9.50/3.95 YES 11.44/4.49 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 11.44/4.49 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.44/4.49 11.44/4.49 11.44/4.49 H-Termination with start terms of the given HASKELL could be proven: 11.44/4.49 11.44/4.49 (0) HASKELL 11.44/4.49 (1) IFR [EQUIVALENT, 0 ms] 11.44/4.49 (2) HASKELL 11.44/4.49 (3) BR [EQUIVALENT, 0 ms] 11.44/4.49 (4) HASKELL 11.44/4.49 (5) COR [EQUIVALENT, 0 ms] 11.44/4.49 (6) HASKELL 11.44/4.49 (7) Narrow [SOUND, 0 ms] 11.44/4.49 (8) AND 11.44/4.49 (9) QDP 11.44/4.49 (10) DependencyGraphProof [EQUIVALENT, 0 ms] 11.44/4.49 (11) AND 11.44/4.49 (12) QDP 11.44/4.49 (13) MRRProof [EQUIVALENT, 0 ms] 11.44/4.49 (14) QDP 11.44/4.49 (15) PisEmptyProof [EQUIVALENT, 0 ms] 11.44/4.49 (16) YES 11.44/4.49 (17) QDP 11.44/4.49 (18) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.44/4.49 (19) YES 11.44/4.49 (20) QDP 11.44/4.49 (21) DependencyGraphProof [EQUIVALENT, 0 ms] 11.44/4.49 (22) AND 11.44/4.49 (23) QDP 11.44/4.49 (24) MRRProof [EQUIVALENT, 0 ms] 11.44/4.49 (25) QDP 11.44/4.49 (26) PisEmptyProof [EQUIVALENT, 0 ms] 11.44/4.49 (27) YES 11.44/4.49 (28) QDP 11.44/4.49 (29) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.44/4.49 (30) YES 11.44/4.49 (31) QDP 11.44/4.49 (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.44/4.49 (33) YES 11.44/4.49 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (0) 11.44/4.49 Obligation: 11.44/4.49 mainModule Main 11.44/4.49 module Main where { 11.44/4.49 import qualified Prelude; 11.44/4.49 } 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (1) IFR (EQUIVALENT) 11.44/4.49 If Reductions: 11.44/4.49 The following If expression 11.44/4.49 "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" 11.44/4.49 is transformed to 11.44/4.49 "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); 11.44/4.49 primModNatS0 x y False = Succ x; 11.44/4.49 " 11.44/4.49 The following If expression 11.44/4.49 "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" 11.44/4.49 is transformed to 11.44/4.49 "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); 11.44/4.49 primDivNatS0 x y False = Zero; 11.44/4.49 " 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (2) 11.44/4.49 Obligation: 11.44/4.49 mainModule Main 11.44/4.49 module Main where { 11.44/4.49 import qualified Prelude; 11.44/4.49 } 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (3) BR (EQUIVALENT) 11.44/4.49 Replaced joker patterns by fresh variables and removed binding patterns. 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (4) 11.44/4.49 Obligation: 11.44/4.49 mainModule Main 11.44/4.49 module Main where { 11.44/4.49 import qualified Prelude; 11.44/4.49 } 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (5) COR (EQUIVALENT) 11.44/4.49 Cond Reductions: 11.44/4.49 The following Function with conditions 11.44/4.49 "undefined |Falseundefined; 11.44/4.49 " 11.44/4.49 is transformed to 11.44/4.49 "undefined = undefined1; 11.44/4.49 " 11.44/4.49 "undefined0 True = undefined; 11.44/4.49 " 11.44/4.49 "undefined1 = undefined0 False; 11.44/4.49 " 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (6) 11.44/4.49 Obligation: 11.44/4.49 mainModule Main 11.44/4.49 module Main where { 11.44/4.49 import qualified Prelude; 11.44/4.49 } 11.44/4.49 11.44/4.49 ---------------------------------------- 11.44/4.49 11.44/4.49 (7) Narrow (SOUND) 11.44/4.49 Haskell To QDPs 11.44/4.49 11.44/4.49 digraph dp_graph { 11.44/4.49 node [outthreshold=100, inthreshold=100];1[label="quotRem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.44/4.49 3[label="quotRem wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.44/4.49 4[label="quotRem wv3 wv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 11.44/4.49 5[label="primQrmInt wv3 wv4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 11.44/4.49 6[label="(primQuotInt wv3 wv4,primRemInt wv3 wv4)",fontsize=16,color="green",shape="box"];6 -> 7[label="",style="dashed", color="green", weight=3]; 11.44/4.49 6 -> 8[label="",style="dashed", color="green", weight=3]; 11.44/4.49 7[label="primQuotInt wv3 wv4",fontsize=16,color="burlywood",shape="box"];667[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];7 -> 667[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 667 -> 9[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 668[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];7 -> 668[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 668 -> 10[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 8[label="primRemInt wv3 wv4",fontsize=16,color="burlywood",shape="box"];669[label="wv3/Pos wv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 669[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 669 -> 11[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 670[label="wv3/Neg wv30",fontsize=10,color="white",style="solid",shape="box"];8 -> 670[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 670 -> 12[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 9[label="primQuotInt (Pos wv30) wv4",fontsize=16,color="burlywood",shape="box"];671[label="wv4/Pos wv40",fontsize=10,color="white",style="solid",shape="box"];9 -> 671[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 671 -> 13[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 672[label="wv4/Neg wv40",fontsize=10,color="white",style="solid",shape="box"];9 -> 672[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 672 -> 14[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 10[label="primQuotInt (Neg wv30) wv4",fontsize=16,color="burlywood",shape="box"];673[label="wv4/Pos wv40",fontsize=10,color="white",style="solid",shape="box"];10 -> 673[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 673 -> 15[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 674[label="wv4/Neg wv40",fontsize=10,color="white",style="solid",shape="box"];10 -> 674[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 674 -> 16[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 11[label="primRemInt (Pos wv30) wv4",fontsize=16,color="burlywood",shape="box"];675[label="wv4/Pos wv40",fontsize=10,color="white",style="solid",shape="box"];11 -> 675[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 675 -> 17[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 676[label="wv4/Neg wv40",fontsize=10,color="white",style="solid",shape="box"];11 -> 676[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 676 -> 18[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 12[label="primRemInt (Neg wv30) wv4",fontsize=16,color="burlywood",shape="box"];677[label="wv4/Pos wv40",fontsize=10,color="white",style="solid",shape="box"];12 -> 677[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 677 -> 19[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 678[label="wv4/Neg wv40",fontsize=10,color="white",style="solid",shape="box"];12 -> 678[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 678 -> 20[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 13[label="primQuotInt (Pos wv30) (Pos wv40)",fontsize=16,color="burlywood",shape="box"];679[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];13 -> 679[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 679 -> 21[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 680[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 680[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 680 -> 22[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 14[label="primQuotInt (Pos wv30) (Neg wv40)",fontsize=16,color="burlywood",shape="box"];681[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];14 -> 681[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 681 -> 23[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 682[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 682[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 682 -> 24[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 15[label="primQuotInt (Neg wv30) (Pos wv40)",fontsize=16,color="burlywood",shape="box"];683[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];15 -> 683[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 683 -> 25[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 684[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 684[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 684 -> 26[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 16[label="primQuotInt (Neg wv30) (Neg wv40)",fontsize=16,color="burlywood",shape="box"];685[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];16 -> 685[label="",style="solid", color="burlywood", weight=9]; 11.44/4.49 685 -> 27[label="",style="solid", color="burlywood", weight=3]; 11.44/4.49 686[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 686[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 686 -> 28[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 17[label="primRemInt (Pos wv30) (Pos wv40)",fontsize=16,color="burlywood",shape="box"];687[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];17 -> 687[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 687 -> 29[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 688[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 688[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 688 -> 30[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 18[label="primRemInt (Pos wv30) (Neg wv40)",fontsize=16,color="burlywood",shape="box"];689[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];18 -> 689[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 689 -> 31[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 690[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 690[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 690 -> 32[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 19[label="primRemInt (Neg wv30) (Pos wv40)",fontsize=16,color="burlywood",shape="box"];691[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];19 -> 691[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 691 -> 33[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 692[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 692[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 692 -> 34[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 20[label="primRemInt (Neg wv30) (Neg wv40)",fontsize=16,color="burlywood",shape="box"];693[label="wv40/Succ wv400",fontsize=10,color="white",style="solid",shape="box"];20 -> 693[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 693 -> 35[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 694[label="wv40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 694[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 694 -> 36[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 21[label="primQuotInt (Pos wv30) (Pos (Succ wv400))",fontsize=16,color="black",shape="box"];21 -> 37[label="",style="solid", color="black", weight=3]; 11.44/4.50 22[label="primQuotInt (Pos wv30) (Pos Zero)",fontsize=16,color="black",shape="box"];22 -> 38[label="",style="solid", color="black", weight=3]; 11.44/4.50 23[label="primQuotInt (Pos wv30) (Neg (Succ wv400))",fontsize=16,color="black",shape="box"];23 -> 39[label="",style="solid", color="black", weight=3]; 11.44/4.50 24[label="primQuotInt (Pos wv30) (Neg Zero)",fontsize=16,color="black",shape="box"];24 -> 40[label="",style="solid", color="black", weight=3]; 11.44/4.50 25[label="primQuotInt (Neg wv30) (Pos (Succ wv400))",fontsize=16,color="black",shape="box"];25 -> 41[label="",style="solid", color="black", weight=3]; 11.44/4.50 26[label="primQuotInt (Neg wv30) (Pos Zero)",fontsize=16,color="black",shape="box"];26 -> 42[label="",style="solid", color="black", weight=3]; 11.44/4.50 27[label="primQuotInt (Neg wv30) (Neg (Succ wv400))",fontsize=16,color="black",shape="box"];27 -> 43[label="",style="solid", color="black", weight=3]; 11.44/4.50 28[label="primQuotInt (Neg wv30) (Neg Zero)",fontsize=16,color="black",shape="box"];28 -> 44[label="",style="solid", color="black", weight=3]; 11.44/4.50 29[label="primRemInt (Pos wv30) (Pos (Succ wv400))",fontsize=16,color="black",shape="box"];29 -> 45[label="",style="solid", color="black", weight=3]; 11.44/4.50 30[label="primRemInt (Pos wv30) (Pos Zero)",fontsize=16,color="black",shape="box"];30 -> 46[label="",style="solid", color="black", weight=3]; 11.44/4.50 31[label="primRemInt (Pos wv30) (Neg (Succ wv400))",fontsize=16,color="black",shape="box"];31 -> 47[label="",style="solid", color="black", weight=3]; 11.44/4.50 32[label="primRemInt (Pos wv30) (Neg Zero)",fontsize=16,color="black",shape="box"];32 -> 48[label="",style="solid", color="black", weight=3]; 11.44/4.50 33[label="primRemInt (Neg wv30) (Pos (Succ wv400))",fontsize=16,color="black",shape="box"];33 -> 49[label="",style="solid", color="black", weight=3]; 11.44/4.50 34[label="primRemInt (Neg wv30) (Pos Zero)",fontsize=16,color="black",shape="box"];34 -> 50[label="",style="solid", color="black", weight=3]; 11.44/4.50 35[label="primRemInt (Neg wv30) (Neg (Succ wv400))",fontsize=16,color="black",shape="box"];35 -> 51[label="",style="solid", color="black", weight=3]; 11.44/4.50 36[label="primRemInt (Neg wv30) (Neg Zero)",fontsize=16,color="black",shape="box"];36 -> 52[label="",style="solid", color="black", weight=3]; 11.44/4.50 37[label="Pos (primDivNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];37 -> 53[label="",style="dashed", color="green", weight=3]; 11.44/4.50 38[label="error []",fontsize=16,color="black",shape="triangle"];38 -> 54[label="",style="solid", color="black", weight=3]; 11.44/4.50 39[label="Neg (primDivNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];39 -> 55[label="",style="dashed", color="green", weight=3]; 11.44/4.50 40 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 40[label="error []",fontsize=16,color="magenta"];41[label="Neg (primDivNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];41 -> 56[label="",style="dashed", color="green", weight=3]; 11.44/4.50 42 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 42[label="error []",fontsize=16,color="magenta"];43[label="Pos (primDivNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];43 -> 57[label="",style="dashed", color="green", weight=3]; 11.44/4.50 44 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 44[label="error []",fontsize=16,color="magenta"];45[label="Pos (primModNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];45 -> 58[label="",style="dashed", color="green", weight=3]; 11.44/4.50 46 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 46[label="error []",fontsize=16,color="magenta"];47[label="Pos (primModNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];47 -> 59[label="",style="dashed", color="green", weight=3]; 11.44/4.50 48 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 48[label="error []",fontsize=16,color="magenta"];49[label="Neg (primModNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];49 -> 60[label="",style="dashed", color="green", weight=3]; 11.44/4.50 50 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 50[label="error []",fontsize=16,color="magenta"];51[label="Neg (primModNatS wv30 (Succ wv400))",fontsize=16,color="green",shape="box"];51 -> 61[label="",style="dashed", color="green", weight=3]; 11.44/4.50 52 -> 38[label="",style="dashed", color="red", weight=0]; 11.44/4.50 52[label="error []",fontsize=16,color="magenta"];53[label="primDivNatS wv30 (Succ wv400)",fontsize=16,color="burlywood",shape="triangle"];695[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];53 -> 695[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 695 -> 62[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 696[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 696[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 696 -> 63[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 54[label="error []",fontsize=16,color="red",shape="box"];55 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 55[label="primDivNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];55 -> 64[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 56 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 56[label="primDivNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];56 -> 65[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 57 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 57[label="primDivNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];57 -> 66[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 57 -> 67[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 58[label="primModNatS wv30 (Succ wv400)",fontsize=16,color="burlywood",shape="triangle"];697[label="wv30/Succ wv300",fontsize=10,color="white",style="solid",shape="box"];58 -> 697[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 697 -> 68[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 698[label="wv30/Zero",fontsize=10,color="white",style="solid",shape="box"];58 -> 698[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 698 -> 69[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 59 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 59[label="primModNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];59 -> 70[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 60 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 60[label="primModNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];60 -> 71[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 61 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 61[label="primModNatS wv30 (Succ wv400)",fontsize=16,color="magenta"];61 -> 72[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 61 -> 73[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 62[label="primDivNatS (Succ wv300) (Succ wv400)",fontsize=16,color="black",shape="box"];62 -> 74[label="",style="solid", color="black", weight=3]; 11.44/4.50 63[label="primDivNatS Zero (Succ wv400)",fontsize=16,color="black",shape="box"];63 -> 75[label="",style="solid", color="black", weight=3]; 11.44/4.50 64[label="wv400",fontsize=16,color="green",shape="box"];65[label="wv30",fontsize=16,color="green",shape="box"];66[label="wv400",fontsize=16,color="green",shape="box"];67[label="wv30",fontsize=16,color="green",shape="box"];68[label="primModNatS (Succ wv300) (Succ wv400)",fontsize=16,color="black",shape="box"];68 -> 76[label="",style="solid", color="black", weight=3]; 11.44/4.50 69[label="primModNatS Zero (Succ wv400)",fontsize=16,color="black",shape="box"];69 -> 77[label="",style="solid", color="black", weight=3]; 11.44/4.50 70[label="wv400",fontsize=16,color="green",shape="box"];71[label="wv30",fontsize=16,color="green",shape="box"];72[label="wv400",fontsize=16,color="green",shape="box"];73[label="wv30",fontsize=16,color="green",shape="box"];74[label="primDivNatS0 wv300 wv400 (primGEqNatS wv300 wv400)",fontsize=16,color="burlywood",shape="box"];699[label="wv300/Succ wv3000",fontsize=10,color="white",style="solid",shape="box"];74 -> 699[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 699 -> 78[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 700[label="wv300/Zero",fontsize=10,color="white",style="solid",shape="box"];74 -> 700[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 700 -> 79[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 75[label="Zero",fontsize=16,color="green",shape="box"];76[label="primModNatS0 wv300 wv400 (primGEqNatS wv300 wv400)",fontsize=16,color="burlywood",shape="box"];701[label="wv300/Succ wv3000",fontsize=10,color="white",style="solid",shape="box"];76 -> 701[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 701 -> 80[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 702[label="wv300/Zero",fontsize=10,color="white",style="solid",shape="box"];76 -> 702[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 702 -> 81[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 77[label="Zero",fontsize=16,color="green",shape="box"];78[label="primDivNatS0 (Succ wv3000) wv400 (primGEqNatS (Succ wv3000) wv400)",fontsize=16,color="burlywood",shape="box"];703[label="wv400/Succ wv4000",fontsize=10,color="white",style="solid",shape="box"];78 -> 703[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 703 -> 82[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 704[label="wv400/Zero",fontsize=10,color="white",style="solid",shape="box"];78 -> 704[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 704 -> 83[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 79[label="primDivNatS0 Zero wv400 (primGEqNatS Zero wv400)",fontsize=16,color="burlywood",shape="box"];705[label="wv400/Succ wv4000",fontsize=10,color="white",style="solid",shape="box"];79 -> 705[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 705 -> 84[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 706[label="wv400/Zero",fontsize=10,color="white",style="solid",shape="box"];79 -> 706[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 706 -> 85[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 80[label="primModNatS0 (Succ wv3000) wv400 (primGEqNatS (Succ wv3000) wv400)",fontsize=16,color="burlywood",shape="box"];707[label="wv400/Succ wv4000",fontsize=10,color="white",style="solid",shape="box"];80 -> 707[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 707 -> 86[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 708[label="wv400/Zero",fontsize=10,color="white",style="solid",shape="box"];80 -> 708[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 708 -> 87[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 81[label="primModNatS0 Zero wv400 (primGEqNatS Zero wv400)",fontsize=16,color="burlywood",shape="box"];709[label="wv400/Succ wv4000",fontsize=10,color="white",style="solid",shape="box"];81 -> 709[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 709 -> 88[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 710[label="wv400/Zero",fontsize=10,color="white",style="solid",shape="box"];81 -> 710[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 710 -> 89[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 82[label="primDivNatS0 (Succ wv3000) (Succ wv4000) (primGEqNatS (Succ wv3000) (Succ wv4000))",fontsize=16,color="black",shape="box"];82 -> 90[label="",style="solid", color="black", weight=3]; 11.44/4.50 83[label="primDivNatS0 (Succ wv3000) Zero (primGEqNatS (Succ wv3000) Zero)",fontsize=16,color="black",shape="box"];83 -> 91[label="",style="solid", color="black", weight=3]; 11.44/4.50 84[label="primDivNatS0 Zero (Succ wv4000) (primGEqNatS Zero (Succ wv4000))",fontsize=16,color="black",shape="box"];84 -> 92[label="",style="solid", color="black", weight=3]; 11.44/4.50 85[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];85 -> 93[label="",style="solid", color="black", weight=3]; 11.44/4.50 86[label="primModNatS0 (Succ wv3000) (Succ wv4000) (primGEqNatS (Succ wv3000) (Succ wv4000))",fontsize=16,color="black",shape="box"];86 -> 94[label="",style="solid", color="black", weight=3]; 11.44/4.50 87[label="primModNatS0 (Succ wv3000) Zero (primGEqNatS (Succ wv3000) Zero)",fontsize=16,color="black",shape="box"];87 -> 95[label="",style="solid", color="black", weight=3]; 11.44/4.50 88[label="primModNatS0 Zero (Succ wv4000) (primGEqNatS Zero (Succ wv4000))",fontsize=16,color="black",shape="box"];88 -> 96[label="",style="solid", color="black", weight=3]; 11.44/4.50 89[label="primModNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];89 -> 97[label="",style="solid", color="black", weight=3]; 11.44/4.50 90 -> 535[label="",style="dashed", color="red", weight=0]; 11.44/4.50 90[label="primDivNatS0 (Succ wv3000) (Succ wv4000) (primGEqNatS wv3000 wv4000)",fontsize=16,color="magenta"];90 -> 536[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 90 -> 537[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 90 -> 538[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 90 -> 539[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 91[label="primDivNatS0 (Succ wv3000) Zero True",fontsize=16,color="black",shape="box"];91 -> 100[label="",style="solid", color="black", weight=3]; 11.44/4.50 92[label="primDivNatS0 Zero (Succ wv4000) False",fontsize=16,color="black",shape="box"];92 -> 101[label="",style="solid", color="black", weight=3]; 11.44/4.50 93[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];93 -> 102[label="",style="solid", color="black", weight=3]; 11.44/4.50 94 -> 580[label="",style="dashed", color="red", weight=0]; 11.44/4.50 94[label="primModNatS0 (Succ wv3000) (Succ wv4000) (primGEqNatS wv3000 wv4000)",fontsize=16,color="magenta"];94 -> 581[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 94 -> 582[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 94 -> 583[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 94 -> 584[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 95[label="primModNatS0 (Succ wv3000) Zero True",fontsize=16,color="black",shape="box"];95 -> 105[label="",style="solid", color="black", weight=3]; 11.44/4.50 96[label="primModNatS0 Zero (Succ wv4000) False",fontsize=16,color="black",shape="box"];96 -> 106[label="",style="solid", color="black", weight=3]; 11.44/4.50 97[label="primModNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];97 -> 107[label="",style="solid", color="black", weight=3]; 11.44/4.50 536[label="wv4000",fontsize=16,color="green",shape="box"];537[label="wv4000",fontsize=16,color="green",shape="box"];538[label="wv3000",fontsize=16,color="green",shape="box"];539[label="wv3000",fontsize=16,color="green",shape="box"];535[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS wv48 wv49)",fontsize=16,color="burlywood",shape="triangle"];711[label="wv48/Succ wv480",fontsize=10,color="white",style="solid",shape="box"];535 -> 711[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 711 -> 576[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 712[label="wv48/Zero",fontsize=10,color="white",style="solid",shape="box"];535 -> 712[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 712 -> 577[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 100[label="Succ (primDivNatS (primMinusNatS (Succ wv3000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];100 -> 112[label="",style="dashed", color="green", weight=3]; 11.44/4.50 101[label="Zero",fontsize=16,color="green",shape="box"];102[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];102 -> 113[label="",style="dashed", color="green", weight=3]; 11.44/4.50 581[label="wv3000",fontsize=16,color="green",shape="box"];582[label="wv4000",fontsize=16,color="green",shape="box"];583[label="wv3000",fontsize=16,color="green",shape="box"];584[label="wv4000",fontsize=16,color="green",shape="box"];580[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS wv53 wv54)",fontsize=16,color="burlywood",shape="triangle"];713[label="wv53/Succ wv530",fontsize=10,color="white",style="solid",shape="box"];580 -> 713[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 713 -> 621[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 714[label="wv53/Zero",fontsize=10,color="white",style="solid",shape="box"];580 -> 714[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 714 -> 622[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 105 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 105[label="primModNatS (primMinusNatS (Succ wv3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];105 -> 118[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 105 -> 119[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 106[label="Succ Zero",fontsize=16,color="green",shape="box"];107 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 107[label="primModNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];107 -> 120[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 107 -> 121[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 576[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS (Succ wv480) wv49)",fontsize=16,color="burlywood",shape="box"];715[label="wv49/Succ wv490",fontsize=10,color="white",style="solid",shape="box"];576 -> 715[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 715 -> 623[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 716[label="wv49/Zero",fontsize=10,color="white",style="solid",shape="box"];576 -> 716[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 716 -> 624[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 577[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS Zero wv49)",fontsize=16,color="burlywood",shape="box"];717[label="wv49/Succ wv490",fontsize=10,color="white",style="solid",shape="box"];577 -> 717[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 717 -> 625[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 718[label="wv49/Zero",fontsize=10,color="white",style="solid",shape="box"];577 -> 718[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 718 -> 626[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 112 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 112[label="primDivNatS (primMinusNatS (Succ wv3000) Zero) (Succ Zero)",fontsize=16,color="magenta"];112 -> 126[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 112 -> 127[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 113 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 113[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];113 -> 128[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 113 -> 129[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 621[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS (Succ wv530) wv54)",fontsize=16,color="burlywood",shape="box"];719[label="wv54/Succ wv540",fontsize=10,color="white",style="solid",shape="box"];621 -> 719[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 719 -> 627[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 720[label="wv54/Zero",fontsize=10,color="white",style="solid",shape="box"];621 -> 720[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 720 -> 628[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 622[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS Zero wv54)",fontsize=16,color="burlywood",shape="box"];721[label="wv54/Succ wv540",fontsize=10,color="white",style="solid",shape="box"];622 -> 721[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 721 -> 629[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 722[label="wv54/Zero",fontsize=10,color="white",style="solid",shape="box"];622 -> 722[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 722 -> 630[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 118[label="Zero",fontsize=16,color="green",shape="box"];119[label="primMinusNatS (Succ wv3000) Zero",fontsize=16,color="black",shape="triangle"];119 -> 134[label="",style="solid", color="black", weight=3]; 11.44/4.50 120[label="Zero",fontsize=16,color="green",shape="box"];121[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];121 -> 135[label="",style="solid", color="black", weight=3]; 11.44/4.50 623[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS (Succ wv480) (Succ wv490))",fontsize=16,color="black",shape="box"];623 -> 631[label="",style="solid", color="black", weight=3]; 11.44/4.50 624[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS (Succ wv480) Zero)",fontsize=16,color="black",shape="box"];624 -> 632[label="",style="solid", color="black", weight=3]; 11.44/4.50 625[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS Zero (Succ wv490))",fontsize=16,color="black",shape="box"];625 -> 633[label="",style="solid", color="black", weight=3]; 11.44/4.50 626[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];626 -> 634[label="",style="solid", color="black", weight=3]; 11.44/4.50 126[label="Zero",fontsize=16,color="green",shape="box"];127 -> 119[label="",style="dashed", color="red", weight=0]; 11.44/4.50 127[label="primMinusNatS (Succ wv3000) Zero",fontsize=16,color="magenta"];128[label="Zero",fontsize=16,color="green",shape="box"];129 -> 121[label="",style="dashed", color="red", weight=0]; 11.44/4.50 129[label="primMinusNatS Zero Zero",fontsize=16,color="magenta"];627[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS (Succ wv530) (Succ wv540))",fontsize=16,color="black",shape="box"];627 -> 635[label="",style="solid", color="black", weight=3]; 11.44/4.50 628[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS (Succ wv530) Zero)",fontsize=16,color="black",shape="box"];628 -> 636[label="",style="solid", color="black", weight=3]; 11.44/4.50 629[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS Zero (Succ wv540))",fontsize=16,color="black",shape="box"];629 -> 637[label="",style="solid", color="black", weight=3]; 11.44/4.50 630[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];630 -> 638[label="",style="solid", color="black", weight=3]; 11.44/4.50 134[label="Succ wv3000",fontsize=16,color="green",shape="box"];135[label="Zero",fontsize=16,color="green",shape="box"];631 -> 535[label="",style="dashed", color="red", weight=0]; 11.44/4.50 631[label="primDivNatS0 (Succ wv46) (Succ wv47) (primGEqNatS wv480 wv490)",fontsize=16,color="magenta"];631 -> 639[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 631 -> 640[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 632[label="primDivNatS0 (Succ wv46) (Succ wv47) True",fontsize=16,color="black",shape="triangle"];632 -> 641[label="",style="solid", color="black", weight=3]; 11.44/4.50 633[label="primDivNatS0 (Succ wv46) (Succ wv47) False",fontsize=16,color="black",shape="box"];633 -> 642[label="",style="solid", color="black", weight=3]; 11.44/4.50 634 -> 632[label="",style="dashed", color="red", weight=0]; 11.44/4.50 634[label="primDivNatS0 (Succ wv46) (Succ wv47) True",fontsize=16,color="magenta"];635 -> 580[label="",style="dashed", color="red", weight=0]; 11.44/4.50 635[label="primModNatS0 (Succ wv51) (Succ wv52) (primGEqNatS wv530 wv540)",fontsize=16,color="magenta"];635 -> 643[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 635 -> 644[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 636[label="primModNatS0 (Succ wv51) (Succ wv52) True",fontsize=16,color="black",shape="triangle"];636 -> 645[label="",style="solid", color="black", weight=3]; 11.44/4.50 637[label="primModNatS0 (Succ wv51) (Succ wv52) False",fontsize=16,color="black",shape="box"];637 -> 646[label="",style="solid", color="black", weight=3]; 11.44/4.50 638 -> 636[label="",style="dashed", color="red", weight=0]; 11.44/4.50 638[label="primModNatS0 (Succ wv51) (Succ wv52) True",fontsize=16,color="magenta"];639[label="wv490",fontsize=16,color="green",shape="box"];640[label="wv480",fontsize=16,color="green",shape="box"];641[label="Succ (primDivNatS (primMinusNatS (Succ wv46) (Succ wv47)) (Succ (Succ wv47)))",fontsize=16,color="green",shape="box"];641 -> 647[label="",style="dashed", color="green", weight=3]; 11.44/4.50 642[label="Zero",fontsize=16,color="green",shape="box"];643[label="wv540",fontsize=16,color="green",shape="box"];644[label="wv530",fontsize=16,color="green",shape="box"];645 -> 58[label="",style="dashed", color="red", weight=0]; 11.44/4.50 645[label="primModNatS (primMinusNatS (Succ wv51) (Succ wv52)) (Succ (Succ wv52))",fontsize=16,color="magenta"];645 -> 648[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 645 -> 649[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 646[label="Succ (Succ wv51)",fontsize=16,color="green",shape="box"];647 -> 53[label="",style="dashed", color="red", weight=0]; 11.44/4.50 647[label="primDivNatS (primMinusNatS (Succ wv46) (Succ wv47)) (Succ (Succ wv47))",fontsize=16,color="magenta"];647 -> 650[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 647 -> 651[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 648[label="Succ wv52",fontsize=16,color="green",shape="box"];649[label="primMinusNatS (Succ wv51) (Succ wv52)",fontsize=16,color="black",shape="triangle"];649 -> 652[label="",style="solid", color="black", weight=3]; 11.44/4.50 650[label="Succ wv47",fontsize=16,color="green",shape="box"];651 -> 649[label="",style="dashed", color="red", weight=0]; 11.44/4.50 651[label="primMinusNatS (Succ wv46) (Succ wv47)",fontsize=16,color="magenta"];651 -> 653[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 651 -> 654[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 652[label="primMinusNatS wv51 wv52",fontsize=16,color="burlywood",shape="triangle"];723[label="wv51/Succ wv510",fontsize=10,color="white",style="solid",shape="box"];652 -> 723[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 723 -> 655[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 724[label="wv51/Zero",fontsize=10,color="white",style="solid",shape="box"];652 -> 724[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 724 -> 656[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 653[label="wv46",fontsize=16,color="green",shape="box"];654[label="wv47",fontsize=16,color="green",shape="box"];655[label="primMinusNatS (Succ wv510) wv52",fontsize=16,color="burlywood",shape="box"];725[label="wv52/Succ wv520",fontsize=10,color="white",style="solid",shape="box"];655 -> 725[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 725 -> 657[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 726[label="wv52/Zero",fontsize=10,color="white",style="solid",shape="box"];655 -> 726[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 726 -> 658[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 656[label="primMinusNatS Zero wv52",fontsize=16,color="burlywood",shape="box"];727[label="wv52/Succ wv520",fontsize=10,color="white",style="solid",shape="box"];656 -> 727[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 727 -> 659[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 728[label="wv52/Zero",fontsize=10,color="white",style="solid",shape="box"];656 -> 728[label="",style="solid", color="burlywood", weight=9]; 11.44/4.50 728 -> 660[label="",style="solid", color="burlywood", weight=3]; 11.44/4.50 657[label="primMinusNatS (Succ wv510) (Succ wv520)",fontsize=16,color="black",shape="box"];657 -> 661[label="",style="solid", color="black", weight=3]; 11.44/4.50 658[label="primMinusNatS (Succ wv510) Zero",fontsize=16,color="black",shape="box"];658 -> 662[label="",style="solid", color="black", weight=3]; 11.44/4.50 659[label="primMinusNatS Zero (Succ wv520)",fontsize=16,color="black",shape="box"];659 -> 663[label="",style="solid", color="black", weight=3]; 11.44/4.50 660[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];660 -> 664[label="",style="solid", color="black", weight=3]; 11.44/4.50 661 -> 652[label="",style="dashed", color="red", weight=0]; 11.44/4.50 661[label="primMinusNatS wv510 wv520",fontsize=16,color="magenta"];661 -> 665[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 661 -> 666[label="",style="dashed", color="magenta", weight=3]; 11.44/4.50 662[label="Succ wv510",fontsize=16,color="green",shape="box"];663[label="Zero",fontsize=16,color="green",shape="box"];664[label="Zero",fontsize=16,color="green",shape="box"];665[label="wv510",fontsize=16,color="green",shape="box"];666[label="wv520",fontsize=16,color="green",shape="box"];} 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (8) 11.44/4.50 Complex Obligation (AND) 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (9) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primModNatS00(wv51, wv52) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) 11.44/4.50 new_primModNatS(Succ(Succ(wv3000)), Zero) -> new_primModNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 new_primModNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primModNatS0(wv3000, wv4000, wv3000, wv4000) 11.44/4.50 new_primModNatS(Succ(Zero), Zero) -> new_primModNatS(new_primMinusNatS1, Zero) 11.44/4.50 new_primModNatS0(wv51, wv52, Succ(wv530), Succ(wv540)) -> new_primModNatS0(wv51, wv52, wv530, wv540) 11.44/4.50 new_primModNatS0(wv51, wv52, Succ(wv530), Zero) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) 11.44/4.50 new_primModNatS0(wv51, wv52, Zero, Zero) -> new_primModNatS00(wv51, wv52) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (10) DependencyGraphProof (EQUIVALENT) 11.44/4.50 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (11) 11.44/4.50 Complex Obligation (AND) 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (12) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primModNatS(Succ(Succ(wv3000)), Zero) -> new_primModNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (13) MRRProof (EQUIVALENT) 11.44/4.50 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 11.44/4.50 11.44/4.50 Strictly oriented dependency pairs: 11.44/4.50 11.44/4.50 new_primModNatS(Succ(Succ(wv3000)), Zero) -> new_primModNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 11.44/4.50 Strictly oriented rules of the TRS R: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 Used ordering: Polynomial interpretation [POLO]: 11.44/4.50 11.44/4.50 POL(Succ(x_1)) = 1 + x_1 11.44/4.50 POL(Zero) = 2 11.44/4.50 POL(new_primMinusNatS0(x_1)) = 1 + x_1 11.44/4.50 POL(new_primMinusNatS1) = 2 11.44/4.50 POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 11.44/4.50 POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2 11.44/4.50 POL(new_primModNatS(x_1, x_2)) = x_1 + x_2 11.44/4.50 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (14) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 P is empty. 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (15) PisEmptyProof (EQUIVALENT) 11.44/4.50 The TRS P is empty. Hence, there is no (P,Q,R) chain. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (16) 11.44/4.50 YES 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (17) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primModNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primModNatS0(wv3000, wv4000, wv3000, wv4000) 11.44/4.50 new_primModNatS0(wv51, wv52, Succ(wv530), Succ(wv540)) -> new_primModNatS0(wv51, wv52, wv530, wv540) 11.44/4.50 new_primModNatS0(wv51, wv52, Succ(wv530), Zero) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) 11.44/4.50 new_primModNatS0(wv51, wv52, Zero, Zero) -> new_primModNatS00(wv51, wv52) 11.44/4.50 new_primModNatS00(wv51, wv52) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (18) QDPSizeChangeProof (EQUIVALENT) 11.44/4.50 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 11.44/4.50 11.44/4.50 Order:Polynomial interpretation [POLO]: 11.44/4.50 11.44/4.50 POL(Succ(x_1)) = 1 + x_1 11.44/4.50 POL(Zero) = 1 11.44/4.50 POL(new_primMinusNatS2(x_1, x_2)) = x_1 11.44/4.50 POL(new_primMinusNatS3(x_1, x_2)) = x_1 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 From the DPs we obtained the following set of size-change graphs: 11.44/4.50 *new_primModNatS0(wv51, wv52, Succ(wv530), Zero) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1 11.44/4.50 11.44/4.50 11.44/4.50 *new_primModNatS00(wv51, wv52) -> new_primModNatS(new_primMinusNatS2(wv51, wv52), Succ(wv52)) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1 11.44/4.50 11.44/4.50 11.44/4.50 *new_primModNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primModNatS0(wv3000, wv4000, wv3000, wv4000) (allowed arguments on rhs = {1, 2, 3, 4}) 11.44/4.50 The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 11.44/4.50 11.44/4.50 11.44/4.50 *new_primModNatS0(wv51, wv52, Succ(wv530), Succ(wv540)) -> new_primModNatS0(wv51, wv52, wv530, wv540) (allowed arguments on rhs = {1, 2, 3, 4}) 11.44/4.50 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 11.44/4.50 11.44/4.50 11.44/4.50 *new_primModNatS0(wv51, wv52, Zero, Zero) -> new_primModNatS00(wv51, wv52) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1, 2 >= 2 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 We oriented the following set of usable rules [AAECC05,FROCOS05]. 11.44/4.50 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (19) 11.44/4.50 YES 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (20) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primDivNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primDivNatS0(wv3000, wv4000, wv3000, wv4000) 11.44/4.50 new_primDivNatS0(wv46, wv47, Zero, Zero) -> new_primDivNatS00(wv46, wv47) 11.44/4.50 new_primDivNatS(Succ(Succ(wv3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 new_primDivNatS0(wv46, wv47, Succ(wv480), Succ(wv490)) -> new_primDivNatS0(wv46, wv47, wv480, wv490) 11.44/4.50 new_primDivNatS0(wv46, wv47, Succ(wv480), Zero) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) 11.44/4.50 new_primDivNatS00(wv46, wv47) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) 11.44/4.50 new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS1, Zero) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (21) DependencyGraphProof (EQUIVALENT) 11.44/4.50 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (22) 11.44/4.50 Complex Obligation (AND) 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (23) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primDivNatS(Succ(Succ(wv3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (24) MRRProof (EQUIVALENT) 11.44/4.50 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 11.44/4.50 11.44/4.50 Strictly oriented dependency pairs: 11.44/4.50 11.44/4.50 new_primDivNatS(Succ(Succ(wv3000)), Zero) -> new_primDivNatS(new_primMinusNatS0(wv3000), Zero) 11.44/4.50 11.44/4.50 Strictly oriented rules of the TRS R: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 Used ordering: Polynomial interpretation [POLO]: 11.44/4.50 11.44/4.50 POL(Succ(x_1)) = 1 + x_1 11.44/4.50 POL(Zero) = 2 11.44/4.50 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 11.44/4.50 POL(new_primMinusNatS0(x_1)) = 1 + x_1 11.44/4.50 POL(new_primMinusNatS1) = 2 11.44/4.50 POL(new_primMinusNatS2(x_1, x_2)) = 1 + 2*x_1 + 2*x_2 11.44/4.50 POL(new_primMinusNatS3(x_1, x_2)) = 1 + 2*x_1 + x_2 11.44/4.50 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (25) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 P is empty. 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (26) PisEmptyProof (EQUIVALENT) 11.44/4.50 The TRS P is empty. Hence, there is no (P,Q,R) chain. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (27) 11.44/4.50 YES 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (28) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primDivNatS0(wv46, wv47, Zero, Zero) -> new_primDivNatS00(wv46, wv47) 11.44/4.50 new_primDivNatS00(wv46, wv47) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) 11.44/4.50 new_primDivNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primDivNatS0(wv3000, wv4000, wv3000, wv4000) 11.44/4.50 new_primDivNatS0(wv46, wv47, Succ(wv480), Succ(wv490)) -> new_primDivNatS0(wv46, wv47, wv480, wv490) 11.44/4.50 new_primDivNatS0(wv46, wv47, Succ(wv480), Zero) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) 11.44/4.50 11.44/4.50 The TRS R consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS1 -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 new_primMinusNatS0(wv3000) -> Succ(wv3000) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 11.44/4.50 The set Q consists of the following terms: 11.44/4.50 11.44/4.50 new_primMinusNatS3(Succ(x0), Zero) 11.44/4.50 new_primMinusNatS3(Zero, Succ(x0)) 11.44/4.50 new_primMinusNatS3(Zero, Zero) 11.44/4.50 new_primMinusNatS3(Succ(x0), Succ(x1)) 11.44/4.50 new_primMinusNatS1 11.44/4.50 new_primMinusNatS0(x0) 11.44/4.50 new_primMinusNatS2(x0, x1) 11.44/4.50 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (29) QDPSizeChangeProof (EQUIVALENT) 11.44/4.50 We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. 11.44/4.50 11.44/4.50 Order:Polynomial interpretation [POLO]: 11.44/4.50 11.44/4.50 POL(Succ(x_1)) = 1 + x_1 11.44/4.50 POL(Zero) = 1 11.44/4.50 POL(new_primMinusNatS2(x_1, x_2)) = x_1 11.44/4.50 POL(new_primMinusNatS3(x_1, x_2)) = x_1 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 From the DPs we obtained the following set of size-change graphs: 11.44/4.50 *new_primDivNatS00(wv46, wv47) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1 11.44/4.50 11.44/4.50 11.44/4.50 *new_primDivNatS(Succ(Succ(wv3000)), Succ(wv4000)) -> new_primDivNatS0(wv3000, wv4000, wv3000, wv4000) (allowed arguments on rhs = {1, 2, 3, 4}) 11.44/4.50 The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 11.44/4.50 11.44/4.50 11.44/4.50 *new_primDivNatS0(wv46, wv47, Succ(wv480), Succ(wv490)) -> new_primDivNatS0(wv46, wv47, wv480, wv490) (allowed arguments on rhs = {1, 2, 3, 4}) 11.44/4.50 The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 11.44/4.50 11.44/4.50 11.44/4.50 *new_primDivNatS0(wv46, wv47, Zero, Zero) -> new_primDivNatS00(wv46, wv47) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1, 2 >= 2 11.44/4.50 11.44/4.50 11.44/4.50 *new_primDivNatS0(wv46, wv47, Succ(wv480), Zero) -> new_primDivNatS(new_primMinusNatS2(wv46, wv47), Succ(wv47)) (allowed arguments on rhs = {1, 2}) 11.44/4.50 The graph contains the following edges 1 >= 1 11.44/4.50 11.44/4.50 11.44/4.50 11.44/4.50 We oriented the following set of usable rules [AAECC05,FROCOS05]. 11.44/4.50 11.44/4.50 new_primMinusNatS3(Zero, Zero) -> Zero 11.44/4.50 new_primMinusNatS3(Zero, Succ(wv520)) -> Zero 11.44/4.50 new_primMinusNatS3(Succ(wv510), Zero) -> Succ(wv510) 11.44/4.50 new_primMinusNatS3(Succ(wv510), Succ(wv520)) -> new_primMinusNatS3(wv510, wv520) 11.44/4.50 new_primMinusNatS2(wv51, wv52) -> new_primMinusNatS3(wv51, wv52) 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (30) 11.44/4.50 YES 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (31) 11.44/4.50 Obligation: 11.44/4.50 Q DP problem: 11.44/4.50 The TRS P consists of the following rules: 11.44/4.50 11.44/4.50 new_primMinusNatS(Succ(wv510), Succ(wv520)) -> new_primMinusNatS(wv510, wv520) 11.44/4.50 11.44/4.50 R is empty. 11.44/4.50 Q is empty. 11.44/4.50 We have to consider all minimal (P,Q,R)-chains. 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (32) QDPSizeChangeProof (EQUIVALENT) 11.44/4.50 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. 11.44/4.50 11.44/4.50 From the DPs we obtained the following set of size-change graphs: 11.44/4.50 *new_primMinusNatS(Succ(wv510), Succ(wv520)) -> new_primMinusNatS(wv510, wv520) 11.44/4.50 The graph contains the following edges 1 > 1, 2 > 2 11.44/4.50 11.44/4.50 11.44/4.50 ---------------------------------------- 11.44/4.50 11.44/4.50 (33) 11.44/4.50 YES 11.60/5.55 EOF