7.99/3.64 YES 10.19/4.25 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.19/4.25 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.19/4.25 10.19/4.25 10.19/4.25 H-Termination with start terms of the given HASKELL could be proven: 10.19/4.25 10.19/4.25 (0) HASKELL 10.19/4.25 (1) CR [EQUIVALENT, 0 ms] 10.19/4.25 (2) HASKELL 10.19/4.25 (3) BR [EQUIVALENT, 0 ms] 10.19/4.25 (4) HASKELL 10.19/4.25 (5) COR [EQUIVALENT, 0 ms] 10.19/4.25 (6) HASKELL 10.19/4.25 (7) LetRed [EQUIVALENT, 0 ms] 10.19/4.25 (8) HASKELL 10.19/4.25 (9) Narrow [SOUND, 0 ms] 10.19/4.25 (10) QDP 10.19/4.25 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.19/4.25 (12) YES 10.19/4.25 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (0) 10.19/4.25 Obligation: 10.19/4.25 mainModule Main 10.19/4.25 module Maybe where { 10.19/4.25 import qualified Main; 10.19/4.25 import qualified Prelude; 10.19/4.25 mapMaybe :: (b -> Maybe a) -> [b] -> [a]; 10.19/4.25 mapMaybe _ [] = []; 10.19/4.25 mapMaybe f (x : xs) = let { 10.19/4.25 rs = mapMaybe f xs; 10.19/4.25 } in case f x of { 10.19/4.25 Nothing-> rs; 10.19/4.25 Just r-> r : rs; 10.19/4.25 } ; 10.19/4.25 10.19/4.25 } 10.19/4.25 module Main where { 10.19/4.25 import qualified Maybe; 10.19/4.25 import qualified Prelude; 10.19/4.25 } 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (1) CR (EQUIVALENT) 10.19/4.25 Case Reductions: 10.19/4.25 The following Case expression 10.19/4.25 "case f x of { 10.19/4.25 Nothing -> rs; 10.19/4.25 Just r -> r : rs} 10.19/4.25 " 10.19/4.25 is transformed to 10.19/4.25 "mapMaybe0 Nothing = rs; 10.19/4.25 mapMaybe0 (Just r) = r : rs; 10.19/4.25 " 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (2) 10.19/4.25 Obligation: 10.19/4.25 mainModule Main 10.19/4.25 module Maybe where { 10.19/4.25 import qualified Main; 10.19/4.25 import qualified Prelude; 10.19/4.25 mapMaybe :: (b -> Maybe a) -> [b] -> [a]; 10.19/4.25 mapMaybe _ [] = []; 10.19/4.25 mapMaybe f (x : xs) = let { 10.19/4.25 mapMaybe0 Nothing = rs; 10.19/4.25 mapMaybe0 (Just r) = r : rs; 10.19/4.25 rs = mapMaybe f xs; 10.19/4.25 } in mapMaybe0 (f x); 10.19/4.25 10.19/4.25 } 10.19/4.25 module Main where { 10.19/4.25 import qualified Maybe; 10.19/4.25 import qualified Prelude; 10.19/4.25 } 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (3) BR (EQUIVALENT) 10.19/4.25 Replaced joker patterns by fresh variables and removed binding patterns. 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (4) 10.19/4.25 Obligation: 10.19/4.25 mainModule Main 10.19/4.25 module Maybe where { 10.19/4.25 import qualified Main; 10.19/4.25 import qualified Prelude; 10.19/4.25 mapMaybe :: (a -> Maybe b) -> [a] -> [b]; 10.19/4.25 mapMaybe vy [] = []; 10.19/4.25 mapMaybe f (x : xs) = let { 10.19/4.25 mapMaybe0 Nothing = rs; 10.19/4.25 mapMaybe0 (Just r) = r : rs; 10.19/4.25 rs = mapMaybe f xs; 10.19/4.25 } in mapMaybe0 (f x); 10.19/4.25 10.19/4.25 } 10.19/4.25 module Main where { 10.19/4.25 import qualified Maybe; 10.19/4.25 import qualified Prelude; 10.19/4.25 } 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (5) COR (EQUIVALENT) 10.19/4.25 Cond Reductions: 10.19/4.25 The following Function with conditions 10.19/4.25 "undefined |Falseundefined; 10.19/4.25 " 10.19/4.25 is transformed to 10.19/4.25 "undefined = undefined1; 10.19/4.25 " 10.19/4.25 "undefined0 True = undefined; 10.19/4.25 " 10.19/4.25 "undefined1 = undefined0 False; 10.19/4.25 " 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (6) 10.19/4.25 Obligation: 10.19/4.25 mainModule Main 10.19/4.25 module Maybe where { 10.19/4.25 import qualified Main; 10.19/4.25 import qualified Prelude; 10.19/4.25 mapMaybe :: (a -> Maybe b) -> [a] -> [b]; 10.19/4.25 mapMaybe vy [] = []; 10.19/4.25 mapMaybe f (x : xs) = let { 10.19/4.25 mapMaybe0 Nothing = rs; 10.19/4.25 mapMaybe0 (Just r) = r : rs; 10.19/4.25 rs = mapMaybe f xs; 10.19/4.25 } in mapMaybe0 (f x); 10.19/4.25 10.19/4.25 } 10.19/4.25 module Main where { 10.19/4.25 import qualified Maybe; 10.19/4.25 import qualified Prelude; 10.19/4.25 } 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (7) LetRed (EQUIVALENT) 10.19/4.25 Let/Where Reductions: 10.19/4.25 The bindings of the following Let/Where expression 10.19/4.25 "let { 10.19/4.25 mapMaybe0 Nothing = rs; 10.19/4.25 mapMaybe0 (Just r) = r : rs; 10.19/4.25 ; 10.19/4.25 rs = mapMaybe f xs; 10.19/4.25 } in mapMaybe0 (f x)" 10.19/4.25 are unpacked to the following functions on top level 10.19/4.25 "mapMaybeMapMaybe0 vz wu Nothing = mapMaybeRs vz wu; 10.19/4.25 mapMaybeMapMaybe0 vz wu (Just r) = r : mapMaybeRs vz wu; 10.19/4.25 " 10.19/4.25 "mapMaybeRs vz wu = mapMaybe vz wu; 10.19/4.25 " 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (8) 10.19/4.25 Obligation: 10.19/4.25 mainModule Main 10.19/4.25 module Maybe where { 10.19/4.25 import qualified Main; 10.19/4.25 import qualified Prelude; 10.19/4.25 mapMaybe :: (a -> Maybe b) -> [a] -> [b]; 10.19/4.25 mapMaybe vy [] = []; 10.19/4.25 mapMaybe f (x : xs) = mapMaybeMapMaybe0 f xs (f x); 10.19/4.25 10.19/4.25 mapMaybeMapMaybe0 vz wu Nothing = mapMaybeRs vz wu; 10.19/4.25 mapMaybeMapMaybe0 vz wu (Just r) = r : mapMaybeRs vz wu; 10.19/4.25 10.19/4.25 mapMaybeRs vz wu = mapMaybe vz wu; 10.19/4.25 10.19/4.25 } 10.19/4.25 module Main where { 10.19/4.25 import qualified Maybe; 10.19/4.25 import qualified Prelude; 10.19/4.25 } 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (9) Narrow (SOUND) 10.19/4.25 Haskell To QDPs 10.19/4.25 10.19/4.25 digraph dp_graph { 10.19/4.25 node [outthreshold=100, inthreshold=100];1[label="Maybe.mapMaybe",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.19/4.25 3[label="Maybe.mapMaybe wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.19/4.25 4[label="Maybe.mapMaybe wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];20[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9]; 10.19/4.25 20 -> 5[label="",style="solid", color="burlywood", weight=3]; 10.19/4.25 21[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9]; 10.19/4.25 21 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.19/4.25 5[label="Maybe.mapMaybe wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 10.19/4.25 6[label="Maybe.mapMaybe wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 10.19/4.25 7 -> 9[label="",style="dashed", color="red", weight=0]; 10.19/4.25 7[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (wv3 wv40)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3]; 10.19/4.25 8[label="[]",fontsize=16,color="green",shape="box"];10[label="wv3 wv40",fontsize=16,color="green",shape="box"];10 -> 14[label="",style="dashed", color="green", weight=3]; 10.19/4.25 9[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 wv5",fontsize=16,color="burlywood",shape="triangle"];22[label="wv5/Nothing",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9]; 10.19/4.25 22 -> 12[label="",style="solid", color="burlywood", weight=3]; 10.19/4.25 23[label="wv5/Just wv50",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9]; 10.19/4.25 23 -> 13[label="",style="solid", color="burlywood", weight=3]; 10.19/4.25 14[label="wv40",fontsize=16,color="green",shape="box"];12[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 Nothing",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3]; 10.19/4.25 13[label="Maybe.mapMaybeMapMaybe0 wv3 wv41 (Just wv50)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3]; 10.19/4.25 15[label="Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="black",shape="triangle"];15 -> 17[label="",style="solid", color="black", weight=3]; 10.19/4.25 16[label="wv50 : Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3]; 10.19/4.25 17 -> 4[label="",style="dashed", color="red", weight=0]; 10.19/4.25 17[label="Maybe.mapMaybe wv3 wv41",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3]; 10.19/4.25 18 -> 15[label="",style="dashed", color="red", weight=0]; 10.19/4.25 18[label="Maybe.mapMaybeRs wv3 wv41",fontsize=16,color="magenta"];19[label="wv41",fontsize=16,color="green",shape="box"];} 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (10) 10.19/4.25 Obligation: 10.19/4.25 Q DP problem: 10.19/4.25 The TRS P consists of the following rules: 10.19/4.25 10.19/4.25 new_mapMaybeRs(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) 10.19/4.25 new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) 10.19/4.25 new_mapMaybe(wv3, :(wv40, wv41), h, ba) -> new_mapMaybeMapMaybe0(wv3, wv41, h, ba) 10.19/4.25 new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybeRs(wv3, wv41, h, ba) 10.19/4.25 10.19/4.25 R is empty. 10.19/4.25 Q is empty. 10.19/4.25 We have to consider all minimal (P,Q,R)-chains. 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (11) QDPSizeChangeProof (EQUIVALENT) 10.19/4.25 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. 10.19/4.25 10.19/4.25 From the DPs we obtained the following set of size-change graphs: 10.19/4.25 *new_mapMaybe(wv3, :(wv40, wv41), h, ba) -> new_mapMaybeMapMaybe0(wv3, wv41, h, ba) 10.19/4.25 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 10.19/4.25 10.19/4.25 10.19/4.25 *new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybeRs(wv3, wv41, h, ba) 10.19/4.25 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 10.19/4.25 10.19/4.25 10.19/4.25 *new_mapMaybeMapMaybe0(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) 10.19/4.25 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 10.19/4.25 10.19/4.25 10.19/4.25 *new_mapMaybeRs(wv3, wv41, h, ba) -> new_mapMaybe(wv3, wv41, h, ba) 10.19/4.25 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4 10.19/4.25 10.19/4.25 10.19/4.25 ---------------------------------------- 10.19/4.25 10.19/4.25 (12) 10.19/4.25 YES 10.48/4.34 EOF