8.73/3.84 YES 10.70/4.37 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.70/4.37 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.70/4.37 10.70/4.37 10.70/4.37 H-Termination with start terms of the given HASKELL could be proven: 10.70/4.37 10.70/4.37 (0) HASKELL 10.70/4.37 (1) LR [EQUIVALENT, 0 ms] 10.70/4.37 (2) HASKELL 10.70/4.37 (3) BR [EQUIVALENT, 0 ms] 10.70/4.37 (4) HASKELL 10.70/4.37 (5) COR [EQUIVALENT, 0 ms] 10.70/4.37 (6) HASKELL 10.70/4.37 (7) Narrow [SOUND, 0 ms] 10.70/4.37 (8) QDP 10.70/4.37 (9) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.70/4.37 (10) YES 10.70/4.37 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (0) 10.70/4.37 Obligation: 10.70/4.37 mainModule Main 10.70/4.37 module Maybe where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Main where { 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Monad where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Prelude; 10.70/4.37 liftM :: Monad a => (c -> b) -> a c -> a b; 10.70/4.37 liftM f m1 = m1 >>= (\x1 ->return (f x1)); 10.70/4.37 10.70/4.37 } 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (1) LR (EQUIVALENT) 10.70/4.37 Lambda Reductions: 10.70/4.37 The following Lambda expression 10.70/4.37 "\x1->return (f x1)" 10.70/4.37 is transformed to 10.70/4.37 "liftM0 f x1 = return (f x1); 10.70/4.37 " 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (2) 10.70/4.37 Obligation: 10.70/4.37 mainModule Main 10.70/4.37 module Maybe where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Main where { 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Monad where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Prelude; 10.70/4.37 liftM :: Monad c => (a -> b) -> c a -> c b; 10.70/4.37 liftM f m1 = m1 >>= liftM0 f; 10.70/4.37 10.70/4.37 liftM0 f x1 = return (f x1); 10.70/4.37 10.70/4.37 } 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (3) BR (EQUIVALENT) 10.70/4.37 Replaced joker patterns by fresh variables and removed binding patterns. 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (4) 10.70/4.37 Obligation: 10.70/4.37 mainModule Main 10.70/4.37 module Maybe where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Main where { 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Monad where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Prelude; 10.70/4.37 liftM :: Monad b => (c -> a) -> b c -> b a; 10.70/4.37 liftM f m1 = m1 >>= liftM0 f; 10.70/4.37 10.70/4.37 liftM0 f x1 = return (f x1); 10.70/4.37 10.70/4.37 } 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (5) COR (EQUIVALENT) 10.70/4.37 Cond Reductions: 10.70/4.37 The following Function with conditions 10.70/4.37 "undefined |Falseundefined; 10.70/4.37 " 10.70/4.37 is transformed to 10.70/4.37 "undefined = undefined1; 10.70/4.37 " 10.70/4.37 "undefined0 True = undefined; 10.70/4.37 " 10.70/4.37 "undefined1 = undefined0 False; 10.70/4.37 " 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (6) 10.70/4.37 Obligation: 10.70/4.37 mainModule Main 10.70/4.37 module Maybe where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Main where { 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Monad; 10.70/4.37 import qualified Prelude; 10.70/4.37 } 10.70/4.37 module Monad where { 10.70/4.37 import qualified Main; 10.70/4.37 import qualified Maybe; 10.70/4.37 import qualified Prelude; 10.70/4.37 liftM :: Monad c => (a -> b) -> c a -> c b; 10.70/4.37 liftM f m1 = m1 >>= liftM0 f; 10.70/4.37 10.70/4.37 liftM0 f x1 = return (f x1); 10.70/4.37 10.70/4.37 } 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (7) Narrow (SOUND) 10.70/4.37 Haskell To QDPs 10.70/4.37 10.70/4.37 digraph dp_graph { 10.70/4.37 node [outthreshold=100, inthreshold=100];1[label="Monad.liftM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.70/4.37 3[label="Monad.liftM vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.70/4.37 4[label="Monad.liftM vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.70/4.37 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]; 10.70/4.37 45 -> 6[label="",style="solid", color="blue", weight=3]; 10.70/4.37 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]; 10.70/4.37 46 -> 7[label="",style="solid", color="blue", weight=3]; 10.70/4.37 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]; 10.70/4.37 47 -> 8[label="",style="solid", color="blue", weight=3]; 10.70/4.37 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]; 10.70/4.37 48 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 49[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9]; 10.70/4.37 49 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 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]; 10.70/4.37 50 -> 11[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 51[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];7 -> 51[label="",style="solid", color="burlywood", weight=9]; 10.70/4.37 51 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 8[label="vy4 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];8 -> 13[label="",style="solid", color="black", weight=3]; 10.70/4.37 9[label="vy40 : vy41 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];9 -> 14[label="",style="solid", color="black", weight=3]; 10.70/4.37 10[label="[] >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];10 -> 15[label="",style="solid", color="black", weight=3]; 10.70/4.37 11[label="Nothing >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];11 -> 16[label="",style="solid", color="black", weight=3]; 10.70/4.37 12[label="Just vy40 >>= Monad.liftM0 vy3",fontsize=16,color="black",shape="box"];12 -> 17[label="",style="solid", color="black", weight=3]; 10.70/4.37 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]; 10.70/4.37 52 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 53[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 53[label="",style="solid", color="burlywood", weight=9]; 10.70/4.37 53 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 54[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 54[label="",style="solid", color="burlywood", weight=9]; 10.70/4.37 54 -> 20[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 55[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];13 -> 55[label="",style="solid", color="burlywood", weight=9]; 10.70/4.37 55 -> 21[label="",style="solid", color="burlywood", weight=3]; 10.70/4.37 14 -> 22[label="",style="dashed", color="red", weight=0]; 10.70/4.37 14[label="Monad.liftM0 vy3 vy40 ++ (vy41 >>= Monad.liftM0 vy3)",fontsize=16,color="magenta"];14 -> 23[label="",style="dashed", color="magenta", weight=3]; 10.70/4.37 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]; 10.70/4.37 18[label="primbindIO (IO vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3]; 10.70/4.37 19[label="primbindIO (AProVE_IO vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];19 -> 26[label="",style="solid", color="black", weight=3]; 10.70/4.37 20[label="primbindIO (AProVE_Exception vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3]; 10.70/4.37 21[label="primbindIO (AProVE_Error vy40) (Monad.liftM0 vy3)",fontsize=16,color="black",shape="box"];21 -> 28[label="",style="solid", color="black", weight=3]; 10.70/4.37 23 -> 6[label="",style="dashed", color="red", weight=0]; 10.70/4.37 23[label="vy41 >>= Monad.liftM0 vy3",fontsize=16,color="magenta"];23 -> 29[label="",style="dashed", color="magenta", weight=3]; 10.70/4.37 22[label="Monad.liftM0 vy3 vy40 ++ vy5",fontsize=16,color="black",shape="triangle"];22 -> 30[label="",style="solid", color="black", weight=3]; 10.70/4.37 24[label="return (vy3 vy40)",fontsize=16,color="black",shape="box"];24 -> 31[label="",style="solid", color="black", weight=3]; 10.70/4.37 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]; 10.70/4.37 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]; 10.70/4.37 31[label="Just (vy3 vy40)",fontsize=16,color="green",shape="box"];31 -> 34[label="",style="dashed", color="green", weight=3]; 10.70/4.37 32[label="return (vy3 vy40)",fontsize=16,color="black",shape="box"];32 -> 35[label="",style="solid", color="black", weight=3]; 10.70/4.37 33[label="(vy3 vy40 : []) ++ vy5",fontsize=16,color="black",shape="box"];33 -> 36[label="",style="solid", color="black", weight=3]; 10.70/4.37 34[label="vy3 vy40",fontsize=16,color="green",shape="box"];34 -> 37[label="",style="dashed", color="green", weight=3]; 10.70/4.37 35[label="primretIO (vy3 vy40)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3]; 10.70/4.37 36[label="vy3 vy40 : [] ++ vy5",fontsize=16,color="green",shape="box"];36 -> 39[label="",style="dashed", color="green", weight=3]; 10.70/4.37 36 -> 40[label="",style="dashed", color="green", weight=3]; 10.70/4.37 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]; 10.70/4.37 39[label="vy3 vy40",fontsize=16,color="green",shape="box"];39 -> 42[label="",style="dashed", color="green", weight=3]; 10.70/4.37 40[label="[] ++ vy5",fontsize=16,color="black",shape="box"];40 -> 43[label="",style="solid", color="black", weight=3]; 10.70/4.37 41[label="vy3 vy40",fontsize=16,color="green",shape="box"];41 -> 44[label="",style="dashed", color="green", weight=3]; 10.70/4.37 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"];} 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (8) 10.70/4.37 Obligation: 10.70/4.37 Q DP problem: 10.70/4.37 The TRS P consists of the following rules: 10.70/4.37 10.70/4.37 new_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) 10.70/4.37 10.70/4.37 R is empty. 10.70/4.37 Q is empty. 10.70/4.37 We have to consider all minimal (P,Q,R)-chains. 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (9) QDPSizeChangeProof (EQUIVALENT) 10.70/4.37 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. 10.70/4.37 10.70/4.37 From the DPs we obtained the following set of size-change graphs: 10.70/4.37 *new_gtGtEs(:(vy40, vy41), vy3, h, ba) -> new_gtGtEs(vy41, vy3, h, ba) 10.70/4.37 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4 10.70/4.37 10.70/4.37 10.70/4.37 ---------------------------------------- 10.70/4.37 10.70/4.37 (10) 10.70/4.37 YES 10.70/4.41 EOF