/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) QDP (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] (10) YES ---------------------------------------- (0) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; liftM :: Monad a => (b -> c) -> a b -> a c; liftM f m1 = m1 >>= (\x1 ->return (f x1)); } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\x1->return (f x1)" is transformed to "liftM0 f x1 = return (f x1); " ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; liftM :: Monad c => (b -> a) -> c b -> c a; liftM f m1 = m1 >>= liftM0 f; liftM0 f x1 = return (f x1); } ---------------------------------------- (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 Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; liftM :: Monad b => (a -> c) -> b a -> b c; liftM f m1 = m1 >>= liftM0 f; liftM0 f x1 = return (f x1); } ---------------------------------------- (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 Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; liftM :: Monad c => (b -> a) -> c b -> c a; liftM f m1 = m1 >>= liftM0 f; liftM0 f x1 = return (f x1); } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="Monad.liftM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="Monad.liftM vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="Monad.liftM vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="blue",shape="box"];45[label=">>= :: ([] a) -> (a -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];5 -> 45[label="",style="solid", color="blue", weight=9]; 45 -> 6[label="",style="solid", color="blue", weight=3]; 46[label=">>= :: (Maybe a) -> (a -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];5 -> 46[label="",style="solid", color="blue", weight=9]; 46 -> 7[label="",style="solid", color="blue", weight=3]; 47[label=">>= :: (IO a) -> (a -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];5 -> 47[label="",style="solid", color="blue", weight=9]; 47 -> 8[label="",style="solid", color="blue", weight=3]; 6[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="burlywood",shape="triangle"];48[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];6 -> 48[label="",style="solid", color="burlywood", weight=9]; 48 -> 9[label="",style="solid", color="burlywood", weight=3]; 49[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9]; 49 -> 10[label="",style="solid", color="burlywood", weight=3]; 7[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="burlywood",shape="box"];50[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];7 -> 50[label="",style="solid", color="burlywood", weight=9]; 50 -> 11[label="",style="solid", color="burlywood", weight=3]; 51[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];7 -> 51[label="",style="solid", color="burlywood", weight=9]; 51 -> 12[label="",style="solid", color="burlywood", weight=3]; 8[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];8 -> 13[label="",style="solid", color="black", weight=3]; 9[label="vy40 : vy41 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3]; 10[label="[] >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3]; 11[label="Nothing >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3]; 12[label="Just vy40 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3]; 13[label="primbindIO vy4 (Monad.liftM0 vy3)",fontsize=16,color="burlywood",shape="box"];52[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 52[label="",style="solid", color="burlywood", weight=9]; 52 -> 18[label="",style="solid", color="burlywood", weight=3]; 53[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 53[label="",style="solid", color="burlywood", weight=9]; 53 -> 19[label="",style="solid", color="burlywood", weight=3]; 54[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 54[label="",style="solid", color="burlywood", weight=9]; 54 -> 20[label="",style="solid", color="burlywood", weight=3]; 55[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 55[label="",style="solid", color="burlywood", weight=9]; 55 -> 21[label="",style="solid", color="burlywood", weight=3]; 14 -> 22[label="",style="dashed", color="red", weight=0]; 14[label="Monad.liftM0 vy3 vy40 ++ (vy41 >>= Monad.liftM0 vy3)",fontsize=16,color="magenta"];14 -> 23[label="",style="dashed", color="magenta", weight=3]; 15[label="[]",fontsize=16,color="green",shape="box"];16[label="Nothing",fontsize=16,color="green",shape="box"];17[label="Monad.liftM0 vy3 vy40",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3]; 18[label="primbindIO (IO vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3]; 19[label="primbindIO (AProVE_IO vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3]; 20[label="primbindIO (AProVE_Exception vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3]; 21[label="primbindIO (AProVE_Error vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3]; 23 -> 6[label="",style="dashed", color="red", weight=0]; 23[label="vy41 >>= Monad.liftM0 vy3",fontsize=16,color="magenta"];23 -> 29[label="",style="dashed", color="magenta", weight=3]; 22[label="Monad.liftM0 vy3 vy40 ++ vy5",fontsize=16,color="black",shape="triangle"];22 -> 30[label="",style="solid", color="black", weight=3]; 24[label="return (vy3 vy40)",fontsize=16,color="black",shape="box"];24 -> 31[label="",style="solid", color="black", weight=3]; 25[label="error []",fontsize=16,color="red",shape="box"];26[label="Monad.liftM0 vy3 vy40",fontsize=16,color="black",shape="box"];26 -> 32[label="",style="solid", color="black", weight=3]; 27[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];28[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];29[label="vy41",fontsize=16,color="green",shape="box"];30[label="return (vy3 vy40) ++ vy5",fontsize=16,color="black",shape="box"];30 -> 33[label="",style="solid", color="black", weight=3]; 31[label="Just (vy3 vy40)",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3]; 32[label="return (vy3 vy40)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3]; 33[label="(vy3 vy40 : []) ++ vy5",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3]; 34[label="vy3 vy40",fontsize=16,color="green",shape="box"];34 -> 37[label="",style="dashed", color="green", weight=3]; 35[label="primretIO (vy3 vy40)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3]; 36[label="vy3 vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3]; 36 -> 40[label="",style="dashed", color="green", weight=3]; 37[label="vy40",fontsize=16,color="green",shape="box"];38[label="AProVE_IO (vy3 vy40)",fontsize=16,color="green",shape="box"];38 -> 41[label="",style="dashed", color="green", weight=3]; 39[label="vy3 vy40",fontsize=16,color="green",shape="box"];39 -> 42[label="",style="dashed", color="green", weight=3]; 40[label="[] ++ vy5",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3]; 41[label="vy3 vy40",fontsize=16,color="green",shape="box"];41 -> 44[label="",style="dashed", color="green", weight=3]; 42[label="vy40",fontsize=16,color="green",shape="box"];43[label="vy5",fontsize=16,color="green",shape="box"];44[label="vy40",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) 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_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 ---------------------------------------- (10) YES