/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) IFR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4)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, 20 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) QDP (9) DependencyGraphProof [EQUIVALENT, 0 ms] (10) AND (11) QDP (12) QDPSizeChangeProof [EQUIVALENT, 0 ms] (13) YES (14) QDP (15) QDPSizeChangeProof [EQUIVALENT, 0 ms] (16) 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; 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; 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; 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; 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.delete",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="List.delete wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="List.delete wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="List.deleteBy (==) wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];30[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 30[label="",style="solid", color="burlywood", weight=9]; 30 -> 6[label="",style="solid", color="burlywood", weight=3]; 31[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9]; 31 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="List.deleteBy (==) wu3 (wu40 : wu41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="List.deleteBy (==) wu3 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8[label="List.deleteBy0 wu41 wu40 (==) wu3 ((==) wu3 wu40)",fontsize=16,color="burlywood",shape="box"];32[label="wu3/False",fontsize=10,color="white",style="solid",shape="box"];8 -> 32[label="",style="solid", color="burlywood", weight=9]; 32 -> 10[label="",style="solid", color="burlywood", weight=3]; 33[label="wu3/True",fontsize=10,color="white",style="solid",shape="box"];8 -> 33[label="",style="solid", color="burlywood", weight=9]; 33 -> 11[label="",style="solid", color="burlywood", weight=3]; 9[label="[]",fontsize=16,color="green",shape="box"];10[label="List.deleteBy0 wu41 wu40 (==) False ((==) False wu40)",fontsize=16,color="burlywood",shape="box"];34[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];10 -> 34[label="",style="solid", color="burlywood", weight=9]; 34 -> 12[label="",style="solid", color="burlywood", weight=3]; 35[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];10 -> 35[label="",style="solid", color="burlywood", weight=9]; 35 -> 13[label="",style="solid", color="burlywood", weight=3]; 11[label="List.deleteBy0 wu41 wu40 (==) True ((==) True wu40)",fontsize=16,color="burlywood",shape="box"];36[label="wu40/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 36[label="",style="solid", color="burlywood", weight=9]; 36 -> 14[label="",style="solid", color="burlywood", weight=3]; 37[label="wu40/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 37[label="",style="solid", color="burlywood", weight=9]; 37 -> 15[label="",style="solid", color="burlywood", weight=3]; 12[label="List.deleteBy0 wu41 False (==) False ((==) False False)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3]; 13[label="List.deleteBy0 wu41 True (==) False ((==) False True)",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3]; 14[label="List.deleteBy0 wu41 False (==) True ((==) True False)",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 15[label="List.deleteBy0 wu41 True (==) True ((==) True True)",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 16[label="List.deleteBy0 wu41 False (==) False True",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 17[label="List.deleteBy0 wu41 True (==) False False",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 18[label="List.deleteBy0 wu41 False (==) True False",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 19[label="List.deleteBy0 wu41 True (==) True True",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 20[label="wu41",fontsize=16,color="green",shape="box"];21[label="True : List.deleteBy (==) False wu41",fontsize=16,color="green",shape="box"];21 -> 24[label="",style="dashed", color="green", weight=3]; 22[label="False : List.deleteBy (==) True wu41",fontsize=16,color="green",shape="box"];22 -> 25[label="",style="dashed", color="green", weight=3]; 23[label="wu41",fontsize=16,color="green",shape="box"];24 -> 5[label="",style="dashed", color="red", weight=0]; 24[label="List.deleteBy (==) False wu41",fontsize=16,color="magenta"];24 -> 26[label="",style="dashed", color="magenta", weight=3]; 24 -> 27[label="",style="dashed", color="magenta", weight=3]; 25 -> 5[label="",style="dashed", color="red", weight=0]; 25[label="List.deleteBy (==) True wu41",fontsize=16,color="magenta"];25 -> 28[label="",style="dashed", color="magenta", weight=3]; 25 -> 29[label="",style="dashed", color="magenta", weight=3]; 26[label="False",fontsize=16,color="green",shape="box"];27[label="wu41",fontsize=16,color="green",shape="box"];28[label="True",fontsize=16,color="green",shape="box"];29[label="wu41",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Obligation: Q DP problem: The TRS P consists of the following rules: new_deleteBy(True, :(False, wu41)) -> new_deleteBy(True, wu41) new_deleteBy(False, :(True, wu41)) -> new_deleteBy(False, wu41) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (9) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (10) Complex Obligation (AND) ---------------------------------------- (11) Obligation: Q DP problem: The TRS P consists of the following rules: new_deleteBy(False, :(True, wu41)) -> new_deleteBy(False, wu41) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (12) 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_deleteBy(False, :(True, wu41)) -> new_deleteBy(False, wu41) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (13) YES ---------------------------------------- (14) Obligation: Q DP problem: The TRS P consists of the following rules: new_deleteBy(True, :(False, wu41)) -> new_deleteBy(True, wu41) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (15) 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_deleteBy(True, :(False, wu41)) -> new_deleteBy(True, wu41) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (16) YES