10.39/4.33 YES 12.94/5.03 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 12.94/5.03 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 12.94/5.03 12.94/5.03 12.94/5.03 H-Termination with start terms of the given HASKELL could be proven: 12.94/5.03 12.94/5.03 (0) HASKELL 12.94/5.03 (1) LR [EQUIVALENT, 0 ms] 12.94/5.03 (2) HASKELL 12.94/5.03 (3) CR [EQUIVALENT, 0 ms] 12.94/5.03 (4) HASKELL 12.94/5.03 (5) IFR [EQUIVALENT, 0 ms] 12.94/5.03 (6) HASKELL 12.94/5.03 (7) BR [EQUIVALENT, 0 ms] 12.94/5.03 (8) HASKELL 12.94/5.03 (9) COR [EQUIVALENT, 0 ms] 12.94/5.03 (10) HASKELL 12.94/5.03 (11) Narrow [SOUND, 0 ms] 12.94/5.03 (12) QDP 12.94/5.03 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 12.94/5.03 (14) YES 12.94/5.03 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (0) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (\vv2 ->case vv2 of { 12.94/5.03 x-> if any (eq x) ys then x : [] else []; 12.94/5.03 _-> []; 12.94/5.03 } ) xs; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (1) LR (EQUIVALENT) 12.94/5.03 Lambda Reductions: 12.94/5.03 The following Lambda expression 12.94/5.03 "\vv2->case vv2 of { 12.94/5.03 x -> if any (eq x) ys then x : [] else []; 12.94/5.03 _ -> []} 12.94/5.03 " 12.94/5.03 is transformed to 12.94/5.03 "intersectBy0 eq ys vv2 = case vv2 of { 12.94/5.03 x -> if any (eq x) ys then x : [] else []; 12.94/5.03 _ -> []} 12.94/5.03 ; 12.94/5.03 " 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (2) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; 12.94/5.03 12.94/5.03 intersectBy0 eq ys vv2 = case vv2 of { 12.94/5.03 x-> if any (eq x) ys then x : [] else []; 12.94/5.03 _-> []; 12.94/5.03 } ; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (3) CR (EQUIVALENT) 12.94/5.03 Case Reductions: 12.94/5.03 The following Case expression 12.94/5.03 "case vv2 of { 12.94/5.03 x -> if any (eq x) ys then x : [] else []; 12.94/5.03 _ -> []} 12.94/5.03 " 12.94/5.03 is transformed to 12.94/5.03 "intersectBy00 eq ys x = if any (eq x) ys then x : [] else []; 12.94/5.03 intersectBy00 eq ys _ = []; 12.94/5.03 " 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (4) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; 12.94/5.03 12.94/5.03 intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; 12.94/5.03 12.94/5.03 intersectBy00 eq ys x = if any (eq x) ys then x : [] else []; 12.94/5.03 intersectBy00 eq ys _ = []; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (5) IFR (EQUIVALENT) 12.94/5.03 If Reductions: 12.94/5.03 The following If expression 12.94/5.03 "if any (eq x) ys then x : [] else []" 12.94/5.03 is transformed to 12.94/5.03 "intersectBy000 x True = x : []; 12.94/5.03 intersectBy000 x False = []; 12.94/5.03 " 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (6) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; 12.94/5.03 12.94/5.03 intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; 12.94/5.03 12.94/5.03 intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); 12.94/5.03 intersectBy00 eq ys _ = []; 12.94/5.03 12.94/5.03 intersectBy000 x True = x : []; 12.94/5.03 intersectBy000 x False = []; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (7) BR (EQUIVALENT) 12.94/5.03 Replaced joker patterns by fresh variables and removed binding patterns. 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (8) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; 12.94/5.03 12.94/5.03 intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; 12.94/5.03 12.94/5.03 intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); 12.94/5.03 intersectBy00 eq ys vy = []; 12.94/5.03 12.94/5.03 intersectBy000 x True = x : []; 12.94/5.03 intersectBy000 x False = []; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (9) COR (EQUIVALENT) 12.94/5.03 Cond Reductions: 12.94/5.03 The following Function with conditions 12.94/5.03 "undefined |Falseundefined; 12.94/5.03 " 12.94/5.03 is transformed to 12.94/5.03 "undefined = undefined1; 12.94/5.03 " 12.94/5.03 "undefined0 True = undefined; 12.94/5.03 " 12.94/5.03 "undefined1 = undefined0 False; 12.94/5.03 " 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (10) 12.94/5.03 Obligation: 12.94/5.03 mainModule Main 12.94/5.03 module Maybe where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 module List where { 12.94/5.03 import qualified Main; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 intersect :: Eq a => [a] -> [a] -> [a]; 12.94/5.03 intersect = intersectBy (==); 12.94/5.03 12.94/5.03 intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; 12.94/5.03 intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; 12.94/5.03 12.94/5.03 intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; 12.94/5.03 12.94/5.03 intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); 12.94/5.03 intersectBy00 eq ys vy = []; 12.94/5.03 12.94/5.03 intersectBy000 x True = x : []; 12.94/5.03 intersectBy000 x False = []; 12.94/5.03 12.94/5.03 } 12.94/5.03 module Main where { 12.94/5.03 import qualified List; 12.94/5.03 import qualified Maybe; 12.94/5.03 import qualified Prelude; 12.94/5.03 } 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (11) Narrow (SOUND) 12.94/5.03 Haskell To QDPs 12.94/5.03 12.94/5.03 digraph dp_graph { 12.94/5.03 node [outthreshold=100, inthreshold=100];1[label="List.intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 12.94/5.03 3[label="List.intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 12.94/5.03 4[label="List.intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 12.94/5.03 5[label="List.intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 12.94/5.03 6[label="concatMap (List.intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 12.94/5.03 7[label="concat . map (List.intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 12.94/5.03 8[label="concat (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 12.94/5.03 9[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];39[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 39[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 39 -> 10[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 40[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 40[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 40 -> 11[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 10[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 12.94/5.03 11[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 12.94/5.03 12[label="foldr (++) [] (List.intersectBy0 (==) vz4 vz30 : map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 12.94/5.03 13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3]; 12.94/5.03 14 -> 16[label="",style="dashed", color="red", weight=0]; 12.94/5.03 14[label="(++) List.intersectBy0 (==) vz4 vz30 foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3]; 12.94/5.03 15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0]; 12.94/5.03 17[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3]; 12.94/5.03 16[label="(++) List.intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3]; 12.94/5.03 18[label="vz31",fontsize=16,color="green",shape="box"];19[label="(++) List.intersectBy00 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 12.94/5.03 20[label="(++) List.intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3]; 12.94/5.03 21[label="(++) List.intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3]; 12.94/5.03 22[label="(++) List.intersectBy000 vz30 (or (map ((==) vz30) vz4)) vz5",fontsize=16,color="black",shape="box"];22 -> 23[label="",style="solid", color="black", weight=3]; 12.94/5.03 23[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="box"];41[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 41[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 41 -> 24[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 42[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 42[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 42 -> 25[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 24[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) (vz40 : vz41))) vz5",fontsize=16,color="black",shape="box"];24 -> 26[label="",style="solid", color="black", weight=3]; 12.94/5.03 25[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) [])) vz5",fontsize=16,color="black",shape="box"];25 -> 27[label="",style="solid", color="black", weight=3]; 12.94/5.03 26[label="(++) List.intersectBy000 vz30 (foldr (||) False (((==) vz30 vz40) : map ((==) vz30) vz41)) vz5",fontsize=16,color="black",shape="box"];26 -> 28[label="",style="solid", color="black", weight=3]; 12.94/5.03 27[label="(++) List.intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3]; 12.94/5.03 28[label="(++) List.intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];43[label="vz30/()",fontsize=10,color="white",style="solid",shape="box"];28 -> 43[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 43 -> 30[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3]; 12.94/5.03 30[label="(++) List.intersectBy000 () ((||) (==) () vz40 foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];44[label="vz40/()",fontsize=10,color="white",style="solid",shape="box"];30 -> 44[label="",style="solid", color="burlywood", weight=9]; 12.94/5.03 44 -> 32[label="",style="solid", color="burlywood", weight=3]; 12.94/5.03 31[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];31 -> 33[label="",style="solid", color="black", weight=3]; 12.94/5.03 32[label="(++) List.intersectBy000 () ((||) (==) () () foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3]; 12.94/5.03 33[label="vz5",fontsize=16,color="green",shape="box"];34[label="(++) List.intersectBy000 () ((||) True foldr (||) False (map ((==) ()) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 35[label="",style="solid", color="black", weight=3]; 12.94/5.03 35[label="(++) List.intersectBy000 () True vz5",fontsize=16,color="black",shape="box"];35 -> 36[label="",style="solid", color="black", weight=3]; 12.94/5.03 36[label="(++) (() : []) vz5",fontsize=16,color="black",shape="box"];36 -> 37[label="",style="solid", color="black", weight=3]; 12.94/5.03 37[label="() : [] ++ vz5",fontsize=16,color="green",shape="box"];37 -> 38[label="",style="dashed", color="green", weight=3]; 12.94/5.03 38 -> 31[label="",style="dashed", color="red", weight=0]; 12.94/5.03 38[label="[] ++ vz5",fontsize=16,color="magenta"];} 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (12) 12.94/5.03 Obligation: 12.94/5.03 Q DP problem: 12.94/5.03 The TRS P consists of the following rules: 12.94/5.03 12.94/5.03 new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31) 12.94/5.03 12.94/5.03 R is empty. 12.94/5.03 Q is empty. 12.94/5.03 We have to consider all minimal (P,Q,R)-chains. 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (13) QDPSizeChangeProof (EQUIVALENT) 12.94/5.03 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. 12.94/5.03 12.94/5.03 From the DPs we obtained the following set of size-change graphs: 12.94/5.03 *new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31) 12.94/5.03 The graph contains the following edges 1 >= 1, 2 > 2 12.94/5.03 12.94/5.03 12.94/5.03 ---------------------------------------- 12.94/5.03 12.94/5.03 (14) 12.94/5.03 YES 13.18/5.09 EOF