28.74/16.95 MAYBE 30.83/17.51 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 30.83/17.51 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 30.83/17.51 30.83/17.51 30.83/17.51 H-Termination with start terms of the given HASKELL could not be shown: 30.83/17.51 30.83/17.51 (0) HASKELL 30.83/17.51 (1) IFR [EQUIVALENT, 0 ms] 30.83/17.51 (2) HASKELL 30.83/17.51 (3) BR [EQUIVALENT, 0 ms] 30.83/17.51 (4) HASKELL 30.83/17.51 (5) COR [EQUIVALENT, 0 ms] 30.83/17.51 (6) HASKELL 30.83/17.51 (7) NumRed [SOUND, 2 ms] 30.83/17.51 (8) HASKELL 30.83/17.51 (9) Narrow [SOUND, 0 ms] 30.83/17.51 (10) AND 30.83/17.51 (11) QDP 30.83/17.51 (12) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (13) QDP 30.83/17.51 (14) QDPOrderProof [EQUIVALENT, 10 ms] 30.83/17.51 (15) QDP 30.83/17.51 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (17) QDP 30.83/17.51 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.83/17.51 (19) YES 30.83/17.51 (20) QDP 30.83/17.51 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (22) QDP 30.83/17.51 (23) QDPOrderProof [EQUIVALENT, 8 ms] 30.83/17.51 (24) QDP 30.83/17.51 (25) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (26) QDP 30.83/17.51 (27) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.83/17.51 (28) YES 30.83/17.51 (29) QDP 30.83/17.51 (30) QDPSizeChangeProof [EQUIVALENT, 0 ms] 30.83/17.51 (31) YES 30.83/17.51 (32) QDP 30.83/17.51 (33) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (34) QDP 30.83/17.51 (35) TransformationProof [EQUIVALENT, 0 ms] 30.83/17.51 (36) QDP 30.83/17.51 (37) UsableRulesProof [EQUIVALENT, 0 ms] 30.83/17.51 (38) QDP 30.83/17.51 (39) QReductionProof [EQUIVALENT, 0 ms] 30.83/17.51 (40) QDP 30.83/17.51 (41) MNOCProof [EQUIVALENT, 0 ms] 30.83/17.51 (42) QDP 30.83/17.51 (43) InductionCalculusProof [EQUIVALENT, 0 ms] 30.83/17.51 (44) QDP 30.83/17.51 (45) TransformationProof [EQUIVALENT, 0 ms] 30.83/17.51 (46) QDP 30.83/17.51 (47) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (48) QDP 30.83/17.51 (49) TransformationProof [EQUIVALENT, 0 ms] 30.83/17.51 (50) QDP 30.83/17.51 (51) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (52) QDP 30.83/17.51 (53) TransformationProof [EQUIVALENT, 0 ms] 30.83/17.51 (54) QDP 30.83/17.51 (55) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (56) QDP 30.83/17.51 (57) TransformationProof [EQUIVALENT, 0 ms] 30.83/17.51 (58) QDP 30.83/17.51 (59) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (60) QDP 30.83/17.51 (61) MNOCProof [EQUIVALENT, 0 ms] 30.83/17.51 (62) QDP 30.83/17.51 (63) InductionCalculusProof [EQUIVALENT, 0 ms] 30.83/17.51 (64) QDP 30.83/17.51 (65) Narrow [COMPLETE, 0 ms] 30.83/17.51 (66) QDP 30.83/17.51 (67) DependencyGraphProof [EQUIVALENT, 0 ms] 30.83/17.51 (68) TRUE 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (0) 30.83/17.51 Obligation: 30.83/17.51 mainModule Main 30.83/17.51 module Main where { 30.83/17.51 import qualified Prelude; 30.83/17.51 } 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (1) IFR (EQUIVALENT) 30.83/17.51 If Reductions: 30.83/17.51 The following If expression 30.83/17.51 "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" 30.83/17.51 is transformed to 30.83/17.51 "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); 30.83/17.51 primModNatS0 x y False = Succ x; 30.83/17.51 " 30.83/17.51 The following If expression 30.83/17.51 "if primGEqNatS x y then primModNatP (primMinusNatS x y) (Succ y) else primMinusNatS y x" 30.83/17.51 is transformed to 30.83/17.51 "primModNatP0 x y True = primModNatP (primMinusNatS x y) (Succ y); 30.83/17.51 primModNatP0 x y False = primMinusNatS y x; 30.83/17.51 " 30.83/17.51 The following If expression 30.83/17.51 "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" 30.83/17.51 is transformed to 30.83/17.51 "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); 30.83/17.51 primDivNatS0 x y False = Zero; 30.83/17.51 " 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (2) 30.83/17.51 Obligation: 30.83/17.51 mainModule Main 30.83/17.51 module Main where { 30.83/17.51 import qualified Prelude; 30.83/17.51 } 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (3) BR (EQUIVALENT) 30.83/17.51 Replaced joker patterns by fresh variables and removed binding patterns. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (4) 30.83/17.51 Obligation: 30.83/17.51 mainModule Main 30.83/17.51 module Main where { 30.83/17.51 import qualified Prelude; 30.83/17.51 } 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (5) COR (EQUIVALENT) 30.83/17.51 Cond Reductions: 30.83/17.51 The following Function with conditions 30.83/17.51 "undefined |Falseundefined; 30.83/17.51 " 30.83/17.51 is transformed to 30.83/17.51 "undefined = undefined1; 30.83/17.51 " 30.83/17.51 "undefined0 True = undefined; 30.83/17.51 " 30.83/17.51 "undefined1 = undefined0 False; 30.83/17.51 " 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (6) 30.83/17.51 Obligation: 30.83/17.51 mainModule Main 30.83/17.51 module Main where { 30.83/17.51 import qualified Prelude; 30.83/17.51 } 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (7) NumRed (SOUND) 30.83/17.51 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (8) 30.83/17.51 Obligation: 30.83/17.51 mainModule Main 30.83/17.51 module Main where { 30.83/17.51 import qualified Prelude; 30.83/17.51 } 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (9) Narrow (SOUND) 30.83/17.51 Haskell To QDPs 30.83/17.51 30.83/17.51 digraph dp_graph { 30.83/17.51 node [outthreshold=100, inthreshold=100];1[label="show",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 30.83/17.51 3[label="show wv3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 30.83/17.51 4[label="primShowInt wv3",fontsize=16,color="burlywood",shape="triangle"];852[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 852[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 852 -> 5[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 853[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 853[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 853 -> 6[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 5[label="primShowInt (Pos wv30)",fontsize=16,color="burlywood",shape="box"];854[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];5 -> 854[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 854 -> 7[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 855[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];5 -> 855[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 855 -> 8[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 6[label="primShowInt (Neg wv30)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3]; 30.83/17.51 7[label="primShowInt (Pos (Succ wv300))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 30.83/17.51 8[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 30.83/17.51 9[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))) : primShowInt (Pos wv30)",fontsize=16,color="green",shape="box"];9 -> 12[label="",style="dashed", color="green", weight=3]; 30.83/17.51 10 -> 24[label="",style="dashed", color="red", weight=0]; 30.83/17.51 10[label="primShowInt (div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) ++ toEnum (mod Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) : []",fontsize=16,color="magenta"];10 -> 25[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 10 -> 26[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 10 -> 27[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 11[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))) : []",fontsize=16,color="green",shape="box"];12 -> 4[label="",style="dashed", color="red", weight=0]; 30.83/17.51 12[label="primShowInt (Pos wv30)",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 25[label="wv300",fontsize=16,color="green",shape="box"];26 -> 4[label="",style="dashed", color="red", weight=0]; 30.83/17.51 26[label="primShowInt (div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];26 -> 29[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 27[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];24[label="wv11 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="burlywood",shape="triangle"];856[label="wv11/wv110 : wv111",fontsize=10,color="white",style="solid",shape="box"];24 -> 856[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 856 -> 30[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 857[label="wv11/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 857[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 857 -> 31[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 16[label="Pos wv30",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="red", weight=0]; 30.83/17.51 29[label="div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 29 -> 34[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 30[label="(wv110 : wv111) ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3]; 30.83/17.51 31[label="[] ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3]; 30.83/17.51 33[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];34[label="wv300",fontsize=16,color="green",shape="box"];32[label="div Pos (Succ wv13) Pos (Succ wv14)",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3]; 30.83/17.51 35[label="wv110 : wv111 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="green",shape="box"];35 -> 38[label="",style="dashed", color="green", weight=3]; 30.83/17.51 36[label="toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3]; 30.83/17.51 37[label="primDivInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3]; 30.83/17.51 38 -> 24[label="",style="dashed", color="red", weight=0]; 30.83/17.51 38[label="wv111 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="magenta"];38 -> 41[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 39[label="toEnum (mod Pos (Succ wv8) Pos (Succ wv10))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3]; 30.83/17.51 40[label="Pos (primDivNatS (Succ wv13) (Succ wv14))",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3]; 30.83/17.51 41[label="wv111",fontsize=16,color="green",shape="box"];42[label="primIntToChar (mod Pos (Succ wv8) Pos (Succ wv10))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3]; 30.83/17.51 43[label="primDivNatS (Succ wv13) (Succ wv14)",fontsize=16,color="black",shape="triangle"];43 -> 45[label="",style="solid", color="black", weight=3]; 30.83/17.51 44[label="primIntToChar (primModInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 30.83/17.51 45[label="primDivNatS0 wv13 wv14 (primGEqNatS wv13 wv14)",fontsize=16,color="burlywood",shape="box"];858[label="wv13/Succ wv130",fontsize=10,color="white",style="solid",shape="box"];45 -> 858[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 858 -> 47[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 859[label="wv13/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 859[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 859 -> 48[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 46[label="primIntToChar (Pos (primModNatS (Succ wv8) (Succ wv10)))",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3]; 30.83/17.51 47[label="primDivNatS0 (Succ wv130) wv14 (primGEqNatS (Succ wv130) wv14)",fontsize=16,color="burlywood",shape="box"];860[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];47 -> 860[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 860 -> 50[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 861[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 861[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 861 -> 51[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 48[label="primDivNatS0 Zero wv14 (primGEqNatS Zero wv14)",fontsize=16,color="burlywood",shape="box"];862[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];48 -> 862[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 862 -> 52[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 863[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 863[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 863 -> 53[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 49[label="Char (primModNatS (Succ wv8) (Succ wv10))",fontsize=16,color="green",shape="box"];49 -> 54[label="",style="dashed", color="green", weight=3]; 30.83/17.51 50[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS (Succ wv130) (Succ wv140))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 30.83/17.51 51[label="primDivNatS0 (Succ wv130) Zero (primGEqNatS (Succ wv130) Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3]; 30.83/17.51 52[label="primDivNatS0 Zero (Succ wv140) (primGEqNatS Zero (Succ wv140))",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3]; 30.83/17.51 53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3]; 30.83/17.51 54[label="primModNatS (Succ wv8) (Succ wv10)",fontsize=16,color="black",shape="triangle"];54 -> 59[label="",style="solid", color="black", weight=3]; 30.83/17.51 55 -> 526[label="",style="dashed", color="red", weight=0]; 30.83/17.51 55[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS wv130 wv140)",fontsize=16,color="magenta"];55 -> 527[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 55 -> 528[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 55 -> 529[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 55 -> 530[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 56[label="primDivNatS0 (Succ wv130) Zero True",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3]; 30.83/17.51 57[label="primDivNatS0 Zero (Succ wv140) False",fontsize=16,color="black",shape="box"];57 -> 63[label="",style="solid", color="black", weight=3]; 30.83/17.51 58[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];58 -> 64[label="",style="solid", color="black", weight=3]; 30.83/17.51 59[label="primModNatS0 wv8 wv10 (primGEqNatS wv8 wv10)",fontsize=16,color="burlywood",shape="box"];864[label="wv8/Succ wv80",fontsize=10,color="white",style="solid",shape="box"];59 -> 864[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 864 -> 65[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 865[label="wv8/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 865[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 865 -> 66[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 527[label="wv130",fontsize=16,color="green",shape="box"];528[label="wv140",fontsize=16,color="green",shape="box"];529[label="wv140",fontsize=16,color="green",shape="box"];530[label="wv130",fontsize=16,color="green",shape="box"];526[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS wv56 wv57)",fontsize=16,color="burlywood",shape="triangle"];866[label="wv56/Succ wv560",fontsize=10,color="white",style="solid",shape="box"];526 -> 866[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 866 -> 567[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 867[label="wv56/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 867[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 867 -> 568[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 62[label="Succ (primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 71[label="",style="dashed", color="green", weight=3]; 30.83/17.51 63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 72[label="",style="dashed", color="green", weight=3]; 30.83/17.51 65[label="primModNatS0 (Succ wv80) wv10 (primGEqNatS (Succ wv80) wv10)",fontsize=16,color="burlywood",shape="box"];868[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];65 -> 868[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 868 -> 73[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 869[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 869[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 869 -> 74[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 66[label="primModNatS0 Zero wv10 (primGEqNatS Zero wv10)",fontsize=16,color="burlywood",shape="box"];870[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];66 -> 870[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 870 -> 75[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 871[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];66 -> 871[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 871 -> 76[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 567[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) wv57)",fontsize=16,color="burlywood",shape="box"];872[label="wv57/Succ wv570",fontsize=10,color="white",style="solid",shape="box"];567 -> 872[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 872 -> 597[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 873[label="wv57/Zero",fontsize=10,color="white",style="solid",shape="box"];567 -> 873[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 873 -> 598[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 568[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero wv57)",fontsize=16,color="burlywood",shape="box"];874[label="wv57/Succ wv570",fontsize=10,color="white",style="solid",shape="box"];568 -> 874[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 874 -> 599[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 875[label="wv57/Zero",fontsize=10,color="white",style="solid",shape="box"];568 -> 875[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 875 -> 600[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 71 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.51 71[label="primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero)",fontsize=16,color="magenta"];71 -> 813[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 71 -> 814[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 71 -> 815[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 72 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.51 72[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 816[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 72 -> 817[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 72 -> 818[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 73[label="primModNatS0 (Succ wv80) (Succ wv100) (primGEqNatS (Succ wv80) (Succ wv100))",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3]; 30.83/17.51 74[label="primModNatS0 (Succ wv80) Zero (primGEqNatS (Succ wv80) Zero)",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3]; 30.83/17.51 75[label="primModNatS0 Zero (Succ wv100) (primGEqNatS Zero (Succ wv100))",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3]; 30.83/17.51 76[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];76 -> 86[label="",style="solid", color="black", weight=3]; 30.83/17.51 597[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) (Succ wv570))",fontsize=16,color="black",shape="box"];597 -> 613[label="",style="solid", color="black", weight=3]; 30.83/17.51 598[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) Zero)",fontsize=16,color="black",shape="box"];598 -> 614[label="",style="solid", color="black", weight=3]; 30.83/17.51 599[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero (Succ wv570))",fontsize=16,color="black",shape="box"];599 -> 615[label="",style="solid", color="black", weight=3]; 30.83/17.51 600[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];600 -> 616[label="",style="solid", color="black", weight=3]; 30.83/17.51 813[label="Zero",fontsize=16,color="green",shape="box"];814[label="Succ wv130",fontsize=16,color="green",shape="box"];815[label="Zero",fontsize=16,color="green",shape="box"];812[label="primDivNatS (primMinusNatS wv71 wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="triangle"];876[label="wv71/Succ wv710",fontsize=10,color="white",style="solid",shape="box"];812 -> 876[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 876 -> 837[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 877[label="wv71/Zero",fontsize=10,color="white",style="solid",shape="box"];812 -> 877[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 877 -> 838[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 816[label="Zero",fontsize=16,color="green",shape="box"];817[label="Zero",fontsize=16,color="green",shape="box"];818[label="Zero",fontsize=16,color="green",shape="box"];83 -> 637[label="",style="dashed", color="red", weight=0]; 30.83/17.51 83[label="primModNatS0 (Succ wv80) (Succ wv100) (primGEqNatS wv80 wv100)",fontsize=16,color="magenta"];83 -> 638[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 83 -> 639[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 83 -> 640[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 83 -> 641[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 84[label="primModNatS0 (Succ wv80) Zero True",fontsize=16,color="black",shape="box"];84 -> 97[label="",style="solid", color="black", weight=3]; 30.83/17.51 85[label="primModNatS0 Zero (Succ wv100) False",fontsize=16,color="black",shape="box"];85 -> 98[label="",style="solid", color="black", weight=3]; 30.83/17.51 86[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];86 -> 99[label="",style="solid", color="black", weight=3]; 30.83/17.51 613 -> 526[label="",style="dashed", color="red", weight=0]; 30.83/17.51 613[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS wv560 wv570)",fontsize=16,color="magenta"];613 -> 629[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 613 -> 630[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 614[label="primDivNatS0 (Succ wv54) (Succ wv55) True",fontsize=16,color="black",shape="triangle"];614 -> 631[label="",style="solid", color="black", weight=3]; 30.83/17.51 615[label="primDivNatS0 (Succ wv54) (Succ wv55) False",fontsize=16,color="black",shape="box"];615 -> 632[label="",style="solid", color="black", weight=3]; 30.83/17.51 616 -> 614[label="",style="dashed", color="red", weight=0]; 30.83/17.51 616[label="primDivNatS0 (Succ wv54) (Succ wv55) True",fontsize=16,color="magenta"];837[label="primDivNatS (primMinusNatS (Succ wv710) wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="box"];878[label="wv72/Succ wv720",fontsize=10,color="white",style="solid",shape="box"];837 -> 878[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 878 -> 839[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 879[label="wv72/Zero",fontsize=10,color="white",style="solid",shape="box"];837 -> 879[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 879 -> 840[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 838[label="primDivNatS (primMinusNatS Zero wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="box"];880[label="wv72/Succ wv720",fontsize=10,color="white",style="solid",shape="box"];838 -> 880[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 880 -> 841[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 881[label="wv72/Zero",fontsize=10,color="white",style="solid",shape="box"];838 -> 881[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 881 -> 842[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 638[label="wv100",fontsize=16,color="green",shape="box"];639[label="wv80",fontsize=16,color="green",shape="box"];640[label="wv80",fontsize=16,color="green",shape="box"];641[label="wv100",fontsize=16,color="green",shape="box"];637[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS wv64 wv65)",fontsize=16,color="burlywood",shape="triangle"];882[label="wv64/Succ wv640",fontsize=10,color="white",style="solid",shape="box"];637 -> 882[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 882 -> 678[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 883[label="wv64/Zero",fontsize=10,color="white",style="solid",shape="box"];637 -> 883[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 883 -> 679[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 97 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.51 97[label="primModNatS (primMinusNatS (Succ wv80) Zero) (Succ Zero)",fontsize=16,color="magenta"];97 -> 725[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 97 -> 726[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 97 -> 727[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 98[label="Succ Zero",fontsize=16,color="green",shape="box"];99 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.51 99[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];99 -> 728[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 99 -> 729[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 99 -> 730[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 629[label="wv570",fontsize=16,color="green",shape="box"];630[label="wv560",fontsize=16,color="green",shape="box"];631[label="Succ (primDivNatS (primMinusNatS (Succ wv54) (Succ wv55)) (Succ (Succ wv55)))",fontsize=16,color="green",shape="box"];631 -> 680[label="",style="dashed", color="green", weight=3]; 30.83/17.51 632[label="Zero",fontsize=16,color="green",shape="box"];839[label="primDivNatS (primMinusNatS (Succ wv710) (Succ wv720)) (Succ wv73)",fontsize=16,color="black",shape="box"];839 -> 843[label="",style="solid", color="black", weight=3]; 30.83/17.51 840[label="primDivNatS (primMinusNatS (Succ wv710) Zero) (Succ wv73)",fontsize=16,color="black",shape="box"];840 -> 844[label="",style="solid", color="black", weight=3]; 30.83/17.51 841[label="primDivNatS (primMinusNatS Zero (Succ wv720)) (Succ wv73)",fontsize=16,color="black",shape="box"];841 -> 845[label="",style="solid", color="black", weight=3]; 30.83/17.51 842[label="primDivNatS (primMinusNatS Zero Zero) (Succ wv73)",fontsize=16,color="black",shape="box"];842 -> 846[label="",style="solid", color="black", weight=3]; 30.83/17.51 678[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) wv65)",fontsize=16,color="burlywood",shape="box"];884[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];678 -> 884[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 884 -> 685[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 885[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];678 -> 885[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 885 -> 686[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 679[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero wv65)",fontsize=16,color="burlywood",shape="box"];886[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];679 -> 886[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 886 -> 687[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 887[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];679 -> 887[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 887 -> 688[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 725[label="Zero",fontsize=16,color="green",shape="box"];726[label="Succ wv80",fontsize=16,color="green",shape="box"];727[label="Zero",fontsize=16,color="green",shape="box"];724[label="primModNatS (primMinusNatS wv67 wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="triangle"];888[label="wv67/Succ wv670",fontsize=10,color="white",style="solid",shape="box"];724 -> 888[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 888 -> 755[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 889[label="wv67/Zero",fontsize=10,color="white",style="solid",shape="box"];724 -> 889[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 889 -> 756[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 728[label="Zero",fontsize=16,color="green",shape="box"];729[label="Zero",fontsize=16,color="green",shape="box"];730[label="Zero",fontsize=16,color="green",shape="box"];680 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.51 680[label="primDivNatS (primMinusNatS (Succ wv54) (Succ wv55)) (Succ (Succ wv55))",fontsize=16,color="magenta"];680 -> 819[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 680 -> 820[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 680 -> 821[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 843 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.51 843[label="primDivNatS (primMinusNatS wv710 wv720) (Succ wv73)",fontsize=16,color="magenta"];843 -> 847[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 843 -> 848[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 844 -> 43[label="",style="dashed", color="red", weight=0]; 30.83/17.51 844[label="primDivNatS (Succ wv710) (Succ wv73)",fontsize=16,color="magenta"];844 -> 849[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 844 -> 850[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 845[label="primDivNatS Zero (Succ wv73)",fontsize=16,color="black",shape="triangle"];845 -> 851[label="",style="solid", color="black", weight=3]; 30.83/17.51 846 -> 845[label="",style="dashed", color="red", weight=0]; 30.83/17.51 846[label="primDivNatS Zero (Succ wv73)",fontsize=16,color="magenta"];685[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) (Succ wv650))",fontsize=16,color="black",shape="box"];685 -> 696[label="",style="solid", color="black", weight=3]; 30.83/17.51 686[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) Zero)",fontsize=16,color="black",shape="box"];686 -> 697[label="",style="solid", color="black", weight=3]; 30.83/17.51 687[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero (Succ wv650))",fontsize=16,color="black",shape="box"];687 -> 698[label="",style="solid", color="black", weight=3]; 30.83/17.51 688[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];688 -> 699[label="",style="solid", color="black", weight=3]; 30.83/17.51 755[label="primModNatS (primMinusNatS (Succ wv670) wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="box"];890[label="wv68/Succ wv680",fontsize=10,color="white",style="solid",shape="box"];755 -> 890[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 890 -> 763[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 891[label="wv68/Zero",fontsize=10,color="white",style="solid",shape="box"];755 -> 891[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 891 -> 764[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 756[label="primModNatS (primMinusNatS Zero wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="box"];892[label="wv68/Succ wv680",fontsize=10,color="white",style="solid",shape="box"];756 -> 892[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 892 -> 765[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 893[label="wv68/Zero",fontsize=10,color="white",style="solid",shape="box"];756 -> 893[label="",style="solid", color="burlywood", weight=9]; 30.83/17.51 893 -> 766[label="",style="solid", color="burlywood", weight=3]; 30.83/17.51 819[label="Succ wv55",fontsize=16,color="green",shape="box"];820[label="Succ wv54",fontsize=16,color="green",shape="box"];821[label="Succ wv55",fontsize=16,color="green",shape="box"];847[label="wv720",fontsize=16,color="green",shape="box"];848[label="wv710",fontsize=16,color="green",shape="box"];849[label="wv73",fontsize=16,color="green",shape="box"];850[label="wv710",fontsize=16,color="green",shape="box"];851[label="Zero",fontsize=16,color="green",shape="box"];696 -> 637[label="",style="dashed", color="red", weight=0]; 30.83/17.51 696[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS wv640 wv650)",fontsize=16,color="magenta"];696 -> 706[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 696 -> 707[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 697[label="primModNatS0 (Succ wv62) (Succ wv63) True",fontsize=16,color="black",shape="triangle"];697 -> 708[label="",style="solid", color="black", weight=3]; 30.83/17.51 698[label="primModNatS0 (Succ wv62) (Succ wv63) False",fontsize=16,color="black",shape="box"];698 -> 709[label="",style="solid", color="black", weight=3]; 30.83/17.51 699 -> 697[label="",style="dashed", color="red", weight=0]; 30.83/17.51 699[label="primModNatS0 (Succ wv62) (Succ wv63) True",fontsize=16,color="magenta"];763[label="primModNatS (primMinusNatS (Succ wv670) (Succ wv680)) (Succ wv69)",fontsize=16,color="black",shape="box"];763 -> 771[label="",style="solid", color="black", weight=3]; 30.83/17.51 764[label="primModNatS (primMinusNatS (Succ wv670) Zero) (Succ wv69)",fontsize=16,color="black",shape="box"];764 -> 772[label="",style="solid", color="black", weight=3]; 30.83/17.51 765[label="primModNatS (primMinusNatS Zero (Succ wv680)) (Succ wv69)",fontsize=16,color="black",shape="box"];765 -> 773[label="",style="solid", color="black", weight=3]; 30.83/17.51 766[label="primModNatS (primMinusNatS Zero Zero) (Succ wv69)",fontsize=16,color="black",shape="box"];766 -> 774[label="",style="solid", color="black", weight=3]; 30.83/17.51 706[label="wv650",fontsize=16,color="green",shape="box"];707[label="wv640",fontsize=16,color="green",shape="box"];708 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.51 708[label="primModNatS (primMinusNatS (Succ wv62) (Succ wv63)) (Succ (Succ wv63))",fontsize=16,color="magenta"];708 -> 737[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 708 -> 738[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 708 -> 739[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 709[label="Succ (Succ wv62)",fontsize=16,color="green",shape="box"];771 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.51 771[label="primModNatS (primMinusNatS wv670 wv680) (Succ wv69)",fontsize=16,color="magenta"];771 -> 779[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 771 -> 780[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 772 -> 54[label="",style="dashed", color="red", weight=0]; 30.83/17.51 772[label="primModNatS (Succ wv670) (Succ wv69)",fontsize=16,color="magenta"];772 -> 781[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 772 -> 782[label="",style="dashed", color="magenta", weight=3]; 30.83/17.51 773[label="primModNatS Zero (Succ wv69)",fontsize=16,color="black",shape="triangle"];773 -> 783[label="",style="solid", color="black", weight=3]; 30.83/17.51 774 -> 773[label="",style="dashed", color="red", weight=0]; 30.83/17.51 774[label="primModNatS Zero (Succ wv69)",fontsize=16,color="magenta"];737[label="Succ wv63",fontsize=16,color="green",shape="box"];738[label="Succ wv62",fontsize=16,color="green",shape="box"];739[label="Succ wv63",fontsize=16,color="green",shape="box"];779[label="wv670",fontsize=16,color="green",shape="box"];780[label="wv680",fontsize=16,color="green",shape="box"];781[label="wv670",fontsize=16,color="green",shape="box"];782[label="wv69",fontsize=16,color="green",shape="box"];783[label="Zero",fontsize=16,color="green",shape="box"];} 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (10) 30.83/17.51 Complex Obligation (AND) 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (11) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS0(wv54, wv55, Zero, Zero) -> new_primDivNatS00(wv54, wv55) 30.83/17.51 new_primDivNatS00(wv54, wv55) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 new_primDivNatS(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS1(Succ(wv130), Zero) -> new_primDivNatS(Succ(wv130), Zero, Zero) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS0(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 new_primDivNatS1(Succ(wv130), Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS1(Zero, Zero) -> new_primDivNatS(Zero, Zero, Zero) 30.83/17.51 new_primDivNatS(Succ(wv710), Zero, wv73) -> new_primDivNatS1(wv710, wv73) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (12) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (13) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS00(wv54, wv55) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 new_primDivNatS(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS(Succ(wv710), Zero, wv73) -> new_primDivNatS1(wv710, wv73) 30.83/17.51 new_primDivNatS1(Succ(wv130), Zero) -> new_primDivNatS(Succ(wv130), Zero, Zero) 30.83/17.51 new_primDivNatS1(Succ(wv130), Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS0(wv54, wv55, Zero, Zero) -> new_primDivNatS00(wv54, wv55) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS0(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (14) QDPOrderProof (EQUIVALENT) 30.83/17.51 We use the reduction pair processor [LPAR04,JAR06]. 30.83/17.51 30.83/17.51 30.83/17.51 The following pairs can be oriented strictly and are deleted. 30.83/17.51 30.83/17.51 new_primDivNatS(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS1(Succ(wv130), Zero) -> new_primDivNatS(Succ(wv130), Zero, Zero) 30.83/17.51 new_primDivNatS1(Succ(wv130), Succ(wv140)) -> new_primDivNatS0(wv130, wv140, wv130, wv140) 30.83/17.51 The remaining pairs can at least be oriented weakly. 30.83/17.51 Used ordering: Polynomial interpretation [POLO]: 30.83/17.51 30.83/17.51 POL(Succ(x_1)) = 1 + x_1 30.83/17.51 POL(Zero) = 0 30.83/17.51 POL(new_primDivNatS(x_1, x_2, x_3)) = x_1 30.83/17.51 POL(new_primDivNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1 30.83/17.51 POL(new_primDivNatS00(x_1, x_2)) = 1 + x_1 30.83/17.51 POL(new_primDivNatS1(x_1, x_2)) = 1 + x_1 30.83/17.51 30.83/17.51 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.83/17.51 none 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (15) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS00(wv54, wv55) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 new_primDivNatS(Succ(wv710), Zero, wv73) -> new_primDivNatS1(wv710, wv73) 30.83/17.51 new_primDivNatS0(wv54, wv55, Zero, Zero) -> new_primDivNatS00(wv54, wv55) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS0(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS(Succ(wv54), Succ(wv55), Succ(wv55)) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (16) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (17) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS0(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS0(wv54, wv55, wv560, wv570) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (18) QDPSizeChangeProof (EQUIVALENT) 30.83/17.51 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. 30.83/17.51 30.83/17.51 From the DPs we obtained the following set of size-change graphs: 30.83/17.51 *new_primDivNatS0(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS0(wv54, wv55, wv560, wv570) 30.83/17.51 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (19) 30.83/17.51 YES 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (20) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primModNatS(Succ(wv670), Zero, wv69) -> new_primModNatS1(wv670, wv69) 30.83/17.51 new_primModNatS1(Zero, Zero) -> new_primModNatS(Zero, Zero, Zero) 30.83/17.51 new_primModNatS00(wv62, wv63) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Zero) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 new_primModNatS(Succ(wv670), Succ(wv680), wv69) -> new_primModNatS(wv670, wv680, wv69) 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Succ(wv650)) -> new_primModNatS0(wv62, wv63, wv640, wv650) 30.83/17.51 new_primModNatS1(Succ(wv80), Succ(wv100)) -> new_primModNatS0(wv80, wv100, wv80, wv100) 30.83/17.51 new_primModNatS0(wv62, wv63, Zero, Zero) -> new_primModNatS00(wv62, wv63) 30.83/17.51 new_primModNatS1(Succ(wv80), Zero) -> new_primModNatS(Succ(wv80), Zero, Zero) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (21) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (22) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primModNatS1(Succ(wv80), Succ(wv100)) -> new_primModNatS0(wv80, wv100, wv80, wv100) 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Zero) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 new_primModNatS(Succ(wv670), Succ(wv680), wv69) -> new_primModNatS(wv670, wv680, wv69) 30.83/17.51 new_primModNatS(Succ(wv670), Zero, wv69) -> new_primModNatS1(wv670, wv69) 30.83/17.51 new_primModNatS1(Succ(wv80), Zero) -> new_primModNatS(Succ(wv80), Zero, Zero) 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Succ(wv650)) -> new_primModNatS0(wv62, wv63, wv640, wv650) 30.83/17.51 new_primModNatS0(wv62, wv63, Zero, Zero) -> new_primModNatS00(wv62, wv63) 30.83/17.51 new_primModNatS00(wv62, wv63) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (23) QDPOrderProof (EQUIVALENT) 30.83/17.51 We use the reduction pair processor [LPAR04,JAR06]. 30.83/17.51 30.83/17.51 30.83/17.51 The following pairs can be oriented strictly and are deleted. 30.83/17.51 30.83/17.51 new_primModNatS1(Succ(wv80), Succ(wv100)) -> new_primModNatS0(wv80, wv100, wv80, wv100) 30.83/17.51 new_primModNatS(Succ(wv670), Succ(wv680), wv69) -> new_primModNatS(wv670, wv680, wv69) 30.83/17.51 new_primModNatS1(Succ(wv80), Zero) -> new_primModNatS(Succ(wv80), Zero, Zero) 30.83/17.51 The remaining pairs can at least be oriented weakly. 30.83/17.51 Used ordering: Polynomial interpretation [POLO]: 30.83/17.51 30.83/17.51 POL(Succ(x_1)) = 1 + x_1 30.83/17.51 POL(Zero) = 0 30.83/17.51 POL(new_primModNatS(x_1, x_2, x_3)) = x_1 30.83/17.51 POL(new_primModNatS0(x_1, x_2, x_3, x_4)) = 1 + x_1 30.83/17.51 POL(new_primModNatS00(x_1, x_2)) = 1 + x_1 30.83/17.51 POL(new_primModNatS1(x_1, x_2)) = 1 + x_1 30.83/17.51 30.83/17.51 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: 30.83/17.51 none 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (24) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Zero) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 new_primModNatS(Succ(wv670), Zero, wv69) -> new_primModNatS1(wv670, wv69) 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Succ(wv650)) -> new_primModNatS0(wv62, wv63, wv640, wv650) 30.83/17.51 new_primModNatS0(wv62, wv63, Zero, Zero) -> new_primModNatS00(wv62, wv63) 30.83/17.51 new_primModNatS00(wv62, wv63) -> new_primModNatS(Succ(wv62), Succ(wv63), Succ(wv63)) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (25) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (26) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primModNatS0(wv62, wv63, Succ(wv640), Succ(wv650)) -> new_primModNatS0(wv62, wv63, wv640, wv650) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (27) QDPSizeChangeProof (EQUIVALENT) 30.83/17.51 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. 30.83/17.51 30.83/17.51 From the DPs we obtained the following set of size-change graphs: 30.83/17.51 *new_primModNatS0(wv62, wv63, Succ(wv640), Succ(wv650)) -> new_primModNatS0(wv62, wv63, wv640, wv650) 30.83/17.51 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (28) 30.83/17.51 YES 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (29) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_psPs(:(wv110, wv111), wv8, wv10) -> new_psPs(wv111, wv8, wv10) 30.83/17.51 30.83/17.51 R is empty. 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (30) QDPSizeChangeProof (EQUIVALENT) 30.83/17.51 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. 30.83/17.51 30.83/17.51 From the DPs we obtained the following set of size-change graphs: 30.83/17.51 *new_psPs(:(wv110, wv111), wv8, wv10) -> new_psPs(wv111, wv8, wv10) 30.83/17.51 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (31) 30.83/17.51 YES 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (32) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Neg(wv30)) -> new_primShowInt(Pos(wv30)) 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(new_div(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_div(wv13, wv14) -> Pos(new_primDivNatS3(wv13, wv14)) 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_div(x0, x1) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (33) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (34) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(new_div(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_div(wv13, wv14) -> Pos(new_primDivNatS3(wv13, wv14)) 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_div(x0, x1) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (35) TransformationProof (EQUIVALENT) 30.83/17.51 By rewriting [LPAR04] the rule new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(new_div(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0] we obtained the following new rules [LPAR04]: 30.83/17.51 30.83/17.51 (new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))),new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (36) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_div(wv13, wv14) -> Pos(new_primDivNatS3(wv13, wv14)) 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_div(x0, x1) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (37) UsableRulesProof (EQUIVALENT) 30.83/17.51 As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (38) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_div(x0, x1) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (39) QReductionProof (EQUIVALENT) 30.83/17.51 We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. 30.83/17.51 30.83/17.51 new_div(x0, x1) 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (40) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (41) MNOCProof (EQUIVALENT) 30.83/17.51 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (42) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 Q is empty. 30.83/17.51 We have to consider all (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (43) InductionCalculusProof (EQUIVALENT) 30.83/17.51 Note that final constraints are written in bold face. 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 For Pair new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) the following chains were created: 30.83/17.51 *We consider the chain new_primShowInt(Pos(Succ(x0))) -> new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))), new_primShowInt(Pos(Succ(x1))) -> new_primShowInt(Pos(new_primDivNatS3(x1, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) which results in the following constraint: 30.83/17.51 30.83/17.51 (1) (new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))=new_primShowInt(Pos(Succ(x1))) ==> new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint: 30.83/17.51 30.83/17.51 (2) (Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=x2 & new_primDivNatS3(x0, x2)=Succ(x1) ==> new_primShowInt(Pos(Succ(x0)))_>=_new_primShowInt(Pos(new_primDivNatS3(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS3(x0, x2)=Succ(x1) which results in the following new constraints: 30.83/17.51 30.83/17.51 (3) (new_primDivNatS01(x4, x3, x4, x3)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Succ(x3) ==> new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 (4) (Succ(new_primDivNatS2(Succ(x6), Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_primShowInt(Pos(Succ(Succ(x6))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x6), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 (5) (Succ(new_primDivNatS2(Zero, Zero, Zero))=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))=Zero ==> new_primShowInt(Pos(Succ(Zero)))_>=_new_primShowInt(Pos(new_primDivNatS3(Zero, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 We simplified constraint (3) using rules (I), (II), (VII) which results in the following new constraint: 30.83/17.51 30.83/17.51 (6) (x4=x7 & x3=x8 & new_primDivNatS01(x4, x3, x7, x8)=Succ(x1) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3 ==> new_primShowInt(Pos(Succ(Succ(x4))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x4), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 We solved constraint (4) using rules (I), (II).We solved constraint (5) using rules (I), (II).We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x4, x3, x7, x8)=Succ(x1) which results in the following new constraints: 30.83/17.51 30.83/17.51 (7) (new_primDivNatS02(x10, x9)=Succ(x1) & x10=Zero & x9=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x9 ==> new_primShowInt(Pos(Succ(Succ(x10))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x10), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 (8) (new_primDivNatS02(x16, x15)=Succ(x1) & x16=Succ(x14) & x15=Zero & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 ==> new_primShowInt(Pos(Succ(Succ(x16))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x16), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 (9) (new_primDivNatS01(x20, x19, x18, x17)=Succ(x1) & x20=Succ(x18) & x19=Succ(x17) & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x19 & (\/x21:new_primDivNatS01(x20, x19, x18, x17)=Succ(x21) & x20=x18 & x19=x17 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x19 ==> new_primShowInt(Pos(Succ(Succ(x20))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x20), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) ==> new_primShowInt(Pos(Succ(Succ(x20))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(x20), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 We solved constraint (7) using rules (I), (II), (III).We solved constraint (8) using rules (I), (II), (III).We simplified constraint (9) using rules (I), (II), (III), (IV), (VII) which results in the following new constraint: 30.83/17.51 30.83/17.51 (10) (new_primShowInt(Pos(Succ(Succ(Succ(x18)))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(Succ(x18)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 To summarize, we get the following constraints P__>=_ for the following pairs. 30.83/17.51 30.83/17.51 *new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 *(new_primShowInt(Pos(Succ(Succ(Succ(x18)))))_>=_new_primShowInt(Pos(new_primDivNatS3(Succ(Succ(x18)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 30.83/17.51 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. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (44) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (45) TransformationProof (EQUIVALENT) 30.83/17.51 By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(wv300))) -> new_primShowInt(Pos(new_primDivNatS3(wv300, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: 30.83/17.51 30.83/17.51 (new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))),new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))))) 30.83/17.51 (new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero))) 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (46) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 30.83/17.51 new_primShowInt(Pos(Succ(Zero))) -> new_primShowInt(Pos(Zero)) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (47) DependencyGraphProof (EQUIVALENT) 30.83/17.51 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (48) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.51 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.51 new_primDivNatS2(Zero, Zero, x0) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.51 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.51 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.51 new_primDivNatS4(x0) 30.83/17.51 new_primDivNatS3(Zero, Zero) 30.83/17.51 30.83/17.51 We have to consider all minimal (P,Q,R)-chains. 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (49) TransformationProof (EQUIVALENT) 30.83/17.51 By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(x0)))) -> new_primShowInt(Pos(new_primDivNatS01(x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) at position [0,0] we obtained the following new rules [LPAR04]: 30.83/17.51 30.83/17.51 (new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero))) 30.83/17.51 (new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))),new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))))) 30.83/17.51 30.83/17.51 30.83/17.51 ---------------------------------------- 30.83/17.51 30.83/17.51 (50) 30.83/17.51 Obligation: 30.83/17.51 Q DP problem: 30.83/17.51 The TRS P consists of the following rules: 30.83/17.51 30.83/17.51 new_primShowInt(Pos(Succ(Succ(Zero)))) -> new_primShowInt(Pos(Zero)) 30.83/17.51 new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 30.83/17.51 30.83/17.51 The TRS R consists of the following rules: 30.83/17.51 30.83/17.51 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.51 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.51 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.51 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.51 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.51 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.51 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.51 new_primDivNatS4(wv73) -> Zero 30.83/17.51 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.51 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.51 30.83/17.51 The set Q consists of the following terms: 30.83/17.51 30.83/17.51 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.51 new_primDivNatS3(Succ(x0), Zero) 30.83/17.51 new_primDivNatS02(x0, x1) 30.83/17.51 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.51 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.51 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (51) DependencyGraphProof (EQUIVALENT) 30.83/17.52 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (52) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (53) TransformationProof (EQUIVALENT) 30.83/17.52 By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(x2))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(x2), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) at position [0,0] we obtained the following new rules [LPAR04]: 30.83/17.52 30.83/17.52 (new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero))) 30.83/17.52 (new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (54) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Zero))))) -> new_primShowInt(Pos(Zero)) 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (55) DependencyGraphProof (EQUIVALENT) 30.83/17.52 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (56) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (57) TransformationProof (EQUIVALENT) 30.83/17.52 By narrowing [LPAR04] the rule new_primShowInt(Pos(Succ(Succ(Succ(Succ(x2)))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(x2)), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))) at position [0,0] we obtained the following new rules [LPAR04]: 30.83/17.52 30.83/17.52 (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero))) 30.83/17.52 (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))),new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (58) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(Zero)))))) -> new_primShowInt(Pos(Zero)) 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (59) DependencyGraphProof (EQUIVALENT) 30.83/17.52 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (60) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (61) MNOCProof (EQUIVALENT) 30.83/17.52 We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (62) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 Q is empty. 30.83/17.52 We have to consider all (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (63) InductionCalculusProof (EQUIVALENT) 30.83/17.52 Note that final constraints are written in bold face. 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 For Pair new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) the following chains were created: 30.83/17.52 *We consider the chain new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero)))))))), new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x1))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x1, Succ(Succ(Succ(Succ(Succ(Zero)))))))) which results in the following constraint: 30.83/17.52 30.83/17.52 (1) (new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero))))))))=new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x1))))))) ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 We simplified constraint (1) using rules (I), (II), (VII) which results in the following new constraint: 30.83/17.52 30.83/17.52 (2) (Succ(Succ(Succ(x0)))=x2 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x3 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x4 & new_primDivNatS01(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1))))) ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x0)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x0))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x0, Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 We simplified constraint (2) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x2, x3, x0, x4)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints: 30.83/17.52 30.83/17.52 (3) (new_primDivNatS02(x6, x5)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Zero)))=x6 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x5 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Zero)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Zero))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Zero, Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 (4) (new_primDivNatS02(x12, x11)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x10))))=x12 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x11 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Zero ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x10))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x10)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x10), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 (5) (new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x14))))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Succ(Zero)))))=Succ(x13) & (\/x17:new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x17))))) & Succ(Succ(Succ(x14)))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Succ(Zero)))))=x13 ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x14)))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x14))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x14, Succ(Succ(Succ(Succ(Succ(Zero))))))))) ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x14))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x14)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x14), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 We solved constraint (3) using rules (I), (II).We solved constraint (4) using rules (I), (II).We simplified constraint (5) using rules (I), (II), (IV) which results in the following new constraint: 30.83/17.52 30.83/17.52 (6) (new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(x14))))=x16 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x15 & Succ(Succ(Succ(Succ(Zero))))=x13 ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x14))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x14)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x14), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 We simplified constraint (6) using rule (V) (with possible (I) afterwards) using induction on new_primDivNatS01(x16, x15, x14, x13)=Succ(Succ(Succ(Succ(Succ(x1))))) which results in the following new constraints: 30.83/17.52 30.83/17.52 (7) (new_primDivNatS02(x19, x18)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Zero))))=x19 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x18 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Zero)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Zero), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 (8) (new_primDivNatS02(x25, x24)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x23)))))=x25 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x24 & Succ(Succ(Succ(Succ(Zero))))=Zero ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x23)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x23))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x23)), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 (9) (new_primDivNatS01(x29, x28, x27, x26)=Succ(Succ(Succ(Succ(Succ(x1))))) & Succ(Succ(Succ(Succ(Succ(x27)))))=x29 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x28 & Succ(Succ(Succ(Succ(Zero))))=Succ(x26) & (\/x30:new_primDivNatS01(x29, x28, x27, x26)=Succ(Succ(Succ(Succ(Succ(x30))))) & Succ(Succ(Succ(Succ(x27))))=x29 & Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero))))))))=x28 & Succ(Succ(Succ(Succ(Zero))))=x26 ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(x27))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(x27)))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(x27), Succ(Succ(Succ(Succ(Succ(Zero))))))))) ==> new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 We solved constraint (7) using rules (I), (II).We solved constraint (8) using rules (I), (II).We simplified constraint (9) using rules (I), (II), (III), (IV) which results in the following new constraint: 30.83/17.52 30.83/17.52 (10) (new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 To summarize, we get the following constraints P__>=_ for the following pairs. 30.83/17.52 30.83/17.52 *new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 30.83/17.52 30.83/17.52 *(new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(Succ(Succ(x27)))))))))_>=_new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(Succ(Succ(x27))))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), Succ(Succ(x27)), Succ(Succ(Succ(Succ(Succ(Zero))))))))) 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 30.83/17.52 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. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (64) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Pos(Succ(Succ(Succ(Succ(Succ(x2))))))) -> new_primShowInt(Pos(new_primDivNatS01(Succ(Succ(Succ(x2))), Succ(Succ(Succ(Succ(Succ(Succ(Succ(Succ(Zero)))))))), x2, Succ(Succ(Succ(Succ(Succ(Zero)))))))) 30.83/17.52 30.83/17.52 The TRS R consists of the following rules: 30.83/17.52 30.83/17.52 new_primDivNatS3(Succ(wv130), Succ(wv140)) -> new_primDivNatS01(wv130, wv140, wv130, wv140) 30.83/17.52 new_primDivNatS3(Zero, Succ(wv140)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Zero, Succ(wv570)) -> Zero 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Zero) -> new_primDivNatS02(wv54, wv55) 30.83/17.52 new_primDivNatS01(wv54, wv55, Succ(wv560), Succ(wv570)) -> new_primDivNatS01(wv54, wv55, wv560, wv570) 30.83/17.52 new_primDivNatS02(wv54, wv55) -> Succ(new_primDivNatS2(Succ(wv54), Succ(wv55), Succ(wv55))) 30.83/17.52 new_primDivNatS2(Succ(wv710), Succ(wv720), wv73) -> new_primDivNatS2(wv710, wv720, wv73) 30.83/17.52 new_primDivNatS2(Succ(wv710), Zero, wv73) -> new_primDivNatS3(wv710, wv73) 30.83/17.52 new_primDivNatS2(Zero, Zero, wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS2(Zero, Succ(wv720), wv73) -> new_primDivNatS4(wv73) 30.83/17.52 new_primDivNatS4(wv73) -> Zero 30.83/17.52 new_primDivNatS3(Succ(wv130), Zero) -> Succ(new_primDivNatS2(Succ(wv130), Zero, Zero)) 30.83/17.52 new_primDivNatS3(Zero, Zero) -> Succ(new_primDivNatS2(Zero, Zero, Zero)) 30.83/17.52 30.83/17.52 The set Q consists of the following terms: 30.83/17.52 30.83/17.52 new_primDivNatS3(Zero, Succ(x0)) 30.83/17.52 new_primDivNatS3(Succ(x0), Zero) 30.83/17.52 new_primDivNatS02(x0, x1) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Zero) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Zero) 30.83/17.52 new_primDivNatS2(Succ(x0), Zero, x1) 30.83/17.52 new_primDivNatS2(Zero, Succ(x0), x1) 30.83/17.52 new_primDivNatS2(Zero, Zero, x0) 30.83/17.52 new_primDivNatS01(x0, x1, Succ(x2), Succ(x3)) 30.83/17.52 new_primDivNatS01(x0, x1, Zero, Succ(x2)) 30.83/17.52 new_primDivNatS3(Succ(x0), Succ(x1)) 30.83/17.52 new_primDivNatS2(Succ(x0), Succ(x1), x2) 30.83/17.52 new_primDivNatS4(x0) 30.83/17.52 new_primDivNatS3(Zero, Zero) 30.83/17.52 30.83/17.52 We have to consider all minimal (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (65) Narrow (COMPLETE) 30.83/17.52 Haskell To QDPs 30.83/17.52 30.83/17.52 digraph dp_graph { 30.83/17.52 node [outthreshold=100, inthreshold=100];1[label="show",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 30.83/17.52 3[label="show wv3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 30.83/17.52 4[label="primShowInt wv3",fontsize=16,color="burlywood",shape="triangle"];852[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 852[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 852 -> 5[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 853[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];4 -> 853[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 853 -> 6[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 5[label="primShowInt (Pos wv30)",fontsize=16,color="burlywood",shape="box"];854[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];5 -> 854[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 854 -> 7[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 855[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];5 -> 855[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 855 -> 8[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 6[label="primShowInt (Neg wv30)",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3]; 30.83/17.52 7[label="primShowInt (Pos (Succ wv300))",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3]; 30.83/17.52 8[label="primShowInt (Pos Zero)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3]; 30.83/17.52 9[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))))))))))))))))))))))))))))))))))))) : primShowInt (Pos wv30)",fontsize=16,color="green",shape="box"];9 -> 12[label="",style="dashed", color="green", weight=3]; 30.83/17.52 10 -> 24[label="",style="dashed", color="red", weight=0]; 30.83/17.52 10[label="primShowInt (div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) ++ toEnum (mod Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))) : []",fontsize=16,color="magenta"];10 -> 25[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 10 -> 26[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 10 -> 27[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 11[label="Char (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))))))))))))))))))))))))))))))))))))))) : []",fontsize=16,color="green",shape="box"];12 -> 4[label="",style="dashed", color="red", weight=0]; 30.83/17.52 12[label="primShowInt (Pos wv30)",fontsize=16,color="magenta"];12 -> 16[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 25[label="wv300",fontsize=16,color="green",shape="box"];26 -> 4[label="",style="dashed", color="red", weight=0]; 30.83/17.52 26[label="primShowInt (div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero)))))))))))",fontsize=16,color="magenta"];26 -> 29[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 27[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];24[label="wv11 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="burlywood",shape="triangle"];856[label="wv11/wv110 : wv111",fontsize=10,color="white",style="solid",shape="box"];24 -> 856[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 856 -> 30[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 857[label="wv11/[]",fontsize=10,color="white",style="solid",shape="box"];24 -> 857[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 857 -> 31[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 16[label="Pos wv30",fontsize=16,color="green",shape="box"];29 -> 32[label="",style="dashed", color="red", weight=0]; 30.83/17.52 29[label="div Pos (Succ wv300) Pos (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))))",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 29 -> 34[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 30[label="(wv110 : wv111) ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3]; 30.83/17.52 31[label="[] ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3]; 30.83/17.52 33[label="Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ (Succ Zero))))))))",fontsize=16,color="green",shape="box"];34[label="wv300",fontsize=16,color="green",shape="box"];32[label="div Pos (Succ wv13) Pos (Succ wv14)",fontsize=16,color="black",shape="triangle"];32 -> 37[label="",style="solid", color="black", weight=3]; 30.83/17.52 35[label="wv110 : wv111 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="green",shape="box"];35 -> 38[label="",style="dashed", color="green", weight=3]; 30.83/17.52 36[label="toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3]; 30.83/17.52 37[label="primDivInt (Pos (Succ wv13)) (Pos (Succ wv14))",fontsize=16,color="black",shape="box"];37 -> 40[label="",style="solid", color="black", weight=3]; 30.83/17.52 38 -> 24[label="",style="dashed", color="red", weight=0]; 30.83/17.52 38[label="wv111 ++ toEnum (mod Pos (Succ wv8) Pos (Succ wv10)) : []",fontsize=16,color="magenta"];38 -> 41[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 39[label="toEnum (mod Pos (Succ wv8) Pos (Succ wv10))",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3]; 30.83/17.52 40[label="Pos (primDivNatS (Succ wv13) (Succ wv14))",fontsize=16,color="green",shape="box"];40 -> 43[label="",style="dashed", color="green", weight=3]; 30.83/17.52 41[label="wv111",fontsize=16,color="green",shape="box"];42[label="primIntToChar (mod Pos (Succ wv8) Pos (Succ wv10))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3]; 30.83/17.52 43[label="primDivNatS (Succ wv13) (Succ wv14)",fontsize=16,color="black",shape="triangle"];43 -> 45[label="",style="solid", color="black", weight=3]; 30.83/17.52 44[label="primIntToChar (primModInt (Pos (Succ wv8)) (Pos (Succ wv10)))",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 30.83/17.52 45[label="primDivNatS0 wv13 wv14 (primGEqNatS wv13 wv14)",fontsize=16,color="burlywood",shape="box"];858[label="wv13/Succ wv130",fontsize=10,color="white",style="solid",shape="box"];45 -> 858[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 858 -> 47[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 859[label="wv13/Zero",fontsize=10,color="white",style="solid",shape="box"];45 -> 859[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 859 -> 48[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 46[label="primIntToChar (Pos (primModNatS (Succ wv8) (Succ wv10)))",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3]; 30.83/17.52 47[label="primDivNatS0 (Succ wv130) wv14 (primGEqNatS (Succ wv130) wv14)",fontsize=16,color="burlywood",shape="box"];860[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];47 -> 860[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 860 -> 50[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 861[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 861[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 861 -> 51[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 48[label="primDivNatS0 Zero wv14 (primGEqNatS Zero wv14)",fontsize=16,color="burlywood",shape="box"];862[label="wv14/Succ wv140",fontsize=10,color="white",style="solid",shape="box"];48 -> 862[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 862 -> 52[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 863[label="wv14/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 863[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 863 -> 53[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 49[label="Char (primModNatS (Succ wv8) (Succ wv10))",fontsize=16,color="green",shape="box"];49 -> 54[label="",style="dashed", color="green", weight=3]; 30.83/17.52 50[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS (Succ wv130) (Succ wv140))",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 30.83/17.52 51[label="primDivNatS0 (Succ wv130) Zero (primGEqNatS (Succ wv130) Zero)",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3]; 30.83/17.52 52[label="primDivNatS0 Zero (Succ wv140) (primGEqNatS Zero (Succ wv140))",fontsize=16,color="black",shape="box"];52 -> 57[label="",style="solid", color="black", weight=3]; 30.83/17.52 53[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];53 -> 58[label="",style="solid", color="black", weight=3]; 30.83/17.52 54[label="primModNatS (Succ wv8) (Succ wv10)",fontsize=16,color="black",shape="triangle"];54 -> 59[label="",style="solid", color="black", weight=3]; 30.83/17.52 55 -> 526[label="",style="dashed", color="red", weight=0]; 30.83/17.52 55[label="primDivNatS0 (Succ wv130) (Succ wv140) (primGEqNatS wv130 wv140)",fontsize=16,color="magenta"];55 -> 527[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 55 -> 528[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 55 -> 529[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 55 -> 530[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 56[label="primDivNatS0 (Succ wv130) Zero True",fontsize=16,color="black",shape="box"];56 -> 62[label="",style="solid", color="black", weight=3]; 30.83/17.52 57[label="primDivNatS0 Zero (Succ wv140) False",fontsize=16,color="black",shape="box"];57 -> 63[label="",style="solid", color="black", weight=3]; 30.83/17.52 58[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];58 -> 64[label="",style="solid", color="black", weight=3]; 30.83/17.52 59[label="primModNatS0 wv8 wv10 (primGEqNatS wv8 wv10)",fontsize=16,color="burlywood",shape="box"];864[label="wv8/Succ wv80",fontsize=10,color="white",style="solid",shape="box"];59 -> 864[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 864 -> 65[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 865[label="wv8/Zero",fontsize=10,color="white",style="solid",shape="box"];59 -> 865[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 865 -> 66[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 527[label="wv130",fontsize=16,color="green",shape="box"];528[label="wv140",fontsize=16,color="green",shape="box"];529[label="wv140",fontsize=16,color="green",shape="box"];530[label="wv130",fontsize=16,color="green",shape="box"];526[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS wv56 wv57)",fontsize=16,color="burlywood",shape="triangle"];866[label="wv56/Succ wv560",fontsize=10,color="white",style="solid",shape="box"];526 -> 866[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 866 -> 567[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 867[label="wv56/Zero",fontsize=10,color="white",style="solid",shape="box"];526 -> 867[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 867 -> 568[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 62[label="Succ (primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];62 -> 71[label="",style="dashed", color="green", weight=3]; 30.83/17.52 63[label="Zero",fontsize=16,color="green",shape="box"];64[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];64 -> 72[label="",style="dashed", color="green", weight=3]; 30.83/17.52 65[label="primModNatS0 (Succ wv80) wv10 (primGEqNatS (Succ wv80) wv10)",fontsize=16,color="burlywood",shape="box"];868[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];65 -> 868[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 868 -> 73[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 869[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 869[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 869 -> 74[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 66[label="primModNatS0 Zero wv10 (primGEqNatS Zero wv10)",fontsize=16,color="burlywood",shape="box"];870[label="wv10/Succ wv100",fontsize=10,color="white",style="solid",shape="box"];66 -> 870[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 870 -> 75[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 871[label="wv10/Zero",fontsize=10,color="white",style="solid",shape="box"];66 -> 871[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 871 -> 76[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 567[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) wv57)",fontsize=16,color="burlywood",shape="box"];872[label="wv57/Succ wv570",fontsize=10,color="white",style="solid",shape="box"];567 -> 872[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 872 -> 597[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 873[label="wv57/Zero",fontsize=10,color="white",style="solid",shape="box"];567 -> 873[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 873 -> 598[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 568[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero wv57)",fontsize=16,color="burlywood",shape="box"];874[label="wv57/Succ wv570",fontsize=10,color="white",style="solid",shape="box"];568 -> 874[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 874 -> 599[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 875[label="wv57/Zero",fontsize=10,color="white",style="solid",shape="box"];568 -> 875[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 875 -> 600[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 71 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.52 71[label="primDivNatS (primMinusNatS (Succ wv130) Zero) (Succ Zero)",fontsize=16,color="magenta"];71 -> 813[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 71 -> 814[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 71 -> 815[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 72 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.52 72[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];72 -> 816[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 72 -> 817[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 72 -> 818[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 73[label="primModNatS0 (Succ wv80) (Succ wv100) (primGEqNatS (Succ wv80) (Succ wv100))",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3]; 30.83/17.52 74[label="primModNatS0 (Succ wv80) Zero (primGEqNatS (Succ wv80) Zero)",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3]; 30.83/17.52 75[label="primModNatS0 Zero (Succ wv100) (primGEqNatS Zero (Succ wv100))",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3]; 30.83/17.52 76[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];76 -> 86[label="",style="solid", color="black", weight=3]; 30.83/17.52 597[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) (Succ wv570))",fontsize=16,color="black",shape="box"];597 -> 613[label="",style="solid", color="black", weight=3]; 30.83/17.52 598[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS (Succ wv560) Zero)",fontsize=16,color="black",shape="box"];598 -> 614[label="",style="solid", color="black", weight=3]; 30.83/17.52 599[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero (Succ wv570))",fontsize=16,color="black",shape="box"];599 -> 615[label="",style="solid", color="black", weight=3]; 30.83/17.52 600[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];600 -> 616[label="",style="solid", color="black", weight=3]; 30.83/17.52 813[label="Zero",fontsize=16,color="green",shape="box"];814[label="Succ wv130",fontsize=16,color="green",shape="box"];815[label="Zero",fontsize=16,color="green",shape="box"];812[label="primDivNatS (primMinusNatS wv71 wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="triangle"];876[label="wv71/Succ wv710",fontsize=10,color="white",style="solid",shape="box"];812 -> 876[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 876 -> 837[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 877[label="wv71/Zero",fontsize=10,color="white",style="solid",shape="box"];812 -> 877[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 877 -> 838[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 816[label="Zero",fontsize=16,color="green",shape="box"];817[label="Zero",fontsize=16,color="green",shape="box"];818[label="Zero",fontsize=16,color="green",shape="box"];83 -> 637[label="",style="dashed", color="red", weight=0]; 30.83/17.52 83[label="primModNatS0 (Succ wv80) (Succ wv100) (primGEqNatS wv80 wv100)",fontsize=16,color="magenta"];83 -> 638[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 83 -> 639[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 83 -> 640[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 83 -> 641[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 84[label="primModNatS0 (Succ wv80) Zero True",fontsize=16,color="black",shape="box"];84 -> 97[label="",style="solid", color="black", weight=3]; 30.83/17.52 85[label="primModNatS0 Zero (Succ wv100) False",fontsize=16,color="black",shape="box"];85 -> 98[label="",style="solid", color="black", weight=3]; 30.83/17.52 86[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];86 -> 99[label="",style="solid", color="black", weight=3]; 30.83/17.52 613 -> 526[label="",style="dashed", color="red", weight=0]; 30.83/17.52 613[label="primDivNatS0 (Succ wv54) (Succ wv55) (primGEqNatS wv560 wv570)",fontsize=16,color="magenta"];613 -> 629[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 613 -> 630[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 614[label="primDivNatS0 (Succ wv54) (Succ wv55) True",fontsize=16,color="black",shape="triangle"];614 -> 631[label="",style="solid", color="black", weight=3]; 30.83/17.52 615[label="primDivNatS0 (Succ wv54) (Succ wv55) False",fontsize=16,color="black",shape="box"];615 -> 632[label="",style="solid", color="black", weight=3]; 30.83/17.52 616 -> 614[label="",style="dashed", color="red", weight=0]; 30.83/17.52 616[label="primDivNatS0 (Succ wv54) (Succ wv55) True",fontsize=16,color="magenta"];837[label="primDivNatS (primMinusNatS (Succ wv710) wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="box"];878[label="wv72/Succ wv720",fontsize=10,color="white",style="solid",shape="box"];837 -> 878[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 878 -> 839[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 879[label="wv72/Zero",fontsize=10,color="white",style="solid",shape="box"];837 -> 879[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 879 -> 840[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 838[label="primDivNatS (primMinusNatS Zero wv72) (Succ wv73)",fontsize=16,color="burlywood",shape="box"];880[label="wv72/Succ wv720",fontsize=10,color="white",style="solid",shape="box"];838 -> 880[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 880 -> 841[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 881[label="wv72/Zero",fontsize=10,color="white",style="solid",shape="box"];838 -> 881[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 881 -> 842[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 638[label="wv100",fontsize=16,color="green",shape="box"];639[label="wv80",fontsize=16,color="green",shape="box"];640[label="wv80",fontsize=16,color="green",shape="box"];641[label="wv100",fontsize=16,color="green",shape="box"];637[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS wv64 wv65)",fontsize=16,color="burlywood",shape="triangle"];882[label="wv64/Succ wv640",fontsize=10,color="white",style="solid",shape="box"];637 -> 882[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 882 -> 678[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 883[label="wv64/Zero",fontsize=10,color="white",style="solid",shape="box"];637 -> 883[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 883 -> 679[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 97 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.52 97[label="primModNatS (primMinusNatS (Succ wv80) Zero) (Succ Zero)",fontsize=16,color="magenta"];97 -> 725[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 97 -> 726[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 97 -> 727[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 98[label="Succ Zero",fontsize=16,color="green",shape="box"];99 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.52 99[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];99 -> 728[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 99 -> 729[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 99 -> 730[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 629[label="wv570",fontsize=16,color="green",shape="box"];630[label="wv560",fontsize=16,color="green",shape="box"];631[label="Succ (primDivNatS (primMinusNatS (Succ wv54) (Succ wv55)) (Succ (Succ wv55)))",fontsize=16,color="green",shape="box"];631 -> 680[label="",style="dashed", color="green", weight=3]; 30.83/17.52 632[label="Zero",fontsize=16,color="green",shape="box"];839[label="primDivNatS (primMinusNatS (Succ wv710) (Succ wv720)) (Succ wv73)",fontsize=16,color="black",shape="box"];839 -> 843[label="",style="solid", color="black", weight=3]; 30.83/17.52 840[label="primDivNatS (primMinusNatS (Succ wv710) Zero) (Succ wv73)",fontsize=16,color="black",shape="box"];840 -> 844[label="",style="solid", color="black", weight=3]; 30.83/17.52 841[label="primDivNatS (primMinusNatS Zero (Succ wv720)) (Succ wv73)",fontsize=16,color="black",shape="box"];841 -> 845[label="",style="solid", color="black", weight=3]; 30.83/17.52 842[label="primDivNatS (primMinusNatS Zero Zero) (Succ wv73)",fontsize=16,color="black",shape="box"];842 -> 846[label="",style="solid", color="black", weight=3]; 30.83/17.52 678[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) wv65)",fontsize=16,color="burlywood",shape="box"];884[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];678 -> 884[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 884 -> 685[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 885[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];678 -> 885[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 885 -> 686[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 679[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero wv65)",fontsize=16,color="burlywood",shape="box"];886[label="wv65/Succ wv650",fontsize=10,color="white",style="solid",shape="box"];679 -> 886[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 886 -> 687[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 887[label="wv65/Zero",fontsize=10,color="white",style="solid",shape="box"];679 -> 887[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 887 -> 688[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 725[label="Zero",fontsize=16,color="green",shape="box"];726[label="Succ wv80",fontsize=16,color="green",shape="box"];727[label="Zero",fontsize=16,color="green",shape="box"];724[label="primModNatS (primMinusNatS wv67 wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="triangle"];888[label="wv67/Succ wv670",fontsize=10,color="white",style="solid",shape="box"];724 -> 888[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 888 -> 755[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 889[label="wv67/Zero",fontsize=10,color="white",style="solid",shape="box"];724 -> 889[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 889 -> 756[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 728[label="Zero",fontsize=16,color="green",shape="box"];729[label="Zero",fontsize=16,color="green",shape="box"];730[label="Zero",fontsize=16,color="green",shape="box"];680 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.52 680[label="primDivNatS (primMinusNatS (Succ wv54) (Succ wv55)) (Succ (Succ wv55))",fontsize=16,color="magenta"];680 -> 819[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 680 -> 820[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 680 -> 821[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 843 -> 812[label="",style="dashed", color="red", weight=0]; 30.83/17.52 843[label="primDivNatS (primMinusNatS wv710 wv720) (Succ wv73)",fontsize=16,color="magenta"];843 -> 847[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 843 -> 848[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 844 -> 43[label="",style="dashed", color="red", weight=0]; 30.83/17.52 844[label="primDivNatS (Succ wv710) (Succ wv73)",fontsize=16,color="magenta"];844 -> 849[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 844 -> 850[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 845[label="primDivNatS Zero (Succ wv73)",fontsize=16,color="black",shape="triangle"];845 -> 851[label="",style="solid", color="black", weight=3]; 30.83/17.52 846 -> 845[label="",style="dashed", color="red", weight=0]; 30.83/17.52 846[label="primDivNatS Zero (Succ wv73)",fontsize=16,color="magenta"];685[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) (Succ wv650))",fontsize=16,color="black",shape="box"];685 -> 696[label="",style="solid", color="black", weight=3]; 30.83/17.52 686[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS (Succ wv640) Zero)",fontsize=16,color="black",shape="box"];686 -> 697[label="",style="solid", color="black", weight=3]; 30.83/17.52 687[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero (Succ wv650))",fontsize=16,color="black",shape="box"];687 -> 698[label="",style="solid", color="black", weight=3]; 30.83/17.52 688[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];688 -> 699[label="",style="solid", color="black", weight=3]; 30.83/17.52 755[label="primModNatS (primMinusNatS (Succ wv670) wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="box"];890[label="wv68/Succ wv680",fontsize=10,color="white",style="solid",shape="box"];755 -> 890[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 890 -> 763[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 891[label="wv68/Zero",fontsize=10,color="white",style="solid",shape="box"];755 -> 891[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 891 -> 764[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 756[label="primModNatS (primMinusNatS Zero wv68) (Succ wv69)",fontsize=16,color="burlywood",shape="box"];892[label="wv68/Succ wv680",fontsize=10,color="white",style="solid",shape="box"];756 -> 892[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 892 -> 765[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 893[label="wv68/Zero",fontsize=10,color="white",style="solid",shape="box"];756 -> 893[label="",style="solid", color="burlywood", weight=9]; 30.83/17.52 893 -> 766[label="",style="solid", color="burlywood", weight=3]; 30.83/17.52 819[label="Succ wv55",fontsize=16,color="green",shape="box"];820[label="Succ wv54",fontsize=16,color="green",shape="box"];821[label="Succ wv55",fontsize=16,color="green",shape="box"];847[label="wv720",fontsize=16,color="green",shape="box"];848[label="wv710",fontsize=16,color="green",shape="box"];849[label="wv73",fontsize=16,color="green",shape="box"];850[label="wv710",fontsize=16,color="green",shape="box"];851[label="Zero",fontsize=16,color="green",shape="box"];696 -> 637[label="",style="dashed", color="red", weight=0]; 30.83/17.52 696[label="primModNatS0 (Succ wv62) (Succ wv63) (primGEqNatS wv640 wv650)",fontsize=16,color="magenta"];696 -> 706[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 696 -> 707[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 697[label="primModNatS0 (Succ wv62) (Succ wv63) True",fontsize=16,color="black",shape="triangle"];697 -> 708[label="",style="solid", color="black", weight=3]; 30.83/17.52 698[label="primModNatS0 (Succ wv62) (Succ wv63) False",fontsize=16,color="black",shape="box"];698 -> 709[label="",style="solid", color="black", weight=3]; 30.83/17.52 699 -> 697[label="",style="dashed", color="red", weight=0]; 30.83/17.52 699[label="primModNatS0 (Succ wv62) (Succ wv63) True",fontsize=16,color="magenta"];763[label="primModNatS (primMinusNatS (Succ wv670) (Succ wv680)) (Succ wv69)",fontsize=16,color="black",shape="box"];763 -> 771[label="",style="solid", color="black", weight=3]; 30.83/17.52 764[label="primModNatS (primMinusNatS (Succ wv670) Zero) (Succ wv69)",fontsize=16,color="black",shape="box"];764 -> 772[label="",style="solid", color="black", weight=3]; 30.83/17.52 765[label="primModNatS (primMinusNatS Zero (Succ wv680)) (Succ wv69)",fontsize=16,color="black",shape="box"];765 -> 773[label="",style="solid", color="black", weight=3]; 30.83/17.52 766[label="primModNatS (primMinusNatS Zero Zero) (Succ wv69)",fontsize=16,color="black",shape="box"];766 -> 774[label="",style="solid", color="black", weight=3]; 30.83/17.52 706[label="wv650",fontsize=16,color="green",shape="box"];707[label="wv640",fontsize=16,color="green",shape="box"];708 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.52 708[label="primModNatS (primMinusNatS (Succ wv62) (Succ wv63)) (Succ (Succ wv63))",fontsize=16,color="magenta"];708 -> 737[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 708 -> 738[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 708 -> 739[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 709[label="Succ (Succ wv62)",fontsize=16,color="green",shape="box"];771 -> 724[label="",style="dashed", color="red", weight=0]; 30.83/17.52 771[label="primModNatS (primMinusNatS wv670 wv680) (Succ wv69)",fontsize=16,color="magenta"];771 -> 779[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 771 -> 780[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 772 -> 54[label="",style="dashed", color="red", weight=0]; 30.83/17.52 772[label="primModNatS (Succ wv670) (Succ wv69)",fontsize=16,color="magenta"];772 -> 781[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 772 -> 782[label="",style="dashed", color="magenta", weight=3]; 30.83/17.52 773[label="primModNatS Zero (Succ wv69)",fontsize=16,color="black",shape="triangle"];773 -> 783[label="",style="solid", color="black", weight=3]; 30.83/17.52 774 -> 773[label="",style="dashed", color="red", weight=0]; 30.83/17.52 774[label="primModNatS Zero (Succ wv69)",fontsize=16,color="magenta"];737[label="Succ wv63",fontsize=16,color="green",shape="box"];738[label="Succ wv62",fontsize=16,color="green",shape="box"];739[label="Succ wv63",fontsize=16,color="green",shape="box"];779[label="wv670",fontsize=16,color="green",shape="box"];780[label="wv680",fontsize=16,color="green",shape="box"];781[label="wv670",fontsize=16,color="green",shape="box"];782[label="wv69",fontsize=16,color="green",shape="box"];783[label="Zero",fontsize=16,color="green",shape="box"];} 30.83/17.52 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (66) 30.83/17.52 Obligation: 30.83/17.52 Q DP problem: 30.83/17.52 The TRS P consists of the following rules: 30.83/17.52 30.83/17.52 new_primShowInt(Neg(wv30), []) -> new_primShowInt(Pos(wv30), []) 30.83/17.52 30.83/17.52 R is empty. 30.83/17.52 Q is empty. 30.83/17.52 We have to consider all (P,Q,R)-chains. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (67) DependencyGraphProof (EQUIVALENT) 30.83/17.52 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 1 less node. 30.83/17.52 ---------------------------------------- 30.83/17.52 30.83/17.52 (68) 30.83/17.52 TRUE 31.02/17.55 EOF