/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 0 ms] (4) HASKELL (5) NumRed [SOUND, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) QDP (9) TransformationProof [EQUIVALENT, 0 ms] (10) QDP (11) DependencyGraphProof [EQUIVALENT, 0 ms] (12) QDP (13) UsableRulesProof [EQUIVALENT, 0 ms] (14) QDP (15) QReductionProof [EQUIVALENT, 0 ms] (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "drop n xs|n <= 0xs; drop vv [] = []; drop n (vw : xs) = drop (n - 1) xs; " is transformed to "drop n xs = drop3 n xs; drop vv [] = drop1 vv []; drop n (vw : xs) = drop0 n (vw : xs); " "drop0 n (vw : xs) = drop (n - 1) xs; " "drop1 vv [] = []; drop1 wv ww = drop0 wv ww; " "drop2 n xs True = xs; drop2 n xs False = drop1 n xs; " "drop3 n xs = drop2 n xs (n <= 0); drop3 wx wy = drop1 wx wy; " The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) NumRed (SOUND) Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="drop",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="drop wz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="drop wz3 wz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="drop3 wz3 wz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="drop2 wz3 wz4 (wz3 <= Pos Zero)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="drop2 wz3 wz4 (compare wz3 (Pos Zero) /= GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 8[label="drop2 wz3 wz4 (not (compare wz3 (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9[label="drop2 wz3 wz4 (not (primCmpInt wz3 (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];47[label="wz3/Pos wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 47[label="",style="solid", color="burlywood", weight=9]; 47 -> 10[label="",style="solid", color="burlywood", weight=3]; 48[label="wz3/Neg wz30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9]; 48 -> 11[label="",style="solid", color="burlywood", weight=3]; 10[label="drop2 (Pos wz30) wz4 (not (primCmpInt (Pos wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];49[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];10 -> 49[label="",style="solid", color="burlywood", weight=9]; 49 -> 12[label="",style="solid", color="burlywood", weight=3]; 50[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9]; 50 -> 13[label="",style="solid", color="burlywood", weight=3]; 11[label="drop2 (Neg wz30) wz4 (not (primCmpInt (Neg wz30) (Pos Zero) == GT))",fontsize=16,color="burlywood",shape="box"];51[label="wz30/Succ wz300",fontsize=10,color="white",style="solid",shape="box"];11 -> 51[label="",style="solid", color="burlywood", weight=9]; 51 -> 14[label="",style="solid", color="burlywood", weight=3]; 52[label="wz30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9]; 52 -> 15[label="",style="solid", color="burlywood", weight=3]; 12[label="drop2 (Pos (Succ wz300)) wz4 (not (primCmpInt (Pos (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 13[label="drop2 (Pos Zero) wz4 (not (primCmpInt (Pos Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 14[label="drop2 (Neg (Succ wz300)) wz4 (not (primCmpInt (Neg (Succ wz300)) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 15[label="drop2 (Neg Zero) wz4 (not (primCmpInt (Neg Zero) (Pos Zero) == GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 16[label="drop2 (Pos (Succ wz300)) wz4 (not (primCmpNat (Succ wz300) Zero == GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 17[label="drop2 (Pos Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 18[label="drop2 (Neg (Succ wz300)) wz4 (not (LT == GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 19[label="drop2 (Neg Zero) wz4 (not (EQ == GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 20[label="drop2 (Pos (Succ wz300)) wz4 (not (GT == GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 21[label="drop2 (Pos Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 22[label="drop2 (Neg (Succ wz300)) wz4 (not False)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 23[label="drop2 (Neg Zero) wz4 (not False)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 24[label="drop2 (Pos (Succ wz300)) wz4 (not True)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 25[label="drop2 (Pos Zero) wz4 True",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 26[label="drop2 (Neg (Succ wz300)) wz4 True",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 27[label="drop2 (Neg Zero) wz4 True",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3]; 28[label="drop2 (Pos (Succ wz300)) wz4 False",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 29[label="wz4",fontsize=16,color="green",shape="box"];30[label="wz4",fontsize=16,color="green",shape="box"];31[label="wz4",fontsize=16,color="green",shape="box"];32[label="drop1 (Pos (Succ wz300)) wz4",fontsize=16,color="burlywood",shape="box"];53[label="wz4/wz40 : wz41",fontsize=10,color="white",style="solid",shape="box"];32 -> 53[label="",style="solid", color="burlywood", weight=9]; 53 -> 33[label="",style="solid", color="burlywood", weight=3]; 54[label="wz4/[]",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9]; 54 -> 34[label="",style="solid", color="burlywood", weight=3]; 33[label="drop1 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3]; 34[label="drop1 (Pos (Succ wz300)) []",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3]; 35[label="drop0 (Pos (Succ wz300)) (wz40 : wz41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 36[label="[]",fontsize=16,color="green",shape="box"];37 -> 4[label="",style="dashed", color="red", weight=0]; 37[label="drop (Pos (Succ wz300) - Pos (Succ Zero)) wz41",fontsize=16,color="magenta"];37 -> 38[label="",style="dashed", color="magenta", weight=3]; 37 -> 39[label="",style="dashed", color="magenta", weight=3]; 38[label="wz41",fontsize=16,color="green",shape="box"];39[label="Pos (Succ wz300) - Pos (Succ Zero)",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3]; 40[label="primMinusInt (Pos (Succ wz300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3]; 41[label="primMinusNat (Succ wz300) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3]; 42[label="primMinusNat wz300 Zero",fontsize=16,color="burlywood",shape="box"];55[label="wz300/Succ wz3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 55[label="",style="solid", color="burlywood", weight=9]; 55 -> 43[label="",style="solid", color="burlywood", weight=3]; 56[label="wz300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 56[label="",style="solid", color="burlywood", weight=9]; 56 -> 44[label="",style="solid", color="burlywood", weight=3]; 43[label="primMinusNat (Succ wz3000) Zero",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3]; 44[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 45[label="Pos (Succ wz3000)",fontsize=16,color="green",shape="box"];46[label="Pos Zero",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: new_drop(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_drop(new_primMinusNat(wz300), wz41, h) The TRS R consists of the following rules: new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000)) new_primMinusNat(Zero) -> Pos(Zero) The set Q consists of the following terms: new_primMinusNat(Succ(x0)) new_primMinusNat(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) TransformationProof (EQUIVALENT) By narrowing [LPAR04] the rule new_drop(Pos(Succ(wz300)), :(wz40, wz41), h) -> new_drop(new_primMinusNat(wz300), wz41, h) at position [0] we obtained the following new rules [LPAR04]: (new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3),new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3)) (new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3),new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3)) ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3) new_drop(Pos(Succ(Zero)), :(y1, y2), y3) -> new_drop(Pos(Zero), y2, y3) The TRS R consists of the following rules: new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000)) new_primMinusNat(Zero) -> Pos(Zero) The set Q consists of the following terms: new_primMinusNat(Succ(x0)) new_primMinusNat(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3) The TRS R consists of the following rules: new_primMinusNat(Succ(wz3000)) -> Pos(Succ(wz3000)) new_primMinusNat(Zero) -> Pos(Zero) The set Q consists of the following terms: new_primMinusNat(Succ(x0)) new_primMinusNat(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) UsableRulesProof (EQUIVALENT) As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R. ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3) R is empty. The set Q consists of the following terms: new_primMinusNat(Succ(x0)) new_primMinusNat(Zero) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) QReductionProof (EQUIVALENT) We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN]. new_primMinusNat(Succ(x0)) new_primMinusNat(Zero) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: new_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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_drop(Pos(Succ(Succ(x0))), :(y1, y2), y3) -> new_drop(Pos(Succ(x0)), y2, y3) The graph contains the following edges 2 > 2, 3 >= 3 ---------------------------------------- (18) YES