8.04/3.60 YES 10.13/4.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.13/4.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.13/4.16 10.13/4.16 10.13/4.16 H-Termination with start terms of the given HASKELL could be proven: 10.13/4.16 10.13/4.16 (0) HASKELL 10.13/4.16 (1) LR [EQUIVALENT, 0 ms] 10.13/4.16 (2) HASKELL 10.13/4.16 (3) CR [EQUIVALENT, 0 ms] 10.13/4.16 (4) HASKELL 10.13/4.16 (5) BR [EQUIVALENT, 0 ms] 10.13/4.16 (6) HASKELL 10.13/4.16 (7) COR [EQUIVALENT, 0 ms] 10.13/4.16 (8) HASKELL 10.13/4.16 (9) Narrow [SOUND, 0 ms] 10.13/4.16 (10) QDP 10.13/4.16 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.13/4.16 (12) YES 10.13/4.16 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (0) 10.13/4.16 Obligation: 10.13/4.16 mainModule Main 10.13/4.16 module Maybe where { 10.13/4.16 import qualified Main; 10.13/4.16 import qualified Prelude; 10.13/4.16 catMaybes :: [Maybe a] -> [a]; 10.13/4.16 catMaybes ls = concatMap (\vv1 ->case vv1 of { 10.13/4.16 Just x-> x : []; 10.13/4.16 _-> []; 10.13/4.16 } ) ls; 10.13/4.16 10.13/4.16 } 10.13/4.16 module Main where { 10.13/4.16 import qualified Maybe; 10.13/4.16 import qualified Prelude; 10.13/4.16 } 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (1) LR (EQUIVALENT) 10.13/4.16 Lambda Reductions: 10.13/4.16 The following Lambda expression 10.13/4.16 "\vv1->case vv1 of { 10.13/4.16 Just x -> x : []; 10.13/4.16 _ -> []} 10.13/4.16 " 10.13/4.16 is transformed to 10.13/4.16 "catMaybes0 vv1 = case vv1 of { 10.13/4.16 Just x -> x : []; 10.13/4.16 _ -> []} 10.13/4.16 ; 10.13/4.16 " 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (2) 10.13/4.16 Obligation: 10.13/4.16 mainModule Main 10.13/4.16 module Maybe where { 10.13/4.16 import qualified Main; 10.13/4.16 import qualified Prelude; 10.13/4.16 catMaybes :: [Maybe a] -> [a]; 10.13/4.16 catMaybes ls = concatMap catMaybes0 ls; 10.13/4.16 10.13/4.16 catMaybes0 vv1 = case vv1 of { 10.13/4.16 Just x-> x : []; 10.13/4.16 _-> []; 10.13/4.16 } ; 10.13/4.16 10.13/4.16 } 10.13/4.16 module Main where { 10.13/4.16 import qualified Maybe; 10.13/4.16 import qualified Prelude; 10.13/4.16 } 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (3) CR (EQUIVALENT) 10.13/4.16 Case Reductions: 10.13/4.16 The following Case expression 10.13/4.16 "case vv1 of { 10.13/4.16 Just x -> x : []; 10.13/4.16 _ -> []} 10.13/4.16 " 10.13/4.16 is transformed to 10.13/4.16 "catMaybes00 (Just x) = x : []; 10.13/4.16 catMaybes00 _ = []; 10.13/4.16 " 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (4) 10.13/4.16 Obligation: 10.13/4.16 mainModule Main 10.13/4.16 module Maybe where { 10.13/4.16 import qualified Main; 10.13/4.16 import qualified Prelude; 10.13/4.16 catMaybes :: [Maybe a] -> [a]; 10.13/4.16 catMaybes ls = concatMap catMaybes0 ls; 10.13/4.16 10.13/4.16 catMaybes0 vv1 = catMaybes00 vv1; 10.13/4.16 10.13/4.16 catMaybes00 (Just x) = x : []; 10.13/4.16 catMaybes00 _ = []; 10.13/4.16 10.13/4.16 } 10.13/4.16 module Main where { 10.13/4.16 import qualified Maybe; 10.13/4.16 import qualified Prelude; 10.13/4.16 } 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (5) BR (EQUIVALENT) 10.13/4.16 Replaced joker patterns by fresh variables and removed binding patterns. 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (6) 10.13/4.16 Obligation: 10.13/4.16 mainModule Main 10.13/4.16 module Maybe where { 10.13/4.16 import qualified Main; 10.13/4.16 import qualified Prelude; 10.13/4.16 catMaybes :: [Maybe a] -> [a]; 10.13/4.16 catMaybes ls = concatMap catMaybes0 ls; 10.13/4.16 10.13/4.16 catMaybes0 vv1 = catMaybes00 vv1; 10.13/4.16 10.13/4.16 catMaybes00 (Just x) = x : []; 10.13/4.16 catMaybes00 vy = []; 10.13/4.16 10.13/4.16 } 10.13/4.16 module Main where { 10.13/4.16 import qualified Maybe; 10.13/4.16 import qualified Prelude; 10.13/4.16 } 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (7) COR (EQUIVALENT) 10.13/4.16 Cond Reductions: 10.13/4.16 The following Function with conditions 10.13/4.16 "undefined |Falseundefined; 10.13/4.16 " 10.13/4.16 is transformed to 10.13/4.16 "undefined = undefined1; 10.13/4.16 " 10.13/4.16 "undefined0 True = undefined; 10.13/4.16 " 10.13/4.16 "undefined1 = undefined0 False; 10.13/4.16 " 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (8) 10.13/4.16 Obligation: 10.13/4.16 mainModule Main 10.13/4.16 module Maybe where { 10.13/4.16 import qualified Main; 10.13/4.16 import qualified Prelude; 10.13/4.16 catMaybes :: [Maybe a] -> [a]; 10.13/4.16 catMaybes ls = concatMap catMaybes0 ls; 10.13/4.16 10.13/4.16 catMaybes0 vv1 = catMaybes00 vv1; 10.13/4.16 10.13/4.16 catMaybes00 (Just x) = x : []; 10.13/4.16 catMaybes00 vy = []; 10.13/4.16 10.13/4.16 } 10.13/4.16 module Main where { 10.13/4.16 import qualified Maybe; 10.13/4.16 import qualified Prelude; 10.13/4.16 } 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (9) Narrow (SOUND) 10.13/4.16 Haskell To QDPs 10.13/4.16 10.13/4.16 digraph dp_graph { 10.13/4.16 node [outthreshold=100, inthreshold=100];1[label="Maybe.catMaybes",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.13/4.16 3[label="Maybe.catMaybes vz3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 10.13/4.16 4[label="concatMap Maybe.catMaybes0 vz3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.13/4.16 5[label="concat . map Maybe.catMaybes0",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 10.13/4.16 6[label="concat (map Maybe.catMaybes0 vz3)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 10.13/4.16 7[label="foldr (++) [] (map Maybe.catMaybes0 vz3)",fontsize=16,color="burlywood",shape="triangle"];25[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];7 -> 25[label="",style="solid", color="burlywood", weight=9]; 10.13/4.16 25 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.13/4.16 26[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 26[label="",style="solid", color="burlywood", weight=9]; 10.13/4.16 26 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.13/4.16 8[label="foldr (++) [] (map Maybe.catMaybes0 (vz30 : vz31))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 10.13/4.16 9[label="foldr (++) [] (map Maybe.catMaybes0 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10.13/4.16 10[label="foldr (++) [] (Maybe.catMaybes0 vz30 : map Maybe.catMaybes0 vz31)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 10.13/4.16 11[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 10.13/4.16 12 -> 14[label="",style="dashed", color="red", weight=0]; 10.13/4.16 12[label="(++) Maybe.catMaybes0 vz30 foldr (++) [] (map Maybe.catMaybes0 vz31)",fontsize=16,color="magenta"];12 -> 15[label="",style="dashed", color="magenta", weight=3]; 10.13/4.16 13[label="[]",fontsize=16,color="green",shape="box"];15 -> 7[label="",style="dashed", color="red", weight=0]; 10.13/4.16 15[label="foldr (++) [] (map Maybe.catMaybes0 vz31)",fontsize=16,color="magenta"];15 -> 16[label="",style="dashed", color="magenta", weight=3]; 10.13/4.16 14[label="(++) Maybe.catMaybes0 vz30 vz4",fontsize=16,color="black",shape="triangle"];14 -> 17[label="",style="solid", color="black", weight=3]; 10.13/4.16 16[label="vz31",fontsize=16,color="green",shape="box"];17[label="(++) Maybe.catMaybes00 vz30 vz4",fontsize=16,color="burlywood",shape="box"];27[label="vz30/Nothing",fontsize=10,color="white",style="solid",shape="box"];17 -> 27[label="",style="solid", color="burlywood", weight=9]; 10.13/4.16 27 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.13/4.16 28[label="vz30/Just vz300",fontsize=10,color="white",style="solid",shape="box"];17 -> 28[label="",style="solid", color="burlywood", weight=9]; 10.13/4.16 28 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.13/4.16 18[label="(++) Maybe.catMaybes00 Nothing vz4",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3]; 10.13/4.16 19[label="(++) Maybe.catMaybes00 (Just vz300) vz4",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3]; 10.13/4.16 20[label="(++) [] vz4",fontsize=16,color="black",shape="triangle"];20 -> 22[label="",style="solid", color="black", weight=3]; 10.13/4.16 21[label="(++) (vz300 : []) vz4",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3]; 10.13/4.16 22[label="vz4",fontsize=16,color="green",shape="box"];23[label="vz300 : [] ++ vz4",fontsize=16,color="green",shape="box"];23 -> 24[label="",style="dashed", color="green", weight=3]; 10.13/4.16 24 -> 20[label="",style="dashed", color="red", weight=0]; 10.13/4.16 24[label="[] ++ vz4",fontsize=16,color="magenta"];} 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (10) 10.13/4.16 Obligation: 10.13/4.16 Q DP problem: 10.13/4.16 The TRS P consists of the following rules: 10.13/4.16 10.13/4.16 new_foldr(:(vz30, vz31), h) -> new_foldr(vz31, h) 10.13/4.16 10.13/4.16 R is empty. 10.13/4.16 Q is empty. 10.13/4.16 We have to consider all minimal (P,Q,R)-chains. 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (11) QDPSizeChangeProof (EQUIVALENT) 10.13/4.16 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. 10.13/4.16 10.13/4.16 From the DPs we obtained the following set of size-change graphs: 10.13/4.16 *new_foldr(:(vz30, vz31), h) -> new_foldr(vz31, h) 10.13/4.16 The graph contains the following edges 1 > 1, 2 >= 2 10.13/4.16 10.13/4.16 10.13/4.16 ---------------------------------------- 10.13/4.16 10.13/4.16 (12) 10.13/4.16 YES 10.22/4.20 EOF