; 282[label="absReal1 (Pos (Succ vuv300)) (not (primCmpNat (Succ vuv300) Zero == LT))",fontsize=16,color="black",shape="box"];282 -> 288[label="",style="solid", color="black", weight=3]; 283[label="absReal1 (Pos Zero) (not (compare (Pos Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];283 -> 289[label="",style="solid", color="black", weight=3]; 284[label="absReal1 (Neg (Succ vuv300)) (not (LT == 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298[label="",style="solid", color="black", weight=3]; 291[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (fromInt (Pos Zero)) == LT))",fontsize=16,color="black",shape="box"];291 -> 299[label="",style="solid", color="black", weight=3]; 292[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (Succ vuv600) (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (Pos (primModNatS (Succ vuv600) (Succ vuv400)))",fontsize=16,color="black",shape="box"];292 -> 300[label="",style="solid", color="black", weight=3]; 293[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS Zero (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (Pos (primModNatS Zero (Succ vuv400)))",fontsize=16,color="black",shape="box"];293 -> 301[label="",style="solid", color="black", weight=3]; 294[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS (Succ vuv600) (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (Neg (primModNatS (Succ vuv600) (Succ vuv400)))",fontsize=16,color="black",shape="box"];294 -> 302[label="",style="solid", color="black", weight=3]; 295[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS Zero (Succ vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (Neg (primModNatS Zero (Succ vuv400)))",fontsize=16,color="black",shape="box"];295 -> 303[label="",style="solid", color="black", weight=3]; 296[label="absReal1 (Pos (Succ vuv300)) (not False)",fontsize=16,color="black",shape="box"];296 -> 304[label="",style="solid", color="black", weight=3]; 297[label="absReal1 (Pos Zero) (not (primCmpInt (Pos Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];297 -> 305[label="",style="solid", color="black", weight=3]; 298[label="absReal1 (Neg (Succ vuv300)) False",fontsize=16,color="black",shape="box"];298 -> 306[label="",style="solid", color="black", weight=3]; 299[label="absReal1 (Neg Zero) (not (primCmpInt (Neg Zero) (Pos Zero) == LT))",fontsize=16,color="black",shape="box"];299 -> 307[label="",style="solid", color="black", weight=3]; 300[label="gcd0Gcd'1 (primEqInt 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306[label="absReal0 (Neg (Succ vuv300)) otherwise",fontsize=16,color="black",shape="box"];306 -> 316[label="",style="solid", color="black", weight=3]; 307[label="absReal1 (Neg Zero) (not (EQ == LT))",fontsize=16,color="black",shape="box"];307 -> 317[label="",style="solid", color="black", weight=3]; 308[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6000) vuv400 (primGEqNatS (Succ vuv6000) vuv400))) (fromInt (Pos Zero))) (Pos (Succ vuv400)) (Pos (primModNatS0 (Succ vuv6000) vuv400 (primGEqNatS (Succ vuv6000) vuv400)))",fontsize=16,color="burlywood",shape="box"];1549[label="vuv400/Succ vuv4000",fontsize=10,color="white",style="solid",shape="box"];308 -> 1549[label="",style="solid", color="burlywood", weight=9]; 1549 -> 318[label="",style="solid", color="burlywood", weight=3]; 1550[label="vuv400/Zero",fontsize=10,color="white",style="solid",shape="box"];308 -> 1550[label="",style="solid", color="burlywood", weight=9]; 1550 -> 319[label="",style="solid", color="burlywood", 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color="black", weight=3]; 347 -> 232[label="",style="dashed", color="red", weight=0]; 347[label="gcd0Gcd'1 (primEqInt (Pos (Succ Zero)) (fromInt (Pos Zero))) (Pos (Succ (Succ vuv4000))) (Pos (Succ Zero))",fontsize=16,color="magenta"];347 -> 361[label="",style="dashed", color="magenta", weight=3]; 347 -> 362[label="",style="dashed", color="magenta", weight=3]; 348 -> 286[label="",style="dashed", color="red", weight=0]; 348[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (primMinusNatS Zero Zero) (Succ Zero))) (fromInt (Pos Zero))) (Pos (Succ Zero)) (Pos (primModNatS (primMinusNatS Zero Zero) (Succ Zero)))",fontsize=16,color="magenta"];348 -> 363[label="",style="dashed", color="magenta", weight=3]; 348 -> 364[label="",style="dashed", color="magenta", weight=3]; 866[label="vuv6000",fontsize=16,color="green",shape="box"];867[label="Succ 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907[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS0 (Succ vuv40) vuv41 (primGEqNatS Zero vuv43))) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (primModNatS0 (Succ vuv40) vuv41 (primGEqNatS Zero vuv43)))",fontsize=16,color="burlywood",shape="box"];1567[label="vuv43/Succ vuv430",fontsize=10,color="white",style="solid",shape="box"];907 -> 1567[label="",style="solid", color="burlywood", weight=9]; 1567 -> 915[label="",style="solid", color="burlywood", weight=3]; 1568[label="vuv43/Zero",fontsize=10,color="white",style="solid",shape="box"];907 -> 1568[label="",style="solid", color="burlywood", weight=9]; 1568 -> 916[label="",style="solid", color="burlywood", weight=3]; 667 -> 287[label="",style="dashed", color="red", weight=0]; 667[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS (primMinusNatS (Succ vuv26) vuv27) (Succ vuv27))) (fromInt (Pos Zero))) (Pos (Succ vuv27)) (Neg (primModNatS (primMinusNatS (Succ vuv26) vuv27) (Succ vuv27)))",fontsize=16,color="magenta"];667 -> 700[label="",style="dashed", color="magenta", weight=3]; 667 -> 701[label="",style="dashed", color="magenta", weight=3]; 688[label="Zero",fontsize=16,color="green",shape="box"];689[label="Succ vuv4000",fontsize=16,color="green",shape="box"];687[label="gcd0Gcd'1 (primEqInt (Neg (Succ vuv29)) (fromInt (Pos Zero))) (Pos (Succ vuv30)) (Neg (Succ vuv29))",fontsize=16,color="black",shape="triangle"];687 -> 702[label="",style="solid", color="black", weight=3]; 372 -> 363[label="",style="dashed", color="red", weight=0]; 372[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];373[label="Zero",fontsize=16,color="green",shape="box"];908[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS (Succ vuv370) (Succ vuv380)))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS (Succ vuv370) (Succ vuv380))))",fontsize=16,color="black",shape="box"];908 -> 917[label="",style="solid", color="black", weight=3]; 909[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS (Succ vuv370) Zero))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS (Succ vuv370) Zero)))",fontsize=16,color="black",shape="box"];909 -> 918[label="",style="solid", color="black", weight=3]; 910[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS Zero (Succ vuv380)))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS Zero (Succ vuv380))))",fontsize=16,color="black",shape="box"];910 -> 919[label="",style="solid", color="black", weight=3]; 911[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS Zero Zero))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS Zero Zero)))",fontsize=16,color="black",shape="box"];911 -> 920[label="",style="solid", color="black", weight=3]; 668[label="primMinusNatS (Succ vuv23) 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vuv27",fontsize=16,color="magenta"];700 -> 713[label="",style="dashed", color="magenta", weight=3]; 700 -> 714[label="",style="dashed", color="magenta", weight=3]; 701[label="vuv27",fontsize=16,color="green",shape="box"];702[label="gcd0Gcd'1 (primEqInt (Neg (Succ vuv29)) (Pos Zero)) (Pos (Succ vuv30)) (Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];702 -> 715[label="",style="solid", color="black", weight=3]; 917 -> 821[label="",style="dashed", color="red", weight=0]; 917[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS vuv370 vuv380))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (primModNatS0 (Succ vuv35) vuv36 (primGEqNatS vuv370 vuv380)))",fontsize=16,color="magenta"];917 -> 927[label="",style="dashed", color="magenta", weight=3]; 917 -> 928[label="",style="dashed", color="magenta", weight=3]; 918 -> 613[label="",style="dashed", color="red", weight=0]; 918[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv35) vuv36 True)) (fromInt (Pos 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716[label="",style="solid", color="black", weight=3]; 704[label="primMinusNatS (Succ vuv23) Zero",fontsize=16,color="black",shape="box"];704 -> 717[label="",style="solid", color="black", weight=3]; 923 -> 865[label="",style="dashed", color="red", weight=0]; 923[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS0 (Succ vuv40) vuv41 (primGEqNatS vuv420 vuv430))) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (primModNatS0 (Succ vuv40) vuv41 (primGEqNatS vuv420 vuv430)))",fontsize=16,color="magenta"];923 -> 938[label="",style="dashed", color="magenta", weight=3]; 923 -> 939[label="",style="dashed", color="magenta", weight=3]; 924 -> 644[label="",style="dashed", color="red", weight=0]; 924[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS0 (Succ vuv40) vuv41 True)) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (primModNatS0 (Succ vuv40) vuv41 True))",fontsize=16,color="magenta"];924 -> 940[label="",style="dashed", color="magenta", weight=3]; 924 -> 941[label="",style="dashed", color="magenta", weight=3]; 925[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS0 (Succ vuv40) vuv41 False)) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (primModNatS0 (Succ vuv40) vuv41 False))",fontsize=16,color="black",shape="box"];925 -> 942[label="",style="solid", color="black", weight=3]; 926 -> 644[label="",style="dashed", color="red", weight=0]; 926[label="gcd0Gcd'1 (primEqInt (Neg (primModNatS0 (Succ vuv40) vuv41 True)) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (primModNatS0 (Succ vuv40) vuv41 True))",fontsize=16,color="magenta"];926 -> 943[label="",style="dashed", color="magenta", weight=3]; 926 -> 944[label="",style="dashed", color="magenta", weight=3]; 713[label="vuv26",fontsize=16,color="green",shape="box"];714[label="vuv27",fontsize=16,color="green",shape="box"];715[label="gcd0Gcd'1 False (Pos (Succ vuv30)) (Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];715 -> 732[label="",style="solid", color="black", weight=3]; 927[label="vuv370",fontsize=16,color="green",shape="box"];928[label="vuv380",fontsize=16,color="green",shape="box"];929[label="vuv35",fontsize=16,color="green",shape="box"];930[label="vuv36",fontsize=16,color="green",shape="box"];931 -> 232[label="",style="dashed", color="red", weight=0]; 931[label="gcd0Gcd'1 (primEqInt (Pos (Succ (Succ vuv35))) (fromInt (Pos Zero))) (Pos (Succ vuv36)) (Pos (Succ (Succ vuv35)))",fontsize=16,color="magenta"];931 -> 945[label="",style="dashed", color="magenta", weight=3]; 931 -> 946[label="",style="dashed", color="magenta", weight=3]; 932[label="vuv35",fontsize=16,color="green",shape="box"];933[label="vuv36",fontsize=16,color="green",shape="box"];716[label="primMinusNatS vuv23 vuv240",fontsize=16,color="burlywood",shape="triangle"];1571[label="vuv23/Succ vuv230",fontsize=10,color="white",style="solid",shape="box"];716 -> 1571[label="",style="solid", color="burlywood", weight=9]; 1571 -> 733[label="",style="solid", color="burlywood", weight=3]; 1572[label="vuv23/Zero",fontsize=10,color="white",style="solid",shape="box"];716 -> 1572[label="",style="solid", color="burlywood", weight=9]; 1572 -> 734[label="",style="solid", color="burlywood", weight=3]; 717[label="Succ vuv23",fontsize=16,color="green",shape="box"];938[label="vuv420",fontsize=16,color="green",shape="box"];939[label="vuv430",fontsize=16,color="green",shape="box"];940[label="vuv40",fontsize=16,color="green",shape="box"];941[label="vuv41",fontsize=16,color="green",shape="box"];942 -> 687[label="",style="dashed", color="red", weight=0]; 942[label="gcd0Gcd'1 (primEqInt (Neg (Succ (Succ vuv40))) (fromInt (Pos Zero))) (Pos (Succ vuv41)) (Neg (Succ (Succ vuv40)))",fontsize=16,color="magenta"];942 -> 951[label="",style="dashed", color="magenta", weight=3]; 942 -> 952[label="",style="dashed", color="magenta", weight=3]; 943[label="vuv40",fontsize=16,color="green",shape="box"];944[label="vuv41",fontsize=16,color="green",shape="box"];732[label="gcd0Gcd'0 (Pos (Succ vuv30)) (Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];732 -> 747[label="",style="solid", color="black", weight=3]; 945[label="Pos (Succ vuv36)",fontsize=16,color="green",shape="box"];946[label="Succ vuv35",fontsize=16,color="green",shape="box"];733[label="primMinusNatS (Succ vuv230) vuv240",fontsize=16,color="burlywood",shape="box"];1573[label="vuv240/Succ vuv2400",fontsize=10,color="white",style="solid",shape="box"];733 -> 1573[label="",style="solid", color="burlywood", weight=9]; 1573 -> 748[label="",style="solid", color="burlywood", weight=3]; 1574[label="vuv240/Zero",fontsize=10,color="white",style="solid",shape="box"];733 -> 1574[label="",style="solid", color="burlywood", weight=9]; 1574 -> 749[label="",style="solid", color="burlywood", weight=3]; 734[label="primMinusNatS Zero vuv240",fontsize=16,color="burlywood",shape="box"];1575[label="vuv240/Succ vuv2400",fontsize=10,color="white",style="solid",shape="box"];734 -> 1575[label="",style="solid", color="burlywood", weight=9]; 1575 -> 750[label="",style="solid", color="burlywood", weight=3]; 1576[label="vuv240/Zero",fontsize=10,color="white",style="solid",shape="box"];734 -> 1576[label="",style="solid", color="burlywood", weight=9]; 1576 -> 751[label="",style="solid", color="burlywood", weight=3]; 951[label="Succ vuv40",fontsize=16,color="green",shape="box"];952[label="vuv41",fontsize=16,color="green",shape="box"];747[label="gcd0Gcd' (Neg (Succ vuv29)) (Pos (Succ vuv30) `rem` Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];747 -> 760[label="",style="solid", color="black", weight=3]; 748[label="primMinusNatS (Succ vuv230) (Succ vuv2400)",fontsize=16,color="black",shape="box"];748 -> 761[label="",style="solid", color="black", weight=3]; 749[label="primMinusNatS (Succ vuv230) Zero",fontsize=16,color="black",shape="box"];749 -> 762[label="",style="solid", color="black", weight=3]; 750[label="primMinusNatS Zero (Succ vuv2400)",fontsize=16,color="black",shape="box"];750 -> 763[label="",style="solid", color="black", weight=3]; 751[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];751 -> 764[label="",style="solid", color="black", weight=3]; 760[label="gcd0Gcd'2 (Neg (Succ vuv29)) (Pos (Succ vuv30) `rem` Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];760 -> 787[label="",style="solid", color="black", weight=3]; 761 -> 716[label="",style="dashed", color="red", weight=0]; 761[label="primMinusNatS vuv230 vuv2400",fontsize=16,color="magenta"];761 -> 788[label="",style="dashed", color="magenta", weight=3]; 761 -> 789[label="",style="dashed", color="magenta", weight=3]; 762[label="Succ vuv230",fontsize=16,color="green",shape="box"];763[label="Zero",fontsize=16,color="green",shape="box"];764[label="Zero",fontsize=16,color="green",shape="box"];787[label="gcd0Gcd'1 (Pos (Succ vuv30) `rem` Neg (Succ vuv29) == fromInt (Pos Zero)) (Neg (Succ vuv29)) (Pos (Succ vuv30) `rem` Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];787 -> 805[label="",style="solid", color="black", weight=3]; 788[label="vuv230",fontsize=16,color="green",shape="box"];789[label="vuv2400",fontsize=16,color="green",shape="box"];805[label="gcd0Gcd'1 (primEqInt (Pos (Succ vuv30) `rem` Neg (Succ vuv29)) (fromInt (Pos Zero))) (Neg (Succ vuv29)) (Pos (Succ vuv30) `rem` Neg (Succ vuv29))",fontsize=16,color="black",shape="box"];805 -> 816[label="",style="solid", color="black", weight=3]; 816[label="gcd0Gcd'1 (primEqInt (primRemInt (Pos (Succ vuv30)) (Neg (Succ vuv29))) (fromInt (Pos Zero))) (Neg (Succ vuv29)) (primRemInt (Pos (Succ vuv30)) (Neg (Succ vuv29)))",fontsize=16,color="black",shape="box"];816 -> 864[label="",style="solid", color="black", weight=3]; 864 -> 1173[label="",style="dashed", color="red", weight=0]; 864[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (Succ vuv30) (Succ vuv29))) (fromInt (Pos Zero))) (Neg (Succ vuv29)) (Pos (primModNatS (Succ vuv30) (Succ vuv29)))",fontsize=16,color="magenta"];864 -> 1174[label="",style="dashed", color="magenta", weight=3]; 864 -> 1175[label="",style="dashed", color="magenta", weight=3]; 864 -> 1176[label="",style="dashed", color="magenta", weight=3]; 1174[label="Succ vuv30",fontsize=16,color="green",shape="box"];1175[label="vuv29",fontsize=16,color="green",shape="box"];1176[label="Succ vuv30",fontsize=16,color="green",shape="box"];1173[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS vuv64 (Succ vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS vuv63 (Succ vuv62)))",fontsize=16,color="burlywood",shape="triangle"];1577[label="vuv64/Succ vuv640",fontsize=10,color="white",style="solid",shape="box"];1173 -> 1577[label="",style="solid", color="burlywood", weight=9]; 1577 -> 1185[label="",style="solid", color="burlywood", weight=3]; 1578[label="vuv64/Zero",fontsize=10,color="white",style="solid",shape="box"];1173 -> 1578[label="",style="solid", color="burlywood", weight=9]; 1578 -> 1186[label="",style="solid", color="burlywood", weight=3]; 1185[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (Succ vuv640) (Succ vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS vuv63 (Succ vuv62)))",fontsize=16,color="black",shape="box"];1185 -> 1187[label="",style="solid", color="black", weight=3]; 1186[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS Zero (Succ vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS vuv63 (Succ vuv62)))",fontsize=16,color="black",shape="box"];1186 -> 1188[label="",style="solid", color="black", weight=3]; 1187[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 vuv640 vuv62 (primGEqNatS vuv640 vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS0 vuv640 vuv62 (primGEqNatS vuv640 vuv62)))",fontsize=16,color="burlywood",shape="box"];1579[label="vuv640/Succ vuv6400",fontsize=10,color="white",style="solid",shape="box"];1187 -> 1579[label="",style="solid", color="burlywood", weight=9]; 1579 -> 1189[label="",style="solid", color="burlywood", weight=3]; 1580[label="vuv640/Zero",fontsize=10,color="white",style="solid",shape="box"];1187 -> 1580[label="",style="solid", color="burlywood", weight=9]; 1580 -> 1190[label="",style="solid", color="burlywood", weight=3]; 1188[label="gcd0Gcd'1 (primEqInt (Pos Zero) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos Zero)",fontsize=16,color="black",shape="box"];1188 -> 1191[label="",style="solid", color="black", weight=3]; 1189[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6400) vuv62 (primGEqNatS (Succ vuv6400) vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS0 (Succ vuv6400) vuv62 (primGEqNatS (Succ vuv6400) vuv62)))",fontsize=16,color="burlywood",shape="box"];1581[label="vuv62/Succ vuv620",fontsize=10,color="white",style="solid",shape="box"];1189 -> 1581[label="",style="solid", color="burlywood", weight=9]; 1581 -> 1192[label="",style="solid", color="burlywood", weight=3]; 1582[label="vuv62/Zero",fontsize=10,color="white",style="solid",shape="box"];1189 -> 1582[label="",style="solid", color="burlywood", weight=9]; 1582 -> 1193[label="",style="solid", color="burlywood", weight=3]; 1190[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 Zero vuv62 (primGEqNatS Zero vuv62))) (fromInt (Pos Zero))) (Neg (Succ vuv62)) (Pos (primModNatS0 Zero vuv62 (primGEqNatS Zero vuv62)))",fontsize=16,color="burlywood",shape="box"];1583[label="vuv62/Succ vuv620",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1583[label="",style="solid", color="burlywood", weight=9]; 1583 -> 1194[label="",style="solid", color="burlywood", weight=3]; 1584[label="vuv62/Zero",fontsize=10,color="white",style="solid",shape="box"];1190 -> 1584[label="",style="solid", color="burlywood", weight=9]; 1584 -> 1195[label="",style="solid", color="burlywood", weight=3]; 1191[label="gcd0Gcd'1 (primEqInt (Pos Zero) (Pos Zero)) (Neg (Succ vuv62)) (Pos Zero)",fontsize=16,color="black",shape="box"];1191 -> 1196[label="",style="solid", color="black", weight=3]; 1192[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6400) (Succ vuv620) (primGEqNatS (Succ vuv6400) (Succ vuv620)))) (fromInt (Pos Zero))) (Neg (Succ (Succ vuv620))) (Pos (primModNatS0 (Succ vuv6400) (Succ vuv620) (primGEqNatS (Succ vuv6400) (Succ vuv620))))",fontsize=16,color="black",shape="box"];1192 -> 1197[label="",style="solid", color="black", weight=3]; 1193[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6400) Zero (primGEqNatS (Succ vuv6400) Zero))) (fromInt (Pos Zero))) (Neg (Succ Zero)) (Pos (primModNatS0 (Succ vuv6400) Zero (primGEqNatS (Succ vuv6400) Zero)))",fontsize=16,color="black",shape="box"];1193 -> 1198[label="",style="solid", color="black", weight=3]; 1194[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 Zero (Succ vuv620) (primGEqNatS Zero (Succ vuv620)))) (fromInt (Pos Zero))) (Neg (Succ (Succ vuv620))) (Pos (primModNatS0 Zero (Succ vuv620) (primGEqNatS Zero (Succ vuv620))))",fontsize=16,color="black",shape="box"];1194 -> 1199[label="",style="solid", color="black", weight=3]; 1195[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 Zero Zero (primGEqNatS Zero Zero))) (fromInt (Pos Zero))) (Neg (Succ Zero)) (Pos (primModNatS0 Zero Zero (primGEqNatS Zero Zero)))",fontsize=16,color="black",shape="box"];1195 -> 1200[label="",style="solid", color="black", weight=3]; 1196[label="gcd0Gcd'1 True (Neg (Succ vuv62)) (Pos Zero)",fontsize=16,color="black",shape="box"];1196 -> 1201[label="",style="solid", color="black", weight=3]; 1197 -> 1449[label="",style="dashed", color="red", weight=0]; 1197[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6400) (Succ vuv620) (primGEqNatS vuv6400 vuv620))) (fromInt (Pos Zero))) (Neg (Succ (Succ vuv620))) (Pos (primModNatS0 (Succ vuv6400) (Succ vuv620) (primGEqNatS vuv6400 vuv620)))",fontsize=16,color="magenta"];1197 -> 1450[label="",style="dashed", color="magenta", weight=3]; 1197 -> 1451[label="",style="dashed", color="magenta", weight=3]; 1197 -> 1452[label="",style="dashed", color="magenta", weight=3]; 1197 -> 1453[label="",style="dashed", color="magenta", weight=3]; 1198 -> 1357[label="",style="dashed", color="red", weight=0]; 1198[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv6400) Zero True)) (fromInt (Pos Zero))) (Neg (Succ Zero)) (Pos (primModNatS0 (Succ vuv6400) Zero True))",fontsize=16,color="magenta"];1198 -> 1358[label="",style="dashed", color="magenta", weight=3]; 1198 -> 1359[label="",style="dashed", color="magenta", weight=3]; 1199[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 Zero (Succ vuv620) False)) (fromInt (Pos Zero))) (Neg (Succ (Succ vuv620))) (Pos (primModNatS0 Zero (Succ vuv620) False))",fontsize=16,color="black",shape="box"];1199 -> 1205[label="",style="solid", color="black", weight=3]; 1200[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 Zero Zero True)) (fromInt (Pos Zero))) (Neg (Succ Zero)) (Pos (primModNatS0 Zero Zero True))",fontsize=16,color="black",shape="box"];1200 -> 1206[label="",style="solid", color="black", weight=3]; 1201[label="Neg (Succ vuv62)",fontsize=16,color="green",shape="box"];1450[label="vuv6400",fontsize=16,color="green",shape="box"];1451[label="Succ vuv620",fontsize=16,color="green",shape="box"];1452[label="vuv6400",fontsize=16,color="green",shape="box"];1453[label="vuv620",fontsize=16,color="green",shape="box"];1449[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS vuv80 vuv81))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS vuv80 vuv81)))",fontsize=16,color="burlywood",shape="triangle"];1585[label="vuv80/Succ vuv800",fontsize=10,color="white",style="solid",shape="box"];1449 -> 1585[label="",style="solid", color="burlywood", weight=9]; 1585 -> 1490[label="",style="solid", color="burlywood", weight=3]; 1586[label="vuv80/Zero",fontsize=10,color="white",style="solid",shape="box"];1449 -> 1586[label="",style="solid", color="burlywood", weight=9]; 1586 -> 1491[label="",style="solid", color="burlywood", weight=3]; 1358[label="vuv6400",fontsize=16,color="green",shape="box"];1359[label="Zero",fontsize=16,color="green",shape="box"];1357[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv72) vuv73 True)) (fromInt (Pos Zero))) (Neg (Succ vuv73)) (Pos (primModNatS0 (Succ vuv72) vuv73 True))",fontsize=16,color="black",shape="triangle"];1357 -> 1380[label="",style="solid", color="black", weight=3]; 1205 -> 232[label="",style="dashed", color="red", weight=0]; 1205[label="gcd0Gcd'1 (primEqInt (Pos (Succ Zero)) (fromInt (Pos Zero))) (Neg (Succ (Succ vuv620))) (Pos (Succ Zero))",fontsize=16,color="magenta"];1205 -> 1214[label="",style="dashed", color="magenta", weight=3]; 1205 -> 1215[label="",style="dashed", color="magenta", weight=3]; 1206 -> 1173[label="",style="dashed", color="red", weight=0]; 1206[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (primMinusNatS Zero Zero) (Succ Zero))) (fromInt (Pos Zero))) (Neg (Succ Zero)) (Pos (primModNatS (primMinusNatS Zero Zero) (Succ Zero)))",fontsize=16,color="magenta"];1206 -> 1216[label="",style="dashed", color="magenta", weight=3]; 1206 -> 1217[label="",style="dashed", color="magenta", weight=3]; 1206 -> 1218[label="",style="dashed", color="magenta", weight=3]; 1490[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) vuv81))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) vuv81)))",fontsize=16,color="burlywood",shape="box"];1587[label="vuv81/Succ vuv810",fontsize=10,color="white",style="solid",shape="box"];1490 -> 1587[label="",style="solid", color="burlywood", weight=9]; 1587 -> 1492[label="",style="solid", color="burlywood", weight=3]; 1588[label="vuv81/Zero",fontsize=10,color="white",style="solid",shape="box"];1490 -> 1588[label="",style="solid", color="burlywood", weight=9]; 1588 -> 1493[label="",style="solid", color="burlywood", weight=3]; 1491[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero vuv81))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero vuv81)))",fontsize=16,color="burlywood",shape="box"];1589[label="vuv81/Succ vuv810",fontsize=10,color="white",style="solid",shape="box"];1491 -> 1589[label="",style="solid", color="burlywood", weight=9]; 1589 -> 1494[label="",style="solid", color="burlywood", weight=3]; 1590[label="vuv81/Zero",fontsize=10,color="white",style="solid",shape="box"];1491 -> 1590[label="",style="solid", color="burlywood", weight=9]; 1590 -> 1495[label="",style="solid", color="burlywood", weight=3]; 1380 -> 1173[label="",style="dashed", color="red", weight=0]; 1380[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS (primMinusNatS (Succ vuv72) vuv73) (Succ vuv73))) (fromInt (Pos Zero))) (Neg (Succ vuv73)) (Pos (primModNatS (primMinusNatS (Succ vuv72) vuv73) (Succ vuv73)))",fontsize=16,color="magenta"];1380 -> 1387[label="",style="dashed", color="magenta", weight=3]; 1380 -> 1388[label="",style="dashed", color="magenta", weight=3]; 1380 -> 1389[label="",style="dashed", color="magenta", weight=3]; 1214[label="Neg (Succ (Succ vuv620))",fontsize=16,color="green",shape="box"];1215[label="Zero",fontsize=16,color="green",shape="box"];1216 -> 716[label="",style="dashed", color="red", weight=0]; 1216[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];1216 -> 1227[label="",style="dashed", color="magenta", weight=3]; 1216 -> 1228[label="",style="dashed", color="magenta", weight=3]; 1217[label="Zero",fontsize=16,color="green",shape="box"];1218 -> 716[label="",style="dashed", color="red", weight=0]; 1218[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];1218 -> 1229[label="",style="dashed", color="magenta", weight=3]; 1218 -> 1230[label="",style="dashed", color="magenta", weight=3]; 1492[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) (Succ vuv810)))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) (Succ vuv810))))",fontsize=16,color="black",shape="box"];1492 -> 1496[label="",style="solid", color="black", weight=3]; 1493[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) Zero))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS (Succ vuv800) Zero)))",fontsize=16,color="black",shape="box"];1493 -> 1497[label="",style="solid", color="black", weight=3]; 1494[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero (Succ vuv810)))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero (Succ vuv810))))",fontsize=16,color="black",shape="box"];1494 -> 1498[label="",style="solid", color="black", weight=3]; 1495[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero Zero))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS Zero Zero)))",fontsize=16,color="black",shape="box"];1495 -> 1499[label="",style="solid", color="black", weight=3]; 1387 -> 716[label="",style="dashed", color="red", weight=0]; 1387[label="primMinusNatS (Succ vuv72) vuv73",fontsize=16,color="magenta"];1387 -> 1394[label="",style="dashed", color="magenta", weight=3]; 1387 -> 1395[label="",style="dashed", color="magenta", weight=3]; 1388[label="vuv73",fontsize=16,color="green",shape="box"];1389 -> 716[label="",style="dashed", color="red", weight=0]; 1389[label="primMinusNatS (Succ vuv72) vuv73",fontsize=16,color="magenta"];1389 -> 1396[label="",style="dashed", color="magenta", weight=3]; 1389 -> 1397[label="",style="dashed", color="magenta", weight=3]; 1227[label="Zero",fontsize=16,color="green",shape="box"];1228[label="Zero",fontsize=16,color="green",shape="box"];1229[label="Zero",fontsize=16,color="green",shape="box"];1230[label="Zero",fontsize=16,color="green",shape="box"];1496 -> 1449[label="",style="dashed", color="red", weight=0]; 1496[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS vuv800 vuv810))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 (primGEqNatS vuv800 vuv810)))",fontsize=16,color="magenta"];1496 -> 1500[label="",style="dashed", color="magenta", weight=3]; 1496 -> 1501[label="",style="dashed", color="magenta", weight=3]; 1497 -> 1357[label="",style="dashed", color="red", weight=0]; 1497[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 True)) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 True))",fontsize=16,color="magenta"];1497 -> 1502[label="",style="dashed", color="magenta", weight=3]; 1497 -> 1503[label="",style="dashed", color="magenta", weight=3]; 1498[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 False)) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 False))",fontsize=16,color="black",shape="box"];1498 -> 1504[label="",style="solid", color="black", weight=3]; 1499 -> 1357[label="",style="dashed", color="red", weight=0]; 1499[label="gcd0Gcd'1 (primEqInt (Pos (primModNatS0 (Succ vuv78) vuv79 True)) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (primModNatS0 (Succ vuv78) vuv79 True))",fontsize=16,color="magenta"];1499 -> 1505[label="",style="dashed", color="magenta", weight=3]; 1499 -> 1506[label="",style="dashed", color="magenta", weight=3]; 1394[label="Succ vuv72",fontsize=16,color="green",shape="box"];1395[label="vuv73",fontsize=16,color="green",shape="box"];1396[label="Succ vuv72",fontsize=16,color="green",shape="box"];1397[label="vuv73",fontsize=16,color="green",shape="box"];1500[label="vuv800",fontsize=16,color="green",shape="box"];1501[label="vuv810",fontsize=16,color="green",shape="box"];1502[label="vuv78",fontsize=16,color="green",shape="box"];1503[label="vuv79",fontsize=16,color="green",shape="box"];1504 -> 232[label="",style="dashed", color="red", weight=0]; 1504[label="gcd0Gcd'1 (primEqInt (Pos (Succ (Succ vuv78))) (fromInt (Pos Zero))) (Neg (Succ vuv79)) (Pos (Succ (Succ vuv78)))",fontsize=16,color="magenta"];1504 -> 1507[label="",style="dashed", color="magenta", weight=3]; 1504 -> 1508[label="",style="dashed", color="magenta", weight=3]; 1505[label="vuv78",fontsize=16,color="green",shape="box"];1506[label="vuv79",fontsize=16,color="green",shape="box"];1507[label="Neg (Succ vuv79)",fontsize=16,color="green",shape="box"];1508[label="Succ vuv78",fontsize=16,color="green",shape="box"];} ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'16(Succ(Zero), Zero) -> new_gcd0Gcd'16(new_primMinusNatS0, Zero) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'17(Succ(Zero), Zero, vuv63) -> new_gcd0Gcd'17(new_primMinusNatS2(Zero, Zero), Zero, new_primMinusNatS2(Zero, Zero)) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'13(vuv40, vuv41, Zero, Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'1(Succ(Zero), Zero) -> new_gcd0Gcd'1(new_primMinusNatS0, Zero) new_gcd0Gcd'12(vuv400, vuv5) -> new_gcd0Gcd'0(vuv5, vuv400) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'14(vuv26, vuv27) -> new_gcd0Gcd'16(new_primMinusNatS1(vuv26, vuv27), vuv27) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'17(Succ(Zero), Succ(vuv620), vuv63) -> new_gcd0Gcd'12(Zero, Neg(Succ(Succ(vuv620)))) new_gcd0Gcd'0(Neg(Succ(Zero)), Zero) -> new_gcd0Gcd'16(new_primMinusNatS0, Zero) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'16(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Zero)), Zero) -> new_gcd0Gcd'1(new_primMinusNatS0, Zero) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes. ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(vuv400, vuv5) -> new_gcd0Gcd'0(vuv5, vuv400) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'14(vuv26, vuv27) -> new_gcd0Gcd'16(new_primMinusNatS1(vuv26, vuv27), vuv27) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'17(Succ(Zero), Succ(vuv620), vuv63) -> new_gcd0Gcd'12(Zero, Neg(Succ(Succ(vuv620)))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'13(vuv40, vuv41, Zero, Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'16(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) new_gcd0Gcd'13(vuv40, vuv41, Zero, Zero) -> new_gcd0Gcd'14(vuv40, vuv41) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'14(vuv6000, Zero) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = x_1 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(new_gcd0Gcd'0(x_1, x_2)) = x_1 POL(new_gcd0Gcd'1(x_1, x_2)) = 0 POL(new_gcd0Gcd'10(x_1, x_2, x_3, x_4)) = 0 POL(new_gcd0Gcd'11(x_1, x_2)) = 0 POL(new_gcd0Gcd'12(x_1, x_2)) = x_2 POL(new_gcd0Gcd'13(x_1, x_2, x_3, x_4)) = 2 + x_1 POL(new_gcd0Gcd'14(x_1, x_2)) = 1 + x_1 POL(new_gcd0Gcd'15(x_1, x_2)) = 1 + x_1 POL(new_gcd0Gcd'16(x_1, x_2)) = x_1 POL(new_gcd0Gcd'17(x_1, x_2, x_3)) = 1 + x_2 POL(new_gcd0Gcd'18(x_1, x_2, x_3, x_4)) = 1 + x_2 POL(new_gcd0Gcd'19(x_1, x_2)) = 1 + x_2 POL(new_primMinusNatS1(x_1, x_2)) = 1 + x_1 POL(new_primMinusNatS2(x_1, x_2)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(vuv400, vuv5) -> new_gcd0Gcd'0(vuv5, vuv400) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'14(vuv26, vuv27) -> new_gcd0Gcd'16(new_primMinusNatS1(vuv26, vuv27), vuv27) new_gcd0Gcd'16(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'17(Succ(Zero), Succ(vuv620), vuv63) -> new_gcd0Gcd'12(Zero, Neg(Succ(Succ(vuv620)))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'16(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'12(vuv400, vuv5) -> new_gcd0Gcd'0(vuv5, vuv400) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'17(Succ(Zero), Succ(vuv620), vuv63) -> new_gcd0Gcd'12(Zero, Neg(Succ(Succ(vuv620)))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_gcd0Gcd'12(vuv400, vuv5) -> new_gcd0Gcd'0(vuv5, vuv400) we obtained the following new rules [LPAR04]: (new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero),new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero)) (new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)),new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0))) (new_gcd0Gcd'12(Zero, Neg(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Neg(Succ(Succ(z0))), Zero),new_gcd0Gcd'12(Zero, Neg(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Neg(Succ(Succ(z0))), Zero)) (new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)),new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0))) ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'17(Succ(Zero), Succ(vuv620), vuv63) -> new_gcd0Gcd'12(Zero, Neg(Succ(Succ(vuv620)))) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'12(Zero, Neg(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Neg(Succ(Succ(z0))), Zero) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 2 less nodes. ---------------------------------------- (23) Complex Obligation (AND) ---------------------------------------- (24) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (25) 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. ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) 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_primMinusNatS1(x0, Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS0 ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_gcd0Gcd'18(vuv78, vuv79, Zero, Zero) -> new_gcd0Gcd'19(vuv78, vuv79) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Zero, vuv63) -> new_gcd0Gcd'19(vuv6400, Zero) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Zero) -> new_gcd0Gcd'19(vuv78, vuv79) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Neg(x_1)) = 0 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(new_gcd0Gcd'0(x_1, x_2)) = 1 + x_2 POL(new_gcd0Gcd'12(x_1, x_2)) = 1 + x_1 POL(new_gcd0Gcd'13(x_1, x_2, x_3, x_4)) = 1 + x_2 POL(new_gcd0Gcd'15(x_1, x_2)) = 1 + x_2 POL(new_gcd0Gcd'17(x_1, x_2, x_3)) = x_1 POL(new_gcd0Gcd'18(x_1, x_2, x_3, x_4)) = 2 + x_1 POL(new_gcd0Gcd'19(x_1, x_2)) = 1 + x_1 POL(new_primMinusNatS2(x_1, x_2)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero ---------------------------------------- (30) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'19(vuv72, vuv73) -> new_gcd0Gcd'17(new_primMinusNatS2(Succ(vuv72), vuv73), vuv73, new_primMinusNatS2(Succ(vuv72), vuv73)) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (31) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (32) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (33) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_gcd0Gcd'15(vuv29, vuv30) -> new_gcd0Gcd'17(Succ(vuv30), vuv29, Succ(vuv30)) we obtained the following new rules [LPAR04]: (new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)),new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1))) (new_gcd0Gcd'15(Zero, Succ(z0)) -> new_gcd0Gcd'17(Succ(Succ(z0)), Zero, Succ(Succ(z0))),new_gcd0Gcd'15(Zero, Succ(z0)) -> new_gcd0Gcd'17(Succ(Succ(z0)), Zero, Succ(Succ(z0)))) ---------------------------------------- (34) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'0(Neg(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'15(Zero, Succ(vuv4000)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'15(Zero, Succ(z0)) -> new_gcd0Gcd'17(Succ(Succ(z0)), Zero, Succ(Succ(z0))) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (35) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (36) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (37) 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. ---------------------------------------- (38) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) R is empty. The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (39) 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_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) ---------------------------------------- (40) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (41) TransformationProof (EQUIVALENT) By instantiating [LPAR04] the rule new_gcd0Gcd'17(Succ(Succ(vuv6400)), Succ(vuv620), vuv63) -> new_gcd0Gcd'18(vuv6400, Succ(vuv620), vuv6400, vuv620) we obtained the following new rules [LPAR04]: (new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0),new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0)) ---------------------------------------- (42) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (43) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) the following chains were created: *We consider the chain new_gcd0Gcd'18(x3, x4, Zero, Succ(x5)) -> new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4))), new_gcd0Gcd'12(Succ(x6), Neg(Succ(x7))) -> new_gcd0Gcd'0(Neg(Succ(x7)), Succ(x6)) which results in the following constraint: (1) (new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))=new_gcd0Gcd'12(Succ(x6), Neg(Succ(x7))) ==> new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) For Pair new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) the following chains were created: *We consider the chain new_gcd0Gcd'12(Succ(x30), Neg(Succ(x31))) -> new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30)), new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x33)) -> new_gcd0Gcd'13(x32, Succ(x33), x32, x33) which results in the following constraint: (1) (new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30))=new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x33)) ==> new_gcd0Gcd'12(Succ(x30), Neg(Succ(x31)))_>=_new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'12(Succ(x30), Neg(Succ(Succ(x32))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x30))) For Pair new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) the following chains were created: *We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(x50))), Succ(x51)) -> new_gcd0Gcd'13(x50, Succ(x51), x50, x51), new_gcd0Gcd'13(x52, x53, Zero, Succ(x54)) -> new_gcd0Gcd'15(Succ(x52), x53) which results in the following constraint: (1) (new_gcd0Gcd'13(x50, Succ(x51), x50, x51)=new_gcd0Gcd'13(x52, x53, Zero, Succ(x54)) ==> new_gcd0Gcd'0(Neg(Succ(Succ(x50))), Succ(x51))_>=_new_gcd0Gcd'13(x50, Succ(x51), x50, x51)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Succ(Succ(x54)))_>=_new_gcd0Gcd'13(Zero, Succ(Succ(x54)), Zero, Succ(x54))) *We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(x59))), Succ(x60)) -> new_gcd0Gcd'13(x59, Succ(x60), x59, x60), new_gcd0Gcd'13(x61, x62, Succ(x63), Succ(x64)) -> new_gcd0Gcd'13(x61, x62, x63, x64) which results in the following constraint: (1) (new_gcd0Gcd'13(x59, Succ(x60), x59, x60)=new_gcd0Gcd'13(x61, x62, Succ(x63), Succ(x64)) ==> new_gcd0Gcd'0(Neg(Succ(Succ(x59))), Succ(x60))_>=_new_gcd0Gcd'13(x59, Succ(x60), x59, x60)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x63)))), Succ(Succ(x64)))_>=_new_gcd0Gcd'13(Succ(x63), Succ(Succ(x64)), Succ(x63), Succ(x64))) For Pair new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) the following chains were created: *We consider the chain new_gcd0Gcd'13(x79, x80, Zero, Succ(x81)) -> new_gcd0Gcd'15(Succ(x79), x80), new_gcd0Gcd'15(Succ(x82), x83) -> new_gcd0Gcd'17(Succ(x83), Succ(x82), Succ(x83)) which results in the following constraint: (1) (new_gcd0Gcd'15(Succ(x79), x80)=new_gcd0Gcd'15(Succ(x82), x83) ==> new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) For Pair new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) the following chains were created: *We consider the chain new_gcd0Gcd'15(Succ(x107), x108) -> new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108)), new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x110), Succ(Succ(x109))) -> new_gcd0Gcd'18(x109, Succ(x110), x109, x110) which results in the following constraint: (1) (new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108))=new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x110), Succ(Succ(x109))) ==> new_gcd0Gcd'15(Succ(x107), x108)_>=_new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'15(Succ(x107), Succ(x109))_>=_new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x107), Succ(Succ(x109)))) For Pair new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) the following chains were created: *We consider the chain new_gcd0Gcd'18(x111, x112, Succ(x113), Succ(x114)) -> new_gcd0Gcd'18(x111, x112, x113, x114), new_gcd0Gcd'18(x115, x116, Zero, Succ(x117)) -> new_gcd0Gcd'12(Succ(x115), Neg(Succ(x116))) which results in the following constraint: (1) (new_gcd0Gcd'18(x111, x112, x113, x114)=new_gcd0Gcd'18(x115, x116, Zero, Succ(x117)) ==> new_gcd0Gcd'18(x111, x112, Succ(x113), Succ(x114))_>=_new_gcd0Gcd'18(x111, x112, x113, x114)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x111, x112, Succ(Zero), Succ(Succ(x117)))_>=_new_gcd0Gcd'18(x111, x112, Zero, Succ(x117))) *We consider the chain new_gcd0Gcd'18(x134, x135, Succ(x136), Succ(x137)) -> new_gcd0Gcd'18(x134, x135, x136, x137), new_gcd0Gcd'18(x138, x139, Succ(x140), Succ(x141)) -> new_gcd0Gcd'18(x138, x139, x140, x141) which results in the following constraint: (1) (new_gcd0Gcd'18(x134, x135, x136, x137)=new_gcd0Gcd'18(x138, x139, Succ(x140), Succ(x141)) ==> new_gcd0Gcd'18(x134, x135, Succ(x136), Succ(x137))_>=_new_gcd0Gcd'18(x134, x135, x136, x137)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x134, x135, Succ(Succ(x140)), Succ(Succ(x141)))_>=_new_gcd0Gcd'18(x134, x135, Succ(x140), Succ(x141))) For Pair new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) the following chains were created: *We consider the chain new_gcd0Gcd'13(x162, x163, Succ(x164), Succ(x165)) -> new_gcd0Gcd'13(x162, x163, x164, x165), new_gcd0Gcd'13(x166, x167, Zero, Succ(x168)) -> new_gcd0Gcd'15(Succ(x166), x167) which results in the following constraint: (1) (new_gcd0Gcd'13(x162, x163, x164, x165)=new_gcd0Gcd'13(x166, x167, Zero, Succ(x168)) ==> new_gcd0Gcd'13(x162, x163, Succ(x164), Succ(x165))_>=_new_gcd0Gcd'13(x162, x163, x164, x165)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x162, x163, Succ(Zero), Succ(Succ(x168)))_>=_new_gcd0Gcd'13(x162, x163, Zero, Succ(x168))) *We consider the chain new_gcd0Gcd'13(x177, x178, Succ(x179), Succ(x180)) -> new_gcd0Gcd'13(x177, x178, x179, x180), new_gcd0Gcd'13(x181, x182, Succ(x183), Succ(x184)) -> new_gcd0Gcd'13(x181, x182, x183, x184) which results in the following constraint: (1) (new_gcd0Gcd'13(x177, x178, x179, x180)=new_gcd0Gcd'13(x181, x182, Succ(x183), Succ(x184)) ==> new_gcd0Gcd'13(x177, x178, Succ(x179), Succ(x180))_>=_new_gcd0Gcd'13(x177, x178, x179, x180)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x177, x178, Succ(Succ(x183)), Succ(Succ(x184)))_>=_new_gcd0Gcd'13(x177, x178, Succ(x183), Succ(x184))) For Pair new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) the following chains were created: *We consider the chain new_gcd0Gcd'17(Succ(Succ(x189)), Succ(x190), Succ(Succ(x189))) -> new_gcd0Gcd'18(x189, Succ(x190), x189, x190), new_gcd0Gcd'18(x191, x192, Zero, Succ(x193)) -> new_gcd0Gcd'12(Succ(x191), Neg(Succ(x192))) which results in the following constraint: (1) (new_gcd0Gcd'18(x189, Succ(x190), x189, x190)=new_gcd0Gcd'18(x191, x192, Zero, Succ(x193)) ==> new_gcd0Gcd'17(Succ(Succ(x189)), Succ(x190), Succ(Succ(x189)))_>=_new_gcd0Gcd'18(x189, Succ(x190), x189, x190)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'17(Succ(Succ(Zero)), Succ(Succ(x193)), Succ(Succ(Zero)))_>=_new_gcd0Gcd'18(Zero, Succ(Succ(x193)), Zero, Succ(x193))) *We consider the chain new_gcd0Gcd'17(Succ(Succ(x202)), Succ(x203), Succ(Succ(x202))) -> new_gcd0Gcd'18(x202, Succ(x203), x202, x203), new_gcd0Gcd'18(x204, x205, Succ(x206), Succ(x207)) -> new_gcd0Gcd'18(x204, x205, x206, x207) which results in the following constraint: (1) (new_gcd0Gcd'18(x202, Succ(x203), x202, x203)=new_gcd0Gcd'18(x204, x205, Succ(x206), Succ(x207)) ==> new_gcd0Gcd'17(Succ(Succ(x202)), Succ(x203), Succ(Succ(x202)))_>=_new_gcd0Gcd'18(x202, Succ(x203), x202, x203)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'17(Succ(Succ(Succ(x206))), Succ(Succ(x207)), Succ(Succ(Succ(x206))))_>=_new_gcd0Gcd'18(Succ(x206), Succ(Succ(x207)), Succ(x206), Succ(x207))) To summarize, we get the following constraints P__>=_ for the following pairs. *new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) *(new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) *new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) *(new_gcd0Gcd'12(Succ(x30), Neg(Succ(Succ(x32))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x30))) *new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) *(new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Succ(Succ(x54)))_>=_new_gcd0Gcd'13(Zero, Succ(Succ(x54)), Zero, Succ(x54))) *(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x63)))), Succ(Succ(x64)))_>=_new_gcd0Gcd'13(Succ(x63), Succ(Succ(x64)), Succ(x63), Succ(x64))) *new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) *(new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) *new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) *(new_gcd0Gcd'15(Succ(x107), Succ(x109))_>=_new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x107), Succ(Succ(x109)))) *new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) *(new_gcd0Gcd'18(x111, x112, Succ(Zero), Succ(Succ(x117)))_>=_new_gcd0Gcd'18(x111, x112, Zero, Succ(x117))) *(new_gcd0Gcd'18(x134, x135, Succ(Succ(x140)), Succ(Succ(x141)))_>=_new_gcd0Gcd'18(x134, x135, Succ(x140), Succ(x141))) *new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) *(new_gcd0Gcd'13(x162, x163, Succ(Zero), Succ(Succ(x168)))_>=_new_gcd0Gcd'13(x162, x163, Zero, Succ(x168))) *(new_gcd0Gcd'13(x177, x178, Succ(Succ(x183)), Succ(Succ(x184)))_>=_new_gcd0Gcd'13(x177, x178, Succ(x183), Succ(x184))) *new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) *(new_gcd0Gcd'17(Succ(Succ(Zero)), Succ(Succ(x193)), Succ(Succ(Zero)))_>=_new_gcd0Gcd'18(Zero, Succ(Succ(x193)), Zero, Succ(x193))) *(new_gcd0Gcd'17(Succ(Succ(Succ(x206))), Succ(Succ(x207)), Succ(Succ(Succ(x206))))_>=_new_gcd0Gcd'18(Succ(x206), Succ(Succ(x207)), Succ(x206), Succ(x207))) 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_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (45) NonInfProof (EQUIVALENT) The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) the following chains were created: *We consider the chain new_gcd0Gcd'18(x3, x4, Zero, Succ(x5)) -> new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4))), new_gcd0Gcd'12(Succ(x6), Neg(Succ(x7))) -> new_gcd0Gcd'0(Neg(Succ(x7)), Succ(x6)) which results in the following constraint: (1) (new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))=new_gcd0Gcd'12(Succ(x6), Neg(Succ(x7))) ==> new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) For Pair new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) the following chains were created: *We consider the chain new_gcd0Gcd'12(Succ(x30), Neg(Succ(x31))) -> new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30)), new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x33)) -> new_gcd0Gcd'13(x32, Succ(x33), x32, x33) which results in the following constraint: (1) (new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30))=new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x33)) ==> new_gcd0Gcd'12(Succ(x30), Neg(Succ(x31)))_>=_new_gcd0Gcd'0(Neg(Succ(x31)), Succ(x30))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'12(Succ(x30), Neg(Succ(Succ(x32))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x30))) For Pair new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) the following chains were created: *We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(x50))), Succ(x51)) -> new_gcd0Gcd'13(x50, Succ(x51), x50, x51), new_gcd0Gcd'13(x52, x53, Zero, Succ(x54)) -> new_gcd0Gcd'15(Succ(x52), x53) which results in the following constraint: (1) (new_gcd0Gcd'13(x50, Succ(x51), x50, x51)=new_gcd0Gcd'13(x52, x53, Zero, Succ(x54)) ==> new_gcd0Gcd'0(Neg(Succ(Succ(x50))), Succ(x51))_>=_new_gcd0Gcd'13(x50, Succ(x51), x50, x51)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Succ(Succ(x54)))_>=_new_gcd0Gcd'13(Zero, Succ(Succ(x54)), Zero, Succ(x54))) *We consider the chain new_gcd0Gcd'0(Neg(Succ(Succ(x59))), Succ(x60)) -> new_gcd0Gcd'13(x59, Succ(x60), x59, x60), new_gcd0Gcd'13(x61, x62, Succ(x63), Succ(x64)) -> new_gcd0Gcd'13(x61, x62, x63, x64) which results in the following constraint: (1) (new_gcd0Gcd'13(x59, Succ(x60), x59, x60)=new_gcd0Gcd'13(x61, x62, Succ(x63), Succ(x64)) ==> new_gcd0Gcd'0(Neg(Succ(Succ(x59))), Succ(x60))_>=_new_gcd0Gcd'13(x59, Succ(x60), x59, x60)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x63)))), Succ(Succ(x64)))_>=_new_gcd0Gcd'13(Succ(x63), Succ(Succ(x64)), Succ(x63), Succ(x64))) For Pair new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) the following chains were created: *We consider the chain new_gcd0Gcd'13(x79, x80, Zero, Succ(x81)) -> new_gcd0Gcd'15(Succ(x79), x80), new_gcd0Gcd'15(Succ(x82), x83) -> new_gcd0Gcd'17(Succ(x83), Succ(x82), Succ(x83)) which results in the following constraint: (1) (new_gcd0Gcd'15(Succ(x79), x80)=new_gcd0Gcd'15(Succ(x82), x83) ==> new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) For Pair new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) the following chains were created: *We consider the chain new_gcd0Gcd'15(Succ(x107), x108) -> new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108)), new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x110), Succ(Succ(x109))) -> new_gcd0Gcd'18(x109, Succ(x110), x109, x110) which results in the following constraint: (1) (new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108))=new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x110), Succ(Succ(x109))) ==> new_gcd0Gcd'15(Succ(x107), x108)_>=_new_gcd0Gcd'17(Succ(x108), Succ(x107), Succ(x108))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'15(Succ(x107), Succ(x109))_>=_new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x107), Succ(Succ(x109)))) For Pair new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) the following chains were created: *We consider the chain new_gcd0Gcd'18(x111, x112, Succ(x113), Succ(x114)) -> new_gcd0Gcd'18(x111, x112, x113, x114), new_gcd0Gcd'18(x115, x116, Zero, Succ(x117)) -> new_gcd0Gcd'12(Succ(x115), Neg(Succ(x116))) which results in the following constraint: (1) (new_gcd0Gcd'18(x111, x112, x113, x114)=new_gcd0Gcd'18(x115, x116, Zero, Succ(x117)) ==> new_gcd0Gcd'18(x111, x112, Succ(x113), Succ(x114))_>=_new_gcd0Gcd'18(x111, x112, x113, x114)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x111, x112, Succ(Zero), Succ(Succ(x117)))_>=_new_gcd0Gcd'18(x111, x112, Zero, Succ(x117))) *We consider the chain new_gcd0Gcd'18(x134, x135, Succ(x136), Succ(x137)) -> new_gcd0Gcd'18(x134, x135, x136, x137), new_gcd0Gcd'18(x138, x139, Succ(x140), Succ(x141)) -> new_gcd0Gcd'18(x138, x139, x140, x141) which results in the following constraint: (1) (new_gcd0Gcd'18(x134, x135, x136, x137)=new_gcd0Gcd'18(x138, x139, Succ(x140), Succ(x141)) ==> new_gcd0Gcd'18(x134, x135, Succ(x136), Succ(x137))_>=_new_gcd0Gcd'18(x134, x135, x136, x137)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'18(x134, x135, Succ(Succ(x140)), Succ(Succ(x141)))_>=_new_gcd0Gcd'18(x134, x135, Succ(x140), Succ(x141))) For Pair new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) the following chains were created: *We consider the chain new_gcd0Gcd'13(x162, x163, Succ(x164), Succ(x165)) -> new_gcd0Gcd'13(x162, x163, x164, x165), new_gcd0Gcd'13(x166, x167, Zero, Succ(x168)) -> new_gcd0Gcd'15(Succ(x166), x167) which results in the following constraint: (1) (new_gcd0Gcd'13(x162, x163, x164, x165)=new_gcd0Gcd'13(x166, x167, Zero, Succ(x168)) ==> new_gcd0Gcd'13(x162, x163, Succ(x164), Succ(x165))_>=_new_gcd0Gcd'13(x162, x163, x164, x165)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x162, x163, Succ(Zero), Succ(Succ(x168)))_>=_new_gcd0Gcd'13(x162, x163, Zero, Succ(x168))) *We consider the chain new_gcd0Gcd'13(x177, x178, Succ(x179), Succ(x180)) -> new_gcd0Gcd'13(x177, x178, x179, x180), new_gcd0Gcd'13(x181, x182, Succ(x183), Succ(x184)) -> new_gcd0Gcd'13(x181, x182, x183, x184) which results in the following constraint: (1) (new_gcd0Gcd'13(x177, x178, x179, x180)=new_gcd0Gcd'13(x181, x182, Succ(x183), Succ(x184)) ==> new_gcd0Gcd'13(x177, x178, Succ(x179), Succ(x180))_>=_new_gcd0Gcd'13(x177, x178, x179, x180)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'13(x177, x178, Succ(Succ(x183)), Succ(Succ(x184)))_>=_new_gcd0Gcd'13(x177, x178, Succ(x183), Succ(x184))) For Pair new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) the following chains were created: *We consider the chain new_gcd0Gcd'17(Succ(Succ(x189)), Succ(x190), Succ(Succ(x189))) -> new_gcd0Gcd'18(x189, Succ(x190), x189, x190), new_gcd0Gcd'18(x191, x192, Zero, Succ(x193)) -> new_gcd0Gcd'12(Succ(x191), Neg(Succ(x192))) which results in the following constraint: (1) (new_gcd0Gcd'18(x189, Succ(x190), x189, x190)=new_gcd0Gcd'18(x191, x192, Zero, Succ(x193)) ==> new_gcd0Gcd'17(Succ(Succ(x189)), Succ(x190), Succ(Succ(x189)))_>=_new_gcd0Gcd'18(x189, Succ(x190), x189, x190)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'17(Succ(Succ(Zero)), Succ(Succ(x193)), Succ(Succ(Zero)))_>=_new_gcd0Gcd'18(Zero, Succ(Succ(x193)), Zero, Succ(x193))) *We consider the chain new_gcd0Gcd'17(Succ(Succ(x202)), Succ(x203), Succ(Succ(x202))) -> new_gcd0Gcd'18(x202, Succ(x203), x202, x203), new_gcd0Gcd'18(x204, x205, Succ(x206), Succ(x207)) -> new_gcd0Gcd'18(x204, x205, x206, x207) which results in the following constraint: (1) (new_gcd0Gcd'18(x202, Succ(x203), x202, x203)=new_gcd0Gcd'18(x204, x205, Succ(x206), Succ(x207)) ==> new_gcd0Gcd'17(Succ(Succ(x202)), Succ(x203), Succ(Succ(x202)))_>=_new_gcd0Gcd'18(x202, Succ(x203), x202, x203)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'17(Succ(Succ(Succ(x206))), Succ(Succ(x207)), Succ(Succ(Succ(x206))))_>=_new_gcd0Gcd'18(Succ(x206), Succ(Succ(x207)), Succ(x206), Succ(x207))) To summarize, we get the following constraints P__>=_ for the following pairs. *new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) *(new_gcd0Gcd'18(x3, x4, Zero, Succ(x5))_>=_new_gcd0Gcd'12(Succ(x3), Neg(Succ(x4)))) *new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) *(new_gcd0Gcd'12(Succ(x30), Neg(Succ(Succ(x32))))_>=_new_gcd0Gcd'0(Neg(Succ(Succ(x32))), Succ(x30))) *new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) *(new_gcd0Gcd'0(Neg(Succ(Succ(Zero))), Succ(Succ(x54)))_>=_new_gcd0Gcd'13(Zero, Succ(Succ(x54)), Zero, Succ(x54))) *(new_gcd0Gcd'0(Neg(Succ(Succ(Succ(x63)))), Succ(Succ(x64)))_>=_new_gcd0Gcd'13(Succ(x63), Succ(Succ(x64)), Succ(x63), Succ(x64))) *new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) *(new_gcd0Gcd'13(x79, x80, Zero, Succ(x81))_>=_new_gcd0Gcd'15(Succ(x79), x80)) *new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) *(new_gcd0Gcd'15(Succ(x107), Succ(x109))_>=_new_gcd0Gcd'17(Succ(Succ(x109)), Succ(x107), Succ(Succ(x109)))) *new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) *(new_gcd0Gcd'18(x111, x112, Succ(Zero), Succ(Succ(x117)))_>=_new_gcd0Gcd'18(x111, x112, Zero, Succ(x117))) *(new_gcd0Gcd'18(x134, x135, Succ(Succ(x140)), Succ(Succ(x141)))_>=_new_gcd0Gcd'18(x134, x135, Succ(x140), Succ(x141))) *new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) *(new_gcd0Gcd'13(x162, x163, Succ(Zero), Succ(Succ(x168)))_>=_new_gcd0Gcd'13(x162, x163, Zero, Succ(x168))) *(new_gcd0Gcd'13(x177, x178, Succ(Succ(x183)), Succ(Succ(x184)))_>=_new_gcd0Gcd'13(x177, x178, Succ(x183), Succ(x184))) *new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) *(new_gcd0Gcd'17(Succ(Succ(Zero)), Succ(Succ(x193)), Succ(Succ(Zero)))_>=_new_gcd0Gcd'18(Zero, Succ(Succ(x193)), Zero, Succ(x193))) *(new_gcd0Gcd'17(Succ(Succ(Succ(x206))), Succ(Succ(x207)), Succ(Succ(Succ(x206))))_>=_new_gcd0Gcd'18(Succ(x206), Succ(Succ(x207)), Succ(x206), Succ(x207))) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation [NONINF]: POL(Neg(x_1)) = 0 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(c) = -1 POL(new_gcd0Gcd'0(x_1, x_2)) = -x_1 POL(new_gcd0Gcd'12(x_1, x_2)) = -x_2 POL(new_gcd0Gcd'13(x_1, x_2, x_3, x_4)) = 1 + x_1 - x_2 - x_3 + x_4 POL(new_gcd0Gcd'15(x_1, x_2)) = 1 + x_1 - x_2 POL(new_gcd0Gcd'17(x_1, x_2, x_3)) = 1 + x_2 - x_3 POL(new_gcd0Gcd'18(x_1, x_2, x_3, x_4)) = -x_3 + x_4 The following pairs are in P_>: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) The following pairs are in P_bound: new_gcd0Gcd'18(vuv78, vuv79, Zero, Succ(vuv810)) -> new_gcd0Gcd'12(Succ(vuv78), Neg(Succ(vuv79))) new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) There are no usable rules ---------------------------------------- (46) Complex Obligation (AND) ---------------------------------------- (47) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(Succ(z0), Neg(Succ(z1))) -> new_gcd0Gcd'0(Neg(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Neg(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'13(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (48) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (49) Complex Obligation (AND) ---------------------------------------- (50) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (51) 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_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (52) YES ---------------------------------------- (53) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (54) 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_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (55) YES ---------------------------------------- (56) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Zero, Succ(vuv430)) -> new_gcd0Gcd'15(Succ(vuv40), vuv41) new_gcd0Gcd'15(Succ(z0), z1) -> new_gcd0Gcd'17(Succ(z1), Succ(z0), Succ(z1)) new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) new_gcd0Gcd'17(Succ(Succ(x0)), Succ(z0), Succ(Succ(x0))) -> new_gcd0Gcd'18(x0, Succ(z0), x0, z0) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (57) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes. ---------------------------------------- (58) Complex Obligation (AND) ---------------------------------------- (59) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (60) 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_gcd0Gcd'18(vuv78, vuv79, Succ(vuv800), Succ(vuv810)) -> new_gcd0Gcd'18(vuv78, vuv79, vuv800, vuv810) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (61) YES ---------------------------------------- (62) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (63) 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_gcd0Gcd'13(vuv40, vuv41, Succ(vuv420), Succ(vuv430)) -> new_gcd0Gcd'13(vuv40, vuv41, vuv420, vuv430) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (64) YES ---------------------------------------- (65) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero new_primMinusNatS0 -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (66) 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. ---------------------------------------- (67) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) The TRS R consists of the following rules: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS0 new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (68) 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_primMinusNatS0 ---------------------------------------- (69) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) The TRS R consists of the following rules: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (70) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule new_gcd0Gcd'11(vuv23, vuv24) -> new_gcd0Gcd'1(new_primMinusNatS1(vuv23, vuv24), vuv24) at position [0] we obtained the following new rules [LPAR04]: (new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)),new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1))) (new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero),new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero)) ---------------------------------------- (71) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'12(Zero, Pos(Succ(Succ(z0)))) -> new_gcd0Gcd'0(Pos(Succ(Succ(z0))), Zero) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Zero)), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) new_gcd0Gcd'1(Succ(Zero), Succ(vuv4000)) -> new_gcd0Gcd'12(Zero, Pos(Succ(Succ(vuv4000)))) new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)) new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero) The TRS R consists of the following rules: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (72) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes. ---------------------------------------- (73) Complex Obligation (AND) ---------------------------------------- (74) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) The TRS R consists of the following rules: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) 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_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) R is empty. The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) 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_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) ---------------------------------------- (78) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (79) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: new_gcd0Gcd'1(Succ(Succ(vuv6000)), Zero) -> new_gcd0Gcd'11(vuv6000, Zero) Used ordering: Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 1 POL(new_gcd0Gcd'1(x_1, x_2)) = 1 + 2*x_1 + x_2 POL(new_gcd0Gcd'11(x_1, x_2)) = 2 + 2*x_1 + 2*x_2 ---------------------------------------- (80) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'11(x0, Zero) -> new_gcd0Gcd'1(Succ(x0), Zero) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (81) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. ---------------------------------------- (82) TRUE ---------------------------------------- (83) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) The TRS R consists of the following rules: new_primMinusNatS1(vuv23, Succ(vuv240)) -> new_primMinusNatS2(vuv23, vuv240) new_primMinusNatS1(vuv23, Zero) -> Succ(vuv23) new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (84) 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. ---------------------------------------- (85) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS1(x0, Zero) new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS1(x0, Succ(x1)) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (86) 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_primMinusNatS1(x0, Zero) new_primMinusNatS1(x0, Succ(x1)) ---------------------------------------- (87) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (88) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_gcd0Gcd'11(x0, Succ(x1)) -> new_gcd0Gcd'1(new_primMinusNatS2(x0, x1), Succ(x1)) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(Pos(x_1)) = x_1 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(new_gcd0Gcd'0(x_1, x_2)) = x_1 + x_2 POL(new_gcd0Gcd'1(x_1, x_2)) = x_1 + x_2 POL(new_gcd0Gcd'10(x_1, x_2, x_3, x_4)) = 2 + x_1 + x_2 POL(new_gcd0Gcd'11(x_1, x_2)) = 2 + x_1 + x_2 POL(new_gcd0Gcd'12(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS2(x_1, x_2)) = x_1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero ---------------------------------------- (89) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Zero) -> new_gcd0Gcd'11(vuv35, vuv36) new_gcd0Gcd'1(Succ(Succ(vuv6000)), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Zero) -> new_gcd0Gcd'11(vuv35, vuv36) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (90) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes. ---------------------------------------- (91) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (92) InductionCalculusProof (EQUIVALENT) Note that final constraints are written in bold face. For Pair new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) the following chains were created: *We consider the chain new_gcd0Gcd'10(x0, x1, Succ(x2), Succ(x3)) -> new_gcd0Gcd'10(x0, x1, x2, x3), new_gcd0Gcd'10(x4, x5, Succ(x6), Succ(x7)) -> new_gcd0Gcd'10(x4, x5, x6, x7) which results in the following constraint: (1) (new_gcd0Gcd'10(x0, x1, x2, x3)=new_gcd0Gcd'10(x4, x5, Succ(x6), Succ(x7)) ==> new_gcd0Gcd'10(x0, x1, Succ(x2), Succ(x3))_>=_new_gcd0Gcd'10(x0, x1, x2, x3)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x0, x1, Succ(Succ(x6)), Succ(Succ(x7)))_>=_new_gcd0Gcd'10(x0, x1, Succ(x6), Succ(x7))) *We consider the chain new_gcd0Gcd'10(x8, x9, Succ(x10), Succ(x11)) -> new_gcd0Gcd'10(x8, x9, x10, x11), new_gcd0Gcd'10(x12, x13, Zero, Succ(x14)) -> new_gcd0Gcd'12(Succ(x12), Pos(Succ(x13))) which results in the following constraint: (1) (new_gcd0Gcd'10(x8, x9, x10, x11)=new_gcd0Gcd'10(x12, x13, Zero, Succ(x14)) ==> new_gcd0Gcd'10(x8, x9, Succ(x10), Succ(x11))_>=_new_gcd0Gcd'10(x8, x9, x10, x11)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x8, x9, Succ(Zero), Succ(Succ(x14)))_>=_new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))) For Pair new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) the following chains were created: *We consider the chain new_gcd0Gcd'10(x29, x30, Zero, Succ(x31)) -> new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30))), new_gcd0Gcd'12(Succ(x32), Pos(Succ(x33))) -> new_gcd0Gcd'0(Pos(Succ(x33)), Succ(x32)) which results in the following constraint: (1) (new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))=new_gcd0Gcd'12(Succ(x32), Pos(Succ(x33))) ==> new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) For Pair new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) the following chains were created: *We consider the chain new_gcd0Gcd'12(Succ(x43), Pos(Succ(x44))) -> new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43)), new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x46)) -> new_gcd0Gcd'10(x45, Succ(x46), x45, x46) which results in the following constraint: (1) (new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43))=new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x46)) ==> new_gcd0Gcd'12(Succ(x43), Pos(Succ(x44)))_>=_new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'12(Succ(x43), Pos(Succ(Succ(x45))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x43))) For Pair new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) the following chains were created: *We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(x47))), Succ(x48)) -> new_gcd0Gcd'10(x47, Succ(x48), x47, x48), new_gcd0Gcd'10(x49, x50, Succ(x51), Succ(x52)) -> new_gcd0Gcd'10(x49, x50, x51, x52) which results in the following constraint: (1) (new_gcd0Gcd'10(x47, Succ(x48), x47, x48)=new_gcd0Gcd'10(x49, x50, Succ(x51), Succ(x52)) ==> new_gcd0Gcd'0(Pos(Succ(Succ(x47))), Succ(x48))_>=_new_gcd0Gcd'10(x47, Succ(x48), x47, x48)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x51)))), Succ(Succ(x52)))_>=_new_gcd0Gcd'10(Succ(x51), Succ(Succ(x52)), Succ(x51), Succ(x52))) *We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(x53))), Succ(x54)) -> new_gcd0Gcd'10(x53, Succ(x54), x53, x54), new_gcd0Gcd'10(x55, x56, Zero, Succ(x57)) -> new_gcd0Gcd'12(Succ(x55), Pos(Succ(x56))) which results in the following constraint: (1) (new_gcd0Gcd'10(x53, Succ(x54), x53, x54)=new_gcd0Gcd'10(x55, x56, Zero, Succ(x57)) ==> new_gcd0Gcd'0(Pos(Succ(Succ(x53))), Succ(x54))_>=_new_gcd0Gcd'10(x53, Succ(x54), x53, x54)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Succ(Succ(x57)))_>=_new_gcd0Gcd'10(Zero, Succ(Succ(x57)), Zero, Succ(x57))) To summarize, we get the following constraints P__>=_ for the following pairs. *new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) *(new_gcd0Gcd'10(x0, x1, Succ(Succ(x6)), Succ(Succ(x7)))_>=_new_gcd0Gcd'10(x0, x1, Succ(x6), Succ(x7))) *(new_gcd0Gcd'10(x8, x9, Succ(Zero), Succ(Succ(x14)))_>=_new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))) *new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) *(new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) *new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) *(new_gcd0Gcd'12(Succ(x43), Pos(Succ(Succ(x45))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x43))) *new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) *(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x51)))), Succ(Succ(x52)))_>=_new_gcd0Gcd'10(Succ(x51), Succ(Succ(x52)), Succ(x51), Succ(x52))) *(new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Succ(Succ(x57)))_>=_new_gcd0Gcd'10(Zero, Succ(Succ(x57)), Zero, Succ(x57))) 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. ---------------------------------------- (93) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (94) NonInfProof (EQUIVALENT) The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps: Note that final constraints are written in bold face. For Pair new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) the following chains were created: *We consider the chain new_gcd0Gcd'10(x0, x1, Succ(x2), Succ(x3)) -> new_gcd0Gcd'10(x0, x1, x2, x3), new_gcd0Gcd'10(x4, x5, Succ(x6), Succ(x7)) -> new_gcd0Gcd'10(x4, x5, x6, x7) which results in the following constraint: (1) (new_gcd0Gcd'10(x0, x1, x2, x3)=new_gcd0Gcd'10(x4, x5, Succ(x6), Succ(x7)) ==> new_gcd0Gcd'10(x0, x1, Succ(x2), Succ(x3))_>=_new_gcd0Gcd'10(x0, x1, x2, x3)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x0, x1, Succ(Succ(x6)), Succ(Succ(x7)))_>=_new_gcd0Gcd'10(x0, x1, Succ(x6), Succ(x7))) *We consider the chain new_gcd0Gcd'10(x8, x9, Succ(x10), Succ(x11)) -> new_gcd0Gcd'10(x8, x9, x10, x11), new_gcd0Gcd'10(x12, x13, Zero, Succ(x14)) -> new_gcd0Gcd'12(Succ(x12), Pos(Succ(x13))) which results in the following constraint: (1) (new_gcd0Gcd'10(x8, x9, x10, x11)=new_gcd0Gcd'10(x12, x13, Zero, Succ(x14)) ==> new_gcd0Gcd'10(x8, x9, Succ(x10), Succ(x11))_>=_new_gcd0Gcd'10(x8, x9, x10, x11)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x8, x9, Succ(Zero), Succ(Succ(x14)))_>=_new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))) For Pair new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) the following chains were created: *We consider the chain new_gcd0Gcd'10(x29, x30, Zero, Succ(x31)) -> new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30))), new_gcd0Gcd'12(Succ(x32), Pos(Succ(x33))) -> new_gcd0Gcd'0(Pos(Succ(x33)), Succ(x32)) which results in the following constraint: (1) (new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))=new_gcd0Gcd'12(Succ(x32), Pos(Succ(x33))) ==> new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) We simplified constraint (1) using rules (I), (II), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) For Pair new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) the following chains were created: *We consider the chain new_gcd0Gcd'12(Succ(x43), Pos(Succ(x44))) -> new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43)), new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x46)) -> new_gcd0Gcd'10(x45, Succ(x46), x45, x46) which results in the following constraint: (1) (new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43))=new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x46)) ==> new_gcd0Gcd'12(Succ(x43), Pos(Succ(x44)))_>=_new_gcd0Gcd'0(Pos(Succ(x44)), Succ(x43))) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'12(Succ(x43), Pos(Succ(Succ(x45))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x43))) For Pair new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) the following chains were created: *We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(x47))), Succ(x48)) -> new_gcd0Gcd'10(x47, Succ(x48), x47, x48), new_gcd0Gcd'10(x49, x50, Succ(x51), Succ(x52)) -> new_gcd0Gcd'10(x49, x50, x51, x52) which results in the following constraint: (1) (new_gcd0Gcd'10(x47, Succ(x48), x47, x48)=new_gcd0Gcd'10(x49, x50, Succ(x51), Succ(x52)) ==> new_gcd0Gcd'0(Pos(Succ(Succ(x47))), Succ(x48))_>=_new_gcd0Gcd'10(x47, Succ(x48), x47, x48)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x51)))), Succ(Succ(x52)))_>=_new_gcd0Gcd'10(Succ(x51), Succ(Succ(x52)), Succ(x51), Succ(x52))) *We consider the chain new_gcd0Gcd'0(Pos(Succ(Succ(x53))), Succ(x54)) -> new_gcd0Gcd'10(x53, Succ(x54), x53, x54), new_gcd0Gcd'10(x55, x56, Zero, Succ(x57)) -> new_gcd0Gcd'12(Succ(x55), Pos(Succ(x56))) which results in the following constraint: (1) (new_gcd0Gcd'10(x53, Succ(x54), x53, x54)=new_gcd0Gcd'10(x55, x56, Zero, Succ(x57)) ==> new_gcd0Gcd'0(Pos(Succ(Succ(x53))), Succ(x54))_>=_new_gcd0Gcd'10(x53, Succ(x54), x53, x54)) We simplified constraint (1) using rules (I), (II), (III), (IV) which results in the following new constraint: (2) (new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Succ(Succ(x57)))_>=_new_gcd0Gcd'10(Zero, Succ(Succ(x57)), Zero, Succ(x57))) To summarize, we get the following constraints P__>=_ for the following pairs. *new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) *(new_gcd0Gcd'10(x0, x1, Succ(Succ(x6)), Succ(Succ(x7)))_>=_new_gcd0Gcd'10(x0, x1, Succ(x6), Succ(x7))) *(new_gcd0Gcd'10(x8, x9, Succ(Zero), Succ(Succ(x14)))_>=_new_gcd0Gcd'10(x8, x9, Zero, Succ(x14))) *new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) *(new_gcd0Gcd'10(x29, x30, Zero, Succ(x31))_>=_new_gcd0Gcd'12(Succ(x29), Pos(Succ(x30)))) *new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) *(new_gcd0Gcd'12(Succ(x43), Pos(Succ(Succ(x45))))_>=_new_gcd0Gcd'0(Pos(Succ(Succ(x45))), Succ(x43))) *new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) *(new_gcd0Gcd'0(Pos(Succ(Succ(Succ(x51)))), Succ(Succ(x52)))_>=_new_gcd0Gcd'10(Succ(x51), Succ(Succ(x52)), Succ(x51), Succ(x52))) *(new_gcd0Gcd'0(Pos(Succ(Succ(Zero))), Succ(Succ(x57)))_>=_new_gcd0Gcd'10(Zero, Succ(Succ(x57)), Zero, Succ(x57))) The constraints for P_> respective P_bound are constructed from P__>=_ where we just replace every occurence of "t _>=_ s" in P__>=_ by "t > s" respective "t _>=_ c". Here c stands for the fresh constant used for P_bound. Using the following integer polynomial ordering the resulting constraints can be solved Polynomial interpretation [NONINF]: POL(Pos(x_1)) = 1 POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 0 POL(c) = -1 POL(new_gcd0Gcd'0(x_1, x_2)) = 1 + x_2 POL(new_gcd0Gcd'10(x_1, x_2, x_3, x_4)) = 1 + x_1 - x_3 + x_4 POL(new_gcd0Gcd'12(x_1, x_2)) = x_1 + x_2 The following pairs are in P_>: new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) The following pairs are in P_bound: new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) new_gcd0Gcd'0(Pos(Succ(Succ(vuv6000))), Succ(vuv4000)) -> new_gcd0Gcd'10(vuv6000, Succ(vuv4000), vuv6000, vuv4000) There are no usable rules ---------------------------------------- (95) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) new_gcd0Gcd'10(vuv35, vuv36, Zero, Succ(vuv380)) -> new_gcd0Gcd'12(Succ(vuv35), Pos(Succ(vuv36))) new_gcd0Gcd'12(Succ(z0), Pos(Succ(z1))) -> new_gcd0Gcd'0(Pos(Succ(z1)), Succ(z0)) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (96) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes. ---------------------------------------- (97) Obligation: Q DP problem: The TRS P consists of the following rules: new_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) The TRS R consists of the following rules: new_primMinusNatS2(Succ(vuv230), Zero) -> Succ(vuv230) new_primMinusNatS2(Zero, Zero) -> Zero new_primMinusNatS2(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS2(vuv230, vuv2400) new_primMinusNatS2(Zero, Succ(vuv2400)) -> Zero The set Q consists of the following terms: new_primMinusNatS2(Succ(x0), Zero) new_primMinusNatS2(Zero, Zero) new_primMinusNatS2(Succ(x0), Succ(x1)) new_primMinusNatS2(Zero, Succ(x0)) 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_gcd0Gcd'10(vuv35, vuv36, Succ(vuv370), Succ(vuv380)) -> new_gcd0Gcd'10(vuv35, vuv36, vuv370, vuv380) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 ---------------------------------------- (99) YES ---------------------------------------- (100) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS(vuv230, vuv2400) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (101) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primMinusNatS(Succ(vuv230), Succ(vuv2400)) -> new_primMinusNatS(vuv230, vuv2400) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (102) YES