/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/benchmark/theBenchmark.hs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) IFR [EQUIVALENT, 0 ms] (4) HASKELL (5) BR [EQUIVALENT, 0 ms] (6) HASKELL (7) COR [EQUIVALENT, 0 ms] (8) HASKELL (9) LetRed [EQUIVALENT, 0 ms] (10) HASKELL (11) NumRed [SOUND, 0 ms] (12) HASKELL (13) Narrow [SOUND, 0 ms] (14) AND (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) AND (18) QDP (19) MRRProof [EQUIVALENT, 65 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) QDPSizeChangeProof [EQUIVALENT, 0 ms] (28) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\(q,_)->q" is transformed to "q1 (q,_) = q; " The following Lambda expression "\(_,r)->r" is transformed to "r0 (_,r) = r; " The following Lambda expression "\(m,_)->m" is transformed to "m0 (m,_) = m; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) IFR (EQUIVALENT) If Reductions: The following If expression "if primGEqNatS x y then Succ (primDivNatS (primMinusNatS x y) (Succ y)) else Zero" is transformed to "primDivNatS0 x y True = Succ (primDivNatS (primMinusNatS x y) (Succ y)); primDivNatS0 x y False = Zero; " The following If expression "if primGEqNatS x y then primModNatS (primMinusNatS x y) (Succ y) else Succ x" is transformed to "primModNatS0 x y True = primModNatS (primMinusNatS x y) (Succ y); primModNatS0 x y False = Succ x; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. Binding Reductions: The bind variable of the following binding Pattern "frac@(Float vv vw)" is replaced by the following term "Float vv vw" The bind variable of the following binding Pattern "frac@(Double ww wx)" is replaced by the following term "Double ww wx" ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (8) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (9) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "(fromIntegral q,r :% y) where { q = q1 vu30; ; q1 (q,wy) = q; ; r = r0 vu30; ; r0 (wz,r) = r; ; vu30 = quotRem x y; } " are unpacked to the following functions on top level "properFractionR xw xx = properFractionR0 xw xx (properFractionVu30 xw xx); " "properFractionQ1 xw xx (q,wy) = q; " "properFractionQ xw xx = properFractionQ1 xw xx (properFractionVu30 xw xx); " "properFractionR0 xw xx (wz,r) = r; " "properFractionVu30 xw xx = quotRem xw xx; " The bindings of the following Let/Where expression "m where { m = m0 vu6; ; m0 (m,xv) = m; ; vu6 = properFraction x; } " are unpacked to the following functions on top level "truncateVu6 xy = properFraction xy; " "truncateM xy = truncateM0 xy (truncateVu6 xy); " "truncateM0 xy (m,xv) = m; " ---------------------------------------- (10) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (11) NumRed (SOUND) Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. ---------------------------------------- (12) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (13) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="fromEnum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="fromEnum xz3",fontsize=16,color="blue",shape="box"];354[label="fromEnum :: Bool -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 354[label="",style="solid", color="blue", weight=9]; 354 -> 4[label="",style="solid", color="blue", weight=3]; 355[label="fromEnum :: Char -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 355[label="",style="solid", color="blue", weight=9]; 355 -> 5[label="",style="solid", color="blue", weight=3]; 356[label="fromEnum :: Integer -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 356[label="",style="solid", color="blue", weight=9]; 356 -> 6[label="",style="solid", color="blue", weight=3]; 357[label="fromEnum :: () -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 357[label="",style="solid", color="blue", weight=9]; 357 -> 7[label="",style="solid", color="blue", weight=3]; 358[label="fromEnum :: Ordering -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 358[label="",style="solid", color="blue", weight=9]; 358 -> 8[label="",style="solid", color="blue", weight=3]; 359[label="fromEnum :: Double -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 359[label="",style="solid", color="blue", weight=9]; 359 -> 9[label="",style="solid", color="blue", weight=3]; 360[label="fromEnum :: Float -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 360[label="",style="solid", color="blue", weight=9]; 360 -> 10[label="",style="solid", color="blue", weight=3]; 361[label="fromEnum :: Int -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 361[label="",style="solid", color="blue", weight=9]; 361 -> 11[label="",style="solid", color="blue", weight=3]; 362[label="fromEnum :: (Ratio a) -> Int",fontsize=10,color="white",style="solid",shape="box"];3 -> 362[label="",style="solid", color="blue", weight=9]; 362 -> 12[label="",style="solid", color="blue", weight=3]; 4[label="fromEnum xz3",fontsize=16,color="burlywood",shape="box"];363[label="xz3/False",fontsize=10,color="white",style="solid",shape="box"];4 -> 363[label="",style="solid", color="burlywood", weight=9]; 363 -> 13[label="",style="solid", color="burlywood", weight=3]; 364[label="xz3/True",fontsize=10,color="white",style="solid",shape="box"];4 -> 364[label="",style="solid", color="burlywood", weight=9]; 364 -> 14[label="",style="solid", color="burlywood", weight=3]; 5[label="fromEnum xz3",fontsize=16,color="black",shape="box"];5 -> 15[label="",style="solid", color="black", weight=3]; 6[label="fromEnum xz3",fontsize=16,color="burlywood",shape="box"];365[label="xz3/Integer xz30",fontsize=10,color="white",style="solid",shape="box"];6 -> 365[label="",style="solid", color="burlywood", weight=9]; 365 -> 16[label="",style="solid", color="burlywood", weight=3]; 7[label="fromEnum xz3",fontsize=16,color="burlywood",shape="box"];366[label="xz3/()",fontsize=10,color="white",style="solid",shape="box"];7 -> 366[label="",style="solid", color="burlywood", weight=9]; 366 -> 17[label="",style="solid", color="burlywood", weight=3]; 8[label="fromEnum xz3",fontsize=16,color="burlywood",shape="box"];367[label="xz3/LT",fontsize=10,color="white",style="solid",shape="box"];8 -> 367[label="",style="solid", color="burlywood", weight=9]; 367 -> 18[label="",style="solid", color="burlywood", weight=3]; 368[label="xz3/EQ",fontsize=10,color="white",style="solid",shape="box"];8 -> 368[label="",style="solid", color="burlywood", weight=9]; 368 -> 19[label="",style="solid", color="burlywood", weight=3]; 369[label="xz3/GT",fontsize=10,color="white",style="solid",shape="box"];8 -> 369[label="",style="solid", color="burlywood", weight=9]; 369 -> 20[label="",style="solid", color="burlywood", weight=3]; 9[label="fromEnum xz3",fontsize=16,color="black",shape="box"];9 -> 21[label="",style="solid", color="black", weight=3]; 10[label="fromEnum xz3",fontsize=16,color="black",shape="box"];10 -> 22[label="",style="solid", color="black", weight=3]; 11[label="fromEnum xz3",fontsize=16,color="black",shape="box"];11 -> 23[label="",style="solid", color="black", weight=3]; 12[label="fromEnum xz3",fontsize=16,color="black",shape="box"];12 -> 24[label="",style="solid", color="black", weight=3]; 13[label="fromEnum False",fontsize=16,color="black",shape="box"];13 -> 25[label="",style="solid", color="black", weight=3]; 14[label="fromEnum True",fontsize=16,color="black",shape="box"];14 -> 26[label="",style="solid", color="black", weight=3]; 15[label="primCharToInt xz3",fontsize=16,color="burlywood",shape="box"];370[label="xz3/Char xz30",fontsize=10,color="white",style="solid",shape="box"];15 -> 370[label="",style="solid", color="burlywood", weight=9]; 370 -> 27[label="",style="solid", color="burlywood", weight=3]; 16[label="fromEnum (Integer xz30)",fontsize=16,color="black",shape="box"];16 -> 28[label="",style="solid", color="black", weight=3]; 17[label="fromEnum ()",fontsize=16,color="black",shape="box"];17 -> 29[label="",style="solid", color="black", weight=3]; 18[label="fromEnum LT",fontsize=16,color="black",shape="box"];18 -> 30[label="",style="solid", color="black", weight=3]; 19[label="fromEnum EQ",fontsize=16,color="black",shape="box"];19 -> 31[label="",style="solid", color="black", weight=3]; 20[label="fromEnum GT",fontsize=16,color="black",shape="box"];20 -> 32[label="",style="solid", color="black", weight=3]; 21[label="truncate xz3",fontsize=16,color="black",shape="box"];21 -> 33[label="",style="solid", color="black", weight=3]; 22[label="truncate xz3",fontsize=16,color="black",shape="box"];22 -> 34[label="",style="solid", color="black", weight=3]; 23[label="id xz3",fontsize=16,color="black",shape="box"];23 -> 35[label="",style="solid", color="black", weight=3]; 24[label="truncate xz3",fontsize=16,color="black",shape="box"];24 -> 36[label="",style="solid", color="black", weight=3]; 25[label="Pos Zero",fontsize=16,color="green",shape="box"];26[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];27[label="primCharToInt (Char xz30)",fontsize=16,color="black",shape="box"];27 -> 37[label="",style="solid", color="black", weight=3]; 28[label="xz30",fontsize=16,color="green",shape="box"];29[label="Pos Zero",fontsize=16,color="green",shape="box"];30[label="Pos Zero",fontsize=16,color="green",shape="box"];31[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];32[label="Pos (Succ (Succ Zero))",fontsize=16,color="green",shape="box"];33[label="truncateM xz3",fontsize=16,color="black",shape="box"];33 -> 38[label="",style="solid", color="black", weight=3]; 34[label="truncateM xz3",fontsize=16,color="black",shape="box"];34 -> 39[label="",style="solid", color="black", weight=3]; 35[label="xz3",fontsize=16,color="green",shape="box"];36[label="truncateM xz3",fontsize=16,color="black",shape="box"];36 -> 40[label="",style="solid", color="black", weight=3]; 37[label="Pos xz30",fontsize=16,color="green",shape="box"];38[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3]; 39[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3]; 40[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3]; 41[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="black",shape="box"];41 -> 44[label="",style="solid", color="black", weight=3]; 42[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="black",shape="box"];42 -> 45[label="",style="solid", color="black", weight=3]; 43[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="burlywood",shape="box"];371[label="xz3/xz30 :% xz31",fontsize=10,color="white",style="solid",shape="box"];43 -> 371[label="",style="solid", color="burlywood", weight=9]; 371 -> 46[label="",style="solid", color="burlywood", weight=3]; 44[label="truncateM0 xz3 (floatProperFractionDouble xz3)",fontsize=16,color="burlywood",shape="box"];372[label="xz3/Double xz30 xz31",fontsize=10,color="white",style="solid",shape="box"];44 -> 372[label="",style="solid", color="burlywood", weight=9]; 372 -> 47[label="",style="solid", color="burlywood", weight=3]; 45[label="truncateM0 xz3 (floatProperFractionFloat xz3)",fontsize=16,color="burlywood",shape="box"];373[label="xz3/Float xz30 xz31",fontsize=10,color="white",style="solid",shape="box"];45 -> 373[label="",style="solid", color="burlywood", weight=9]; 373 -> 48[label="",style="solid", color="burlywood", weight=3]; 46[label="truncateM0 (xz30 :% xz31) (properFraction (xz30 :% xz31))",fontsize=16,color="black",shape="box"];46 -> 49[label="",style="solid", color="black", weight=3]; 47[label="truncateM0 (Double xz30 xz31) (floatProperFractionDouble (Double xz30 xz31))",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3]; 48[label="truncateM0 (Float xz30 xz31) (floatProperFractionFloat (Float xz30 xz31))",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3]; 49[label="truncateM0 (xz30 :% xz31) (fromIntegral (properFractionQ xz30 xz31),properFractionR xz30 xz31 :% xz31)",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3]; 50[label="truncateM0 (Double xz30 xz31) (fromInt (xz30 `quot` xz31),Double xz30 xz31 - fromInt (xz30 `quot` xz31))",fontsize=16,color="black",shape="box"];50 -> 53[label="",style="solid", color="black", weight=3]; 51[label="truncateM0 (Float xz30 xz31) (fromInt (xz30 `quot` xz31),Float xz30 xz31 - fromInt (xz30 `quot` xz31))",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3]; 52[label="fromIntegral (properFractionQ xz30 xz31)",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3]; 53[label="fromInt (xz30 `quot` xz31)",fontsize=16,color="black",shape="triangle"];53 -> 56[label="",style="solid", color="black", weight=3]; 54 -> 53[label="",style="dashed", color="red", weight=0]; 54[label="fromInt (xz30 `quot` xz31)",fontsize=16,color="magenta"];54 -> 57[label="",style="dashed", color="magenta", weight=3]; 54 -> 58[label="",style="dashed", color="magenta", weight=3]; 55[label="fromInteger . toInteger",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 56[label="xz30 `quot` xz31",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 57[label="xz31",fontsize=16,color="green",shape="box"];58[label="xz30",fontsize=16,color="green",shape="box"];59[label="fromInteger (toInteger (properFractionQ xz30 xz31))",fontsize=16,color="blue",shape="box"];374[label="toInteger :: Int -> Integer",fontsize=10,color="white",style="solid",shape="box"];59 -> 374[label="",style="solid", color="blue", weight=9]; 374 -> 61[label="",style="solid", color="blue", weight=3]; 375[label="toInteger :: Integer -> Integer",fontsize=10,color="white",style="solid",shape="box"];59 -> 375[label="",style="solid", color="blue", weight=9]; 375 -> 62[label="",style="solid", color="blue", weight=3]; 60[label="primQuotInt xz30 xz31",fontsize=16,color="burlywood",shape="triangle"];376[label="xz30/Pos xz300",fontsize=10,color="white",style="solid",shape="box"];60 -> 376[label="",style="solid", color="burlywood", weight=9]; 376 -> 63[label="",style="solid", color="burlywood", weight=3]; 377[label="xz30/Neg xz300",fontsize=10,color="white",style="solid",shape="box"];60 -> 377[label="",style="solid", color="burlywood", weight=9]; 377 -> 64[label="",style="solid", color="burlywood", weight=3]; 61[label="fromInteger (toInteger (properFractionQ xz30 xz31))",fontsize=16,color="black",shape="box"];61 -> 65[label="",style="solid", color="black", weight=3]; 62[label="fromInteger (toInteger (properFractionQ xz30 xz31))",fontsize=16,color="black",shape="box"];62 -> 66[label="",style="solid", color="black", weight=3]; 63[label="primQuotInt (Pos xz300) xz31",fontsize=16,color="burlywood",shape="box"];378[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];63 -> 378[label="",style="solid", color="burlywood", weight=9]; 378 -> 67[label="",style="solid", color="burlywood", weight=3]; 379[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];63 -> 379[label="",style="solid", color="burlywood", weight=9]; 379 -> 68[label="",style="solid", color="burlywood", weight=3]; 64[label="primQuotInt (Neg xz300) xz31",fontsize=16,color="burlywood",shape="box"];380[label="xz31/Pos xz310",fontsize=10,color="white",style="solid",shape="box"];64 -> 380[label="",style="solid", color="burlywood", weight=9]; 380 -> 69[label="",style="solid", color="burlywood", weight=3]; 381[label="xz31/Neg xz310",fontsize=10,color="white",style="solid",shape="box"];64 -> 381[label="",style="solid", color="burlywood", weight=9]; 381 -> 70[label="",style="solid", color="burlywood", weight=3]; 65[label="fromInteger (Integer (properFractionQ xz30 xz31))",fontsize=16,color="black",shape="box"];65 -> 71[label="",style="solid", color="black", weight=3]; 66[label="fromInteger (properFractionQ xz30 xz31)",fontsize=16,color="black",shape="box"];66 -> 72[label="",style="solid", color="black", weight=3]; 67[label="primQuotInt (Pos xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];382[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];67 -> 382[label="",style="solid", color="burlywood", weight=9]; 382 -> 73[label="",style="solid", color="burlywood", weight=3]; 383[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];67 -> 383[label="",style="solid", color="burlywood", weight=9]; 383 -> 74[label="",style="solid", color="burlywood", weight=3]; 68[label="primQuotInt (Pos xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];384[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];68 -> 384[label="",style="solid", color="burlywood", weight=9]; 384 -> 75[label="",style="solid", color="burlywood", weight=3]; 385[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];68 -> 385[label="",style="solid", color="burlywood", weight=9]; 385 -> 76[label="",style="solid", color="burlywood", weight=3]; 69[label="primQuotInt (Neg xz300) (Pos xz310)",fontsize=16,color="burlywood",shape="box"];386[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];69 -> 386[label="",style="solid", color="burlywood", weight=9]; 386 -> 77[label="",style="solid", color="burlywood", weight=3]; 387[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];69 -> 387[label="",style="solid", color="burlywood", weight=9]; 387 -> 78[label="",style="solid", color="burlywood", weight=3]; 70[label="primQuotInt (Neg xz300) (Neg xz310)",fontsize=16,color="burlywood",shape="box"];388[label="xz310/Succ xz3100",fontsize=10,color="white",style="solid",shape="box"];70 -> 388[label="",style="solid", color="burlywood", weight=9]; 388 -> 79[label="",style="solid", color="burlywood", weight=3]; 389[label="xz310/Zero",fontsize=10,color="white",style="solid",shape="box"];70 -> 389[label="",style="solid", color="burlywood", weight=9]; 389 -> 80[label="",style="solid", color="burlywood", weight=3]; 71[label="properFractionQ xz30 xz31",fontsize=16,color="black",shape="box"];71 -> 81[label="",style="solid", color="black", weight=3]; 72[label="fromInteger (properFractionQ1 xz30 xz31 (properFractionVu30 xz30 xz31))",fontsize=16,color="black",shape="box"];72 -> 82[label="",style="solid", color="black", weight=3]; 73[label="primQuotInt (Pos xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];73 -> 83[label="",style="solid", color="black", weight=3]; 74[label="primQuotInt (Pos xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];74 -> 84[label="",style="solid", color="black", weight=3]; 75[label="primQuotInt (Pos xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];75 -> 85[label="",style="solid", color="black", weight=3]; 76[label="primQuotInt (Pos xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];76 -> 86[label="",style="solid", color="black", weight=3]; 77[label="primQuotInt (Neg xz300) (Pos (Succ xz3100))",fontsize=16,color="black",shape="box"];77 -> 87[label="",style="solid", color="black", weight=3]; 78[label="primQuotInt (Neg xz300) (Pos Zero)",fontsize=16,color="black",shape="box"];78 -> 88[label="",style="solid", color="black", weight=3]; 79[label="primQuotInt (Neg xz300) (Neg (Succ xz3100))",fontsize=16,color="black",shape="box"];79 -> 89[label="",style="solid", color="black", weight=3]; 80[label="primQuotInt (Neg xz300) (Neg Zero)",fontsize=16,color="black",shape="box"];80 -> 90[label="",style="solid", color="black", weight=3]; 81[label="properFractionQ1 xz30 xz31 (properFractionVu30 xz30 xz31)",fontsize=16,color="black",shape="box"];81 -> 91[label="",style="solid", color="black", weight=3]; 82[label="fromInteger (properFractionQ1 xz30 xz31 (quotRem xz30 xz31))",fontsize=16,color="burlywood",shape="box"];390[label="xz30/Integer xz300",fontsize=10,color="white",style="solid",shape="box"];82 -> 390[label="",style="solid", color="burlywood", weight=9]; 390 -> 92[label="",style="solid", color="burlywood", weight=3]; 83[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];83 -> 93[label="",style="dashed", color="green", weight=3]; 84[label="error []",fontsize=16,color="black",shape="triangle"];84 -> 94[label="",style="solid", color="black", weight=3]; 85[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];85 -> 95[label="",style="dashed", color="green", weight=3]; 86 -> 84[label="",style="dashed", color="red", weight=0]; 86[label="error []",fontsize=16,color="magenta"];87[label="Neg (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];87 -> 96[label="",style="dashed", color="green", weight=3]; 88 -> 84[label="",style="dashed", color="red", weight=0]; 88[label="error []",fontsize=16,color="magenta"];89[label="Pos (primDivNatS xz300 (Succ xz3100))",fontsize=16,color="green",shape="box"];89 -> 97[label="",style="dashed", color="green", weight=3]; 90 -> 84[label="",style="dashed", color="red", weight=0]; 90[label="error []",fontsize=16,color="magenta"];91[label="properFractionQ1 xz30 xz31 (quotRem xz30 xz31)",fontsize=16,color="black",shape="box"];91 -> 98[label="",style="solid", color="black", weight=3]; 92[label="fromInteger (properFractionQ1 (Integer xz300) xz31 (quotRem (Integer xz300) xz31))",fontsize=16,color="burlywood",shape="box"];391[label="xz31/Integer xz310",fontsize=10,color="white",style="solid",shape="box"];92 -> 391[label="",style="solid", color="burlywood", weight=9]; 391 -> 99[label="",style="solid", color="burlywood", weight=3]; 93[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="burlywood",shape="triangle"];392[label="xz300/Succ xz3000",fontsize=10,color="white",style="solid",shape="box"];93 -> 392[label="",style="solid", color="burlywood", weight=9]; 392 -> 100[label="",style="solid", color="burlywood", weight=3]; 393[label="xz300/Zero",fontsize=10,color="white",style="solid",shape="box"];93 -> 393[label="",style="solid", color="burlywood", weight=9]; 393 -> 101[label="",style="solid", color="burlywood", weight=3]; 94[label="error []",fontsize=16,color="red",shape="box"];95 -> 93[label="",style="dashed", color="red", weight=0]; 95[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];95 -> 102[label="",style="dashed", color="magenta", weight=3]; 96 -> 93[label="",style="dashed", color="red", weight=0]; 96[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];96 -> 103[label="",style="dashed", color="magenta", weight=3]; 97 -> 93[label="",style="dashed", color="red", weight=0]; 97[label="primDivNatS xz300 (Succ xz3100)",fontsize=16,color="magenta"];97 -> 104[label="",style="dashed", color="magenta", weight=3]; 97 -> 105[label="",style="dashed", color="magenta", weight=3]; 98[label="properFractionQ1 xz30 xz31 (primQrmInt xz30 xz31)",fontsize=16,color="black",shape="box"];98 -> 106[label="",style="solid", color="black", weight=3]; 99[label="fromInteger (properFractionQ1 (Integer xz300) (Integer xz310) (quotRem (Integer xz300) (Integer xz310)))",fontsize=16,color="black",shape="box"];99 -> 107[label="",style="solid", color="black", weight=3]; 100[label="primDivNatS (Succ xz3000) (Succ xz3100)",fontsize=16,color="black",shape="box"];100 -> 108[label="",style="solid", color="black", weight=3]; 101[label="primDivNatS Zero (Succ xz3100)",fontsize=16,color="black",shape="box"];101 -> 109[label="",style="solid", color="black", weight=3]; 102[label="xz3100",fontsize=16,color="green",shape="box"];103[label="xz300",fontsize=16,color="green",shape="box"];104[label="xz3100",fontsize=16,color="green",shape="box"];105[label="xz300",fontsize=16,color="green",shape="box"];106 -> 110[label="",style="dashed", color="red", weight=0]; 106[label="properFractionQ1 xz30 xz31 (primQuotInt xz30 xz31,primRemInt xz30 xz31)",fontsize=16,color="magenta"];106 -> 111[label="",style="dashed", color="magenta", weight=3]; 107 -> 112[label="",style="dashed", color="red", weight=0]; 107[label="fromInteger (properFractionQ1 (Integer xz300) (Integer xz310) (Integer (primQuotInt xz300 xz310),Integer (primRemInt xz300 xz310)))",fontsize=16,color="magenta"];107 -> 113[label="",style="dashed", color="magenta", weight=3]; 108[label="primDivNatS0 xz3000 xz3100 (primGEqNatS xz3000 xz3100)",fontsize=16,color="burlywood",shape="box"];394[label="xz3000/Succ xz30000",fontsize=10,color="white",style="solid",shape="box"];108 -> 394[label="",style="solid", color="burlywood", weight=9]; 394 -> 114[label="",style="solid", color="burlywood", weight=3]; 395[label="xz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];108 -> 395[label="",style="solid", color="burlywood", weight=9]; 395 -> 115[label="",style="solid", color="burlywood", weight=3]; 109[label="Zero",fontsize=16,color="green",shape="box"];111 -> 60[label="",style="dashed", color="red", weight=0]; 111[label="primQuotInt xz30 xz31",fontsize=16,color="magenta"];111 -> 116[label="",style="dashed", color="magenta", weight=3]; 111 -> 117[label="",style="dashed", color="magenta", weight=3]; 110[label="properFractionQ1 xz30 xz31 (xz6,primRemInt xz30 xz31)",fontsize=16,color="black",shape="triangle"];110 -> 118[label="",style="solid", color="black", weight=3]; 113 -> 60[label="",style="dashed", color="red", weight=0]; 113[label="primQuotInt xz300 xz310",fontsize=16,color="magenta"];113 -> 119[label="",style="dashed", color="magenta", weight=3]; 113 -> 120[label="",style="dashed", color="magenta", weight=3]; 112[label="fromInteger (properFractionQ1 (Integer xz300) (Integer xz310) (Integer xz7,Integer (primRemInt xz300 xz310)))",fontsize=16,color="black",shape="triangle"];112 -> 121[label="",style="solid", color="black", weight=3]; 114[label="primDivNatS0 (Succ xz30000) xz3100 (primGEqNatS (Succ xz30000) xz3100)",fontsize=16,color="burlywood",shape="box"];396[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];114 -> 396[label="",style="solid", color="burlywood", weight=9]; 396 -> 122[label="",style="solid", color="burlywood", weight=3]; 397[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];114 -> 397[label="",style="solid", color="burlywood", weight=9]; 397 -> 123[label="",style="solid", color="burlywood", weight=3]; 115[label="primDivNatS0 Zero xz3100 (primGEqNatS Zero xz3100)",fontsize=16,color="burlywood",shape="box"];398[label="xz3100/Succ xz31000",fontsize=10,color="white",style="solid",shape="box"];115 -> 398[label="",style="solid", color="burlywood", weight=9]; 398 -> 124[label="",style="solid", color="burlywood", weight=3]; 399[label="xz3100/Zero",fontsize=10,color="white",style="solid",shape="box"];115 -> 399[label="",style="solid", color="burlywood", weight=9]; 399 -> 125[label="",style="solid", color="burlywood", weight=3]; 116[label="xz31",fontsize=16,color="green",shape="box"];117[label="xz30",fontsize=16,color="green",shape="box"];118[label="xz6",fontsize=16,color="green",shape="box"];119[label="xz310",fontsize=16,color="green",shape="box"];120[label="xz300",fontsize=16,color="green",shape="box"];121[label="fromInteger (Integer xz7)",fontsize=16,color="black",shape="box"];121 -> 126[label="",style="solid", color="black", weight=3]; 122[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS (Succ xz30000) (Succ xz31000))",fontsize=16,color="black",shape="box"];122 -> 127[label="",style="solid", color="black", weight=3]; 123[label="primDivNatS0 (Succ xz30000) Zero (primGEqNatS (Succ xz30000) Zero)",fontsize=16,color="black",shape="box"];123 -> 128[label="",style="solid", color="black", weight=3]; 124[label="primDivNatS0 Zero (Succ xz31000) (primGEqNatS Zero (Succ xz31000))",fontsize=16,color="black",shape="box"];124 -> 129[label="",style="solid", color="black", weight=3]; 125[label="primDivNatS0 Zero Zero (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];125 -> 130[label="",style="solid", color="black", weight=3]; 126[label="xz7",fontsize=16,color="green",shape="box"];127 -> 291[label="",style="dashed", color="red", weight=0]; 127[label="primDivNatS0 (Succ xz30000) (Succ xz31000) (primGEqNatS xz30000 xz31000)",fontsize=16,color="magenta"];127 -> 292[label="",style="dashed", color="magenta", weight=3]; 127 -> 293[label="",style="dashed", color="magenta", weight=3]; 127 -> 294[label="",style="dashed", color="magenta", weight=3]; 127 -> 295[label="",style="dashed", color="magenta", weight=3]; 128[label="primDivNatS0 (Succ xz30000) Zero True",fontsize=16,color="black",shape="box"];128 -> 133[label="",style="solid", color="black", weight=3]; 129[label="primDivNatS0 Zero (Succ xz31000) False",fontsize=16,color="black",shape="box"];129 -> 134[label="",style="solid", color="black", weight=3]; 130[label="primDivNatS0 Zero Zero True",fontsize=16,color="black",shape="box"];130 -> 135[label="",style="solid", color="black", weight=3]; 292[label="xz30000",fontsize=16,color="green",shape="box"];293[label="xz31000",fontsize=16,color="green",shape="box"];294[label="xz30000",fontsize=16,color="green",shape="box"];295[label="xz31000",fontsize=16,color="green",shape="box"];291[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz26 xz27)",fontsize=16,color="burlywood",shape="triangle"];400[label="xz26/Succ xz260",fontsize=10,color="white",style="solid",shape="box"];291 -> 400[label="",style="solid", color="burlywood", weight=9]; 400 -> 324[label="",style="solid", color="burlywood", weight=3]; 401[label="xz26/Zero",fontsize=10,color="white",style="solid",shape="box"];291 -> 401[label="",style="solid", color="burlywood", weight=9]; 401 -> 325[label="",style="solid", color="burlywood", weight=3]; 133[label="Succ (primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];133 -> 140[label="",style="dashed", color="green", weight=3]; 134[label="Zero",fontsize=16,color="green",shape="box"];135[label="Succ (primDivNatS (primMinusNatS Zero Zero) (Succ Zero))",fontsize=16,color="green",shape="box"];135 -> 141[label="",style="dashed", color="green", weight=3]; 324[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) xz27)",fontsize=16,color="burlywood",shape="box"];402[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];324 -> 402[label="",style="solid", color="burlywood", weight=9]; 402 -> 326[label="",style="solid", color="burlywood", weight=3]; 403[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];324 -> 403[label="",style="solid", color="burlywood", weight=9]; 403 -> 327[label="",style="solid", color="burlywood", weight=3]; 325[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero xz27)",fontsize=16,color="burlywood",shape="box"];404[label="xz27/Succ xz270",fontsize=10,color="white",style="solid",shape="box"];325 -> 404[label="",style="solid", color="burlywood", weight=9]; 404 -> 328[label="",style="solid", color="burlywood", weight=3]; 405[label="xz27/Zero",fontsize=10,color="white",style="solid",shape="box"];325 -> 405[label="",style="solid", color="burlywood", weight=9]; 405 -> 329[label="",style="solid", color="burlywood", weight=3]; 140 -> 93[label="",style="dashed", color="red", weight=0]; 140[label="primDivNatS (primMinusNatS (Succ xz30000) Zero) (Succ Zero)",fontsize=16,color="magenta"];140 -> 146[label="",style="dashed", color="magenta", weight=3]; 140 -> 147[label="",style="dashed", color="magenta", weight=3]; 141 -> 93[label="",style="dashed", color="red", weight=0]; 141[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];141 -> 148[label="",style="dashed", color="magenta", weight=3]; 141 -> 149[label="",style="dashed", color="magenta", weight=3]; 326[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) (Succ xz270))",fontsize=16,color="black",shape="box"];326 -> 330[label="",style="solid", color="black", weight=3]; 327[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS (Succ xz260) Zero)",fontsize=16,color="black",shape="box"];327 -> 331[label="",style="solid", color="black", weight=3]; 328[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero (Succ xz270))",fontsize=16,color="black",shape="box"];328 -> 332[label="",style="solid", color="black", weight=3]; 329[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];329 -> 333[label="",style="solid", color="black", weight=3]; 146[label="Zero",fontsize=16,color="green",shape="box"];147[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];147 -> 155[label="",style="solid", color="black", weight=3]; 148[label="Zero",fontsize=16,color="green",shape="box"];149[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];149 -> 156[label="",style="solid", color="black", weight=3]; 330 -> 291[label="",style="dashed", color="red", weight=0]; 330[label="primDivNatS0 (Succ xz24) (Succ xz25) (primGEqNatS xz260 xz270)",fontsize=16,color="magenta"];330 -> 334[label="",style="dashed", color="magenta", weight=3]; 330 -> 335[label="",style="dashed", color="magenta", weight=3]; 331[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="black",shape="triangle"];331 -> 336[label="",style="solid", color="black", weight=3]; 332[label="primDivNatS0 (Succ xz24) (Succ xz25) False",fontsize=16,color="black",shape="box"];332 -> 337[label="",style="solid", color="black", weight=3]; 333 -> 331[label="",style="dashed", color="red", weight=0]; 333[label="primDivNatS0 (Succ xz24) (Succ xz25) True",fontsize=16,color="magenta"];155[label="Succ xz30000",fontsize=16,color="green",shape="box"];156[label="Zero",fontsize=16,color="green",shape="box"];334[label="xz270",fontsize=16,color="green",shape="box"];335[label="xz260",fontsize=16,color="green",shape="box"];336[label="Succ (primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25)))",fontsize=16,color="green",shape="box"];336 -> 338[label="",style="dashed", color="green", weight=3]; 337[label="Zero",fontsize=16,color="green",shape="box"];338 -> 93[label="",style="dashed", color="red", weight=0]; 338[label="primDivNatS (primMinusNatS (Succ xz24) (Succ xz25)) (Succ (Succ xz25))",fontsize=16,color="magenta"];338 -> 339[label="",style="dashed", color="magenta", weight=3]; 338 -> 340[label="",style="dashed", color="magenta", weight=3]; 339[label="Succ xz25",fontsize=16,color="green",shape="box"];340[label="primMinusNatS (Succ xz24) (Succ xz25)",fontsize=16,color="black",shape="box"];340 -> 341[label="",style="solid", color="black", weight=3]; 341[label="primMinusNatS xz24 xz25",fontsize=16,color="burlywood",shape="triangle"];406[label="xz24/Succ xz240",fontsize=10,color="white",style="solid",shape="box"];341 -> 406[label="",style="solid", color="burlywood", weight=9]; 406 -> 342[label="",style="solid", color="burlywood", weight=3]; 407[label="xz24/Zero",fontsize=10,color="white",style="solid",shape="box"];341 -> 407[label="",style="solid", color="burlywood", weight=9]; 407 -> 343[label="",style="solid", color="burlywood", weight=3]; 342[label="primMinusNatS (Succ xz240) xz25",fontsize=16,color="burlywood",shape="box"];408[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];342 -> 408[label="",style="solid", color="burlywood", weight=9]; 408 -> 344[label="",style="solid", color="burlywood", weight=3]; 409[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];342 -> 409[label="",style="solid", color="burlywood", weight=9]; 409 -> 345[label="",style="solid", color="burlywood", weight=3]; 343[label="primMinusNatS Zero xz25",fontsize=16,color="burlywood",shape="box"];410[label="xz25/Succ xz250",fontsize=10,color="white",style="solid",shape="box"];343 -> 410[label="",style="solid", color="burlywood", weight=9]; 410 -> 346[label="",style="solid", color="burlywood", weight=3]; 411[label="xz25/Zero",fontsize=10,color="white",style="solid",shape="box"];343 -> 411[label="",style="solid", color="burlywood", weight=9]; 411 -> 347[label="",style="solid", color="burlywood", weight=3]; 344[label="primMinusNatS (Succ xz240) (Succ xz250)",fontsize=16,color="black",shape="box"];344 -> 348[label="",style="solid", color="black", weight=3]; 345[label="primMinusNatS (Succ xz240) Zero",fontsize=16,color="black",shape="box"];345 -> 349[label="",style="solid", color="black", weight=3]; 346[label="primMinusNatS Zero (Succ xz250)",fontsize=16,color="black",shape="box"];346 -> 350[label="",style="solid", color="black", weight=3]; 347[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];347 -> 351[label="",style="solid", color="black", weight=3]; 348 -> 341[label="",style="dashed", color="red", weight=0]; 348[label="primMinusNatS xz240 xz250",fontsize=16,color="magenta"];348 -> 352[label="",style="dashed", color="magenta", weight=3]; 348 -> 353[label="",style="dashed", color="magenta", weight=3]; 349[label="Succ xz240",fontsize=16,color="green",shape="box"];350[label="Zero",fontsize=16,color="green",shape="box"];351[label="Zero",fontsize=16,color="green",shape="box"];352[label="xz240",fontsize=16,color="green",shape="box"];353[label="xz250",fontsize=16,color="green",shape="box"];} ---------------------------------------- (14) Complex Obligation (AND) ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 1 less node. ---------------------------------------- (17) Complex Obligation (AND) ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) MRRProof (EQUIVALENT) By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. Strictly oriented dependency pairs: new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) Strictly oriented rules of the TRS R: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) Used ordering: Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + 2*x_1 POL(Zero) = 1 POL(new_primDivNatS(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS0(x_1, x_2)) = x_1 + x_2 POL(new_primMinusNatS1(x_1)) = 2 + 2*x_1 POL(new_primMinusNatS2) = 2 ---------------------------------------- (20) Obligation: Q DP problem: P is empty. R is empty. The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) PisEmptyProof (EQUIVALENT) The TRS P is empty. Hence, there is no (P,Q,R) chain. ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) The set Q consists of the following terms: new_primMinusNatS0(Zero, Succ(x0)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS1(x0) new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Succ(x0), Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) QDPSizeChangeProof (EQUIVALENT) We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. Order:Polynomial interpretation [POLO]: POL(Succ(x_1)) = 1 + x_1 POL(Zero) = 1 POL(new_primMinusNatS0(x_1, x_2)) = x_1 From the DPs we obtained the following set of size-change graphs: *new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 > 1, 2 > 2, 1 > 3, 2 > 4 *new_primDivNatS0(xz24, xz25, Succ(xz260), Succ(xz270)) -> new_primDivNatS0(xz24, xz25, xz260, xz270) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(xz24, xz25, Succ(xz260), Zero) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS0(xz24, xz25, Zero, Zero) -> new_primDivNatS00(xz24, xz25) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS00(xz24, xz25) -> new_primDivNatS(new_primMinusNatS0(xz24, xz25), Succ(xz25)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS0(Zero, Succ(xz250)) -> Zero new_primMinusNatS0(Succ(xz240), Zero) -> Succ(xz240) new_primMinusNatS0(Succ(xz240), Succ(xz250)) -> new_primMinusNatS0(xz240, xz250) ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(xz240), Succ(xz250)) -> new_primMinusNatS(xz240, xz250) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPSizeChangeProof (EQUIVALENT) By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. From the DPs we obtained the following set of size-change graphs: *new_primMinusNatS(Succ(xz240), Succ(xz250)) -> new_primMinusNatS(xz240, xz250) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (28) YES