/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) CR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) LetRed [EQUIVALENT, 0 ms] (8) HASKELL (9) Narrow [SOUND, 0 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Prelude; mapMaybe :: (b -> Maybe a) -> [b] -> [a]; mapMaybe _ [] = []; mapMaybe f (x : xs) = let { rs = mapMaybe f xs; } in case f x of { Nothing-> rs; Just r-> r : rs; } ; } module Main where { import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) CR (EQUIVALENT) Case Reductions: The following Case expression "case f x of { Nothing -> rs; Just r -> r : rs} " is transformed to "mapMaybe0 Nothing = rs; mapMaybe0 (Just r) = r : rs; " ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Prelude; mapMaybe :: (a -> Maybe b) -> [a] -> [b]; mapMaybe _ [] = []; mapMaybe f (x : xs) = let { mapMaybe0 Nothing = rs; mapMaybe0 (Just r) = r : rs; rs = mapMaybe f xs; } in mapMaybe0 (f x); } module Main where { import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (4) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Prelude; mapMaybe :: (b -> Maybe a) -> [b] -> [a]; mapMaybe vy [] = []; mapMaybe f (x : xs) = let { mapMaybe0 Nothing = rs; mapMaybe0 (Just r) = r : rs; rs = mapMaybe f xs; } in mapMaybe0 (f x); } module Main where { import qualified Maybe; import qualified Prelude; } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (6) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Prelude; mapMaybe :: (b -> Maybe a) -> [b] -> [a]; mapMaybe vy [] = []; mapMaybe f (x : xs) = let { mapMaybe0 Nothing = rs; mapMaybe0 (Just r) = r : rs; rs = mapMaybe f xs; } in mapMaybe0 (f x); } module Main where { import qualified Maybe; import qualified Prelude; } ---------------------------------------- (7) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "let { mapMaybe0 Nothing = rs; mapMaybe0 (Just r) = r : rs; ; rs = mapMaybe f xs; } in mapMaybe0 (f x)" are unpacked to the following functions on top level "mapMaybeMapMaybe0 vz wu Nothing = mapMaybeRs vz wu; mapMaybeMapMaybe0 vz wu (Just r) = r : mapMaybeRs vz wu; " "mapMaybeRs vz wu = mapMaybe vz wu; " ---------------------------------------- (8) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Prelude; mapMaybe :: (b -> Maybe a) -> [b] -> [a]; mapMaybe vy [] = []; mapMaybe f (x : xs) = mapMaybeMapMaybe0 f xs (f x); mapMaybeMapMaybe0 vz wu Nothing = mapMaybeRs vz wu; mapMaybeMapMaybe0 vz wu (Just r) = r : mapMaybeRs vz wu; mapMaybeRs vz wu = mapMaybe vz wu; } module Main where { import qualified Maybe; import qualified Prelude; } ---------------------------------------- (9) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="Maybe.mapMaybe",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="Maybe.mapMaybe wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="Maybe.mapMaybe wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];20[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9]; 20 -> 5[label="",style="solid", color="burlywood", weight=3]; 21[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9]; 21 -> 6[label="",style="solid", color="burlywood", weight=3]; 5[label="Maybe.mapMaybe wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 6[label="Maybe.mapMaybe wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7 -> 9[label="",style="dashed", color="red", weight=0]; 7[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (wv3 wv40)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3]; 8[label="[]",fontsize=16,color="green",shape="box"];10[label="wv3 wv40",fontsize=16,color="green",shape="box"];10 -> 14[label="",style="dashed", color="green", weight=3]; 9[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 wv5",fontsize=16,color="burlywood",shape="triangle"];22[label="wv5/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9]; 22 -> 12[label="",style="solid", color="burlywood", weight=3]; 23[label="wv5/Just wv50",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9]; 23 -> 13[label="",style="solid", color="burlywood", weight=3]; 14[label="wv40",fontsize=16,color="green",shape="box"];12[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 Nothing",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3]; 13[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (Just wv50)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3]; 15[label="Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="black",shape="triangle"];15 -> 17[label="",style="solid", color="black", weight=3]; 16[label="wv50 : Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 17 -> 4[label="",style="dashed", color="red", weight=0]; 17[label="Maybe.mapMaybe wv3 wv41",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3]; 18 -> 15[label="",style="dashed", color="red", weight=0]; 18[label="Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="magenta"];19[label="wv41",fontsize=16,color="green",shape="box"];} ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_mapMaybeRs(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) new_mapMaybe(wv3, :(wv40, wv41), h, ba) -> new_mapMaybeMapMaybe0(wv3, wv41, h, ba) new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybeRs(wv3, wv41, h, ba) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) 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_mapMaybe(wv3, :(wv40, wv41), h, ba) -> new_mapMaybeMapMaybe0(wv3, wv41, h, ba) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 *new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybeRs(wv3, wv41, h, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 *new_mapMaybeRs(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 ---------------------------------------- (12) YES