/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) IFR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4)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 List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; infix 5 \\; (\\) :: Eq a => [a] -> [a] -> [a]; (\\) = foldl (flip delete); delete :: Eq a => a -> [a] -> [a]; delete = deleteBy (==); deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]; deleteBy _ _ [] = []; deleteBy eq x (y : ys) = if x `eq` y then ys else y : deleteBy eq x ys; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) IFR (EQUIVALENT) If Reductions: The following If expression "if eq x y then ys else y : deleteBy eq x ys" is transformed to "deleteBy0 ys y eq x True = ys; deleteBy0 ys y eq x False = y : deleteBy eq x ys; " ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; infix 5 \\; (\\) :: Eq a => [a] -> [a] -> [a]; (\\) = foldl (flip delete); delete :: Eq a => a -> [a] -> [a]; delete = deleteBy (==); deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]; deleteBy _ _ [] = []; deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y); deleteBy0 ys y eq x True = ys; deleteBy0 ys y eq x False = y : deleteBy eq x ys; } module Main where { import qualified List; 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 List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; infix 5 \\; (\\) :: Eq a => [a] -> [a] -> [a]; (\\) = foldl (flip delete); delete :: Eq a => a -> [a] -> [a]; delete = deleteBy (==); deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]; deleteBy vy vz [] = []; deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y); deleteBy0 ys y eq x True = ys; deleteBy0 ys y eq x False = y : deleteBy eq x ys; } module Main where { import qualified List; 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 List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; infix 5 \\; (\\) :: Eq a => [a] -> [a] -> [a]; (\\) = foldl (flip delete); delete :: Eq a => a -> [a] -> [a]; delete = deleteBy (==); deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]; deleteBy vy vz [] = []; deleteBy eq x (y : ys) = deleteBy0 ys y eq x (x `eq` y); deleteBy0 ys y eq x True = ys; deleteBy0 ys y eq x False = y : deleteBy eq x ys; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="(List.\\)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="wu3 (List.\\)",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="wu3 (List.\\) wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="foldl (flip List.delete) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];22[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 22[label="",style="solid", color="burlywood", weight=9]; 22 -> 6[label="",style="solid", color="burlywood", weight=3]; 23[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 23[label="",style="solid", color="burlywood", weight=9]; 23 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="foldl (flip List.delete) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="foldl (flip List.delete) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8 -> 5[label="",style="dashed", color="red", weight=0]; 8[label="foldl (flip List.delete) (flip List.delete wu3 wu40) wu41",fontsize=16,color="magenta"];8 -> 10[label="",style="dashed", color="magenta", weight=3]; 8 -> 11[label="",style="dashed", color="magenta", weight=3]; 9[label="wu3",fontsize=16,color="green",shape="box"];10[label="wu41",fontsize=16,color="green",shape="box"];11[label="flip List.delete wu3 wu40",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 12[label="List.delete wu40 wu3",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3]; 13[label="List.deleteBy (==) wu40 wu3",fontsize=16,color="burlywood",shape="box"];24[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];13 -> 24[label="",style="solid", color="burlywood", weight=9]; 24 -> 14[label="",style="solid", color="burlywood", weight=3]; 25[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];13 -> 25[label="",style="solid", color="burlywood", weight=9]; 25 -> 15[label="",style="solid", color="burlywood", weight=3]; 14[label="List.deleteBy (==) wu40 (wu30 : wu31)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3]; 15[label="List.deleteBy (==) wu40 []",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 16[label="List.deleteBy0 wu31 wu30 (==) wu40 ((==) wu40 wu30)",fontsize=16,color="burlywood",shape="box"];26[label="wu40/()",fontsize=10,color="white",style="solid",shape="box"];16 -> 26[label="",style="solid", color="burlywood", weight=9]; 26 -> 18[label="",style="solid", color="burlywood", weight=3]; 17[label="[]",fontsize=16,color="green",shape="box"];18[label="List.deleteBy0 wu31 wu30 (==) () ((==) () wu30)",fontsize=16,color="burlywood",shape="box"];27[label="wu30/()",fontsize=10,color="white",style="solid",shape="box"];18 -> 27[label="",style="solid", color="burlywood", weight=9]; 27 -> 19[label="",style="solid", color="burlywood", weight=3]; 19[label="List.deleteBy0 wu31 () (==) () ((==) () ())",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 20[label="List.deleteBy0 wu31 () (==) () True",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3]; 21[label="wu31",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy(wu40, wu3), wu41) The TRS R consists of the following rules: new_deleteBy(wu40, []) -> [] new_deleteBy(@0, :(@0, wu31)) -> wu31 The set Q consists of the following terms: new_deleteBy(x0, []) new_deleteBy(@0, :(@0, x0)) 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_foldl(wu3, :(wu40, wu41)) -> new_foldl(new_deleteBy(wu40, wu3), wu41) The graph contains the following edges 2 > 2 ---------------------------------------- (10) YES