/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs # AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) IFR [EQUIVALENT, 0 ms] (4)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, 0 ms] (20) QDP (21) PisEmptyProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 6 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; " "properFractionVu30 xw xx = quotRem xw xx; " "properFractionQ xw xx = properFractionQ1 xw xx (properFractionVu30 xw xx); " "properFractionR0 xw xx (wz,r) = r; " 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="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 4[label="truncate xz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="truncateM xz3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="truncateM0 xz3 (truncateVu6 xz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="truncateM0 xz3 (properFraction xz3)",fontsize=16,color="burlywood",shape="box"];288[label="xz3/xz30 :% xz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 288[label="",style="solid", color="burlywood", weight=9]; 288 -> 8[label="",style="solid", color="burlywood", weight=3]; 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75 -> 42[label="",style="dashed", color="red", weight=0]; 75[label="primDivNatS (primMinusNatS Zero Zero) (Succ Zero)",fontsize=16,color="magenta"];75 -> 82[label="",style="dashed", color="magenta", weight=3]; 75 -> 83[label="",style="dashed", color="magenta", weight=3]; 260[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) (Succ xz230))",fontsize=16,color="black",shape="box"];260 -> 264[label="",style="solid", color="black", weight=3]; 261[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS (Succ xz220) Zero)",fontsize=16,color="black",shape="box"];261 -> 265[label="",style="solid", color="black", weight=3]; 262[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero (Succ xz230))",fontsize=16,color="black",shape="box"];262 -> 266[label="",style="solid", color="black", weight=3]; 263[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS Zero Zero)",fontsize=16,color="black",shape="box"];263 -> 267[label="",style="solid", color="black", weight=3]; 80[label="Zero",fontsize=16,color="green",shape="box"];81[label="primMinusNatS (Succ xz30000) Zero",fontsize=16,color="black",shape="triangle"];81 -> 89[label="",style="solid", color="black", weight=3]; 82[label="Zero",fontsize=16,color="green",shape="box"];83[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="triangle"];83 -> 90[label="",style="solid", color="black", weight=3]; 264 -> 225[label="",style="dashed", color="red", weight=0]; 264[label="primDivNatS0 (Succ xz20) (Succ xz21) (primGEqNatS xz220 xz230)",fontsize=16,color="magenta"];264 -> 268[label="",style="dashed", color="magenta", weight=3]; 264 -> 269[label="",style="dashed", color="magenta", weight=3]; 265[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="black",shape="triangle"];265 -> 270[label="",style="solid", color="black", weight=3]; 266[label="primDivNatS0 (Succ xz20) (Succ xz21) False",fontsize=16,color="black",shape="box"];266 -> 271[label="",style="solid", color="black", weight=3]; 267 -> 265[label="",style="dashed", color="red", weight=0]; 267[label="primDivNatS0 (Succ xz20) (Succ xz21) True",fontsize=16,color="magenta"];89[label="Succ xz30000",fontsize=16,color="green",shape="box"];90[label="Zero",fontsize=16,color="green",shape="box"];268[label="xz230",fontsize=16,color="green",shape="box"];269[label="xz220",fontsize=16,color="green",shape="box"];270[label="Succ (primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21)))",fontsize=16,color="green",shape="box"];270 -> 272[label="",style="dashed", color="green", weight=3]; 271[label="Zero",fontsize=16,color="green",shape="box"];272 -> 42[label="",style="dashed", color="red", weight=0]; 272[label="primDivNatS (primMinusNatS (Succ xz20) (Succ xz21)) (Succ (Succ xz21))",fontsize=16,color="magenta"];272 -> 273[label="",style="dashed", color="magenta", weight=3]; 272 -> 274[label="",style="dashed", color="magenta", weight=3]; 273[label="Succ xz21",fontsize=16,color="green",shape="box"];274[label="primMinusNatS (Succ xz20) (Succ xz21)",fontsize=16,color="black",shape="box"];274 -> 275[label="",style="solid", color="black", weight=3]; 275[label="primMinusNatS xz20 xz21",fontsize=16,color="burlywood",shape="triangle"];317[label="xz20/Succ xz200",fontsize=10,color="white",style="solid",shape="box"];275 -> 317[label="",style="solid", color="burlywood", weight=9]; 317 -> 276[label="",style="solid", color="burlywood", weight=3]; 318[label="xz20/Zero",fontsize=10,color="white",style="solid",shape="box"];275 -> 318[label="",style="solid", color="burlywood", weight=9]; 318 -> 277[label="",style="solid", color="burlywood", weight=3]; 276[label="primMinusNatS (Succ xz200) xz21",fontsize=16,color="burlywood",shape="box"];319[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];276 -> 319[label="",style="solid", color="burlywood", weight=9]; 319 -> 278[label="",style="solid", color="burlywood", weight=3]; 320[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];276 -> 320[label="",style="solid", color="burlywood", weight=9]; 320 -> 279[label="",style="solid", color="burlywood", weight=3]; 277[label="primMinusNatS Zero xz21",fontsize=16,color="burlywood",shape="box"];321[label="xz21/Succ xz210",fontsize=10,color="white",style="solid",shape="box"];277 -> 321[label="",style="solid", color="burlywood", weight=9]; 321 -> 280[label="",style="solid", color="burlywood", weight=3]; 322[label="xz21/Zero",fontsize=10,color="white",style="solid",shape="box"];277 -> 322[label="",style="solid", color="burlywood", weight=9]; 322 -> 281[label="",style="solid", color="burlywood", weight=3]; 278[label="primMinusNatS (Succ xz200) (Succ xz210)",fontsize=16,color="black",shape="box"];278 -> 282[label="",style="solid", color="black", weight=3]; 279[label="primMinusNatS (Succ xz200) Zero",fontsize=16,color="black",shape="box"];279 -> 283[label="",style="solid", color="black", weight=3]; 280[label="primMinusNatS Zero (Succ xz210)",fontsize=16,color="black",shape="box"];280 -> 284[label="",style="solid", color="black", weight=3]; 281[label="primMinusNatS Zero Zero",fontsize=16,color="black",shape="box"];281 -> 285[label="",style="solid", color="black", weight=3]; 282 -> 275[label="",style="dashed", color="red", weight=0]; 282[label="primMinusNatS xz200 xz210",fontsize=16,color="magenta"];282 -> 286[label="",style="dashed", color="magenta", weight=3]; 282 -> 287[label="",style="dashed", color="magenta", weight=3]; 283[label="Succ xz200",fontsize=16,color="green",shape="box"];284[label="Zero",fontsize=16,color="green",shape="box"];285[label="Zero",fontsize=16,color="green",shape="box"];286[label="xz210",fontsize=16,color="green",shape="box"];287[label="xz200",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(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) new_primDivNatS(Succ(Zero), Zero) -> new_primDivNatS(new_primMinusNatS2, Zero) new_primDivNatS(Succ(Succ(xz30000)), Zero) -> new_primDivNatS(new_primMinusNatS1(xz30000), Zero) new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) 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(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS(Succ(Succ(xz30000)), Succ(xz31000)) -> new_primDivNatS0(xz30000, xz31000, xz30000, xz31000) new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) new_primDivNatS0(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) The TRS R consists of the following rules: new_primMinusNatS0(Zero, Succ(xz210)) -> Zero new_primMinusNatS0(Zero, Zero) -> Zero new_primMinusNatS1(xz30000) -> Succ(xz30000) new_primMinusNatS2 -> Zero new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) The set Q consists of the following terms: new_primMinusNatS0(Succ(x0), Succ(x1)) new_primMinusNatS0(Zero, Zero) new_primMinusNatS2 new_primMinusNatS0(Succ(x0), Zero) new_primMinusNatS1(x0) new_primMinusNatS0(Zero, Succ(x0)) 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(xz20, xz21, Succ(xz220), Succ(xz230)) -> new_primDivNatS0(xz20, xz21, xz220, xz230) (allowed arguments on rhs = {1, 2, 3, 4}) The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 > 4 *new_primDivNatS0(xz20, xz21, Succ(xz220), Zero) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1 *new_primDivNatS0(xz20, xz21, Zero, Zero) -> new_primDivNatS00(xz20, xz21) (allowed arguments on rhs = {1, 2}) The graph contains the following edges 1 >= 1, 2 >= 2 *new_primDivNatS00(xz20, xz21) -> new_primDivNatS(new_primMinusNatS0(xz20, xz21), Succ(xz21)) (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(xz210)) -> Zero new_primMinusNatS0(Succ(xz200), Zero) -> Succ(xz200) new_primMinusNatS0(Succ(xz200), Succ(xz210)) -> new_primMinusNatS0(xz200, xz210) ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMinusNatS(Succ(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210) 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(xz200), Succ(xz210)) -> new_primMinusNatS(xz200, xz210) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (28) YES