/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) CR [EQUIVALENT, 0 ms] (4) HASKELL (5) IFR [EQUIVALENT, 0 ms] (6) HASKELL (7) BR [EQUIVALENT, 0 ms] (8) HASKELL (9) COR [EQUIVALENT, 22 ms] (10) HASKELL (11) Narrow [SOUND, 0 ms] (12) AND (13) QDP (14) DependencyGraphProof [EQUIVALENT, 0 ms] (15) AND (16) QDP (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] (18) YES (19) QDP (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] (21) YES (22) QDP (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] (24) YES (25) QDP (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] (27) 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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (\vv2 ->case vv2 of { x-> if any (eq x) ys then x : [] else []; _-> []; } ) xs; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\vv2->case vv2 of { x -> if any (eq x) ys then x : [] else []; _ -> []} " is transformed to "intersectBy0 eq ys vv2 = case vv2 of { x -> if any (eq x) ys then x : [] else []; _ -> []} ; " ---------------------------------------- (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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; intersectBy0 eq ys vv2 = case vv2 of { x-> if any (eq x) ys then x : [] else []; _-> []; } ; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) CR (EQUIVALENT) Case Reductions: The following Case expression "case vv2 of { x -> if any (eq x) ys then x : [] else []; _ -> []} " is transformed to "intersectBy00 eq ys x = if any (eq x) ys then x : [] else []; intersectBy00 eq ys _ = []; " ---------------------------------------- (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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; intersectBy00 eq ys x = if any (eq x) ys then x : [] else []; intersectBy00 eq ys _ = []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (5) IFR (EQUIVALENT) If Reductions: The following If expression "if any (eq x) ys then x : [] else []" is transformed to "intersectBy000 x True = x : []; intersectBy000 x 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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); intersectBy00 eq ys _ = []; intersectBy000 x True = x : []; intersectBy000 x False = []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (7) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (8) 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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); intersectBy00 eq ys vy = []; intersectBy000 x True = x : []; intersectBy000 x False = []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (9) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (10) 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; intersect :: Eq a => [a] -> [a] -> [a]; intersect = intersectBy (==); intersectBy :: (a -> a -> Bool) -> [a] -> [a] -> [a]; intersectBy eq xs ys = concatMap (intersectBy0 eq ys) xs; intersectBy0 eq ys vv2 = intersectBy00 eq ys vv2; intersectBy00 eq ys x = intersectBy000 x (any (eq x) ys); intersectBy00 eq ys vy = []; intersectBy000 x True = x : []; intersectBy000 x False = []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (11) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="List.intersect",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="List.intersect vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="List.intersect vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="List.intersectBy (==) vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="concatMap (List.intersectBy0 (==) vz4) vz3",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="concat . map (List.intersectBy0 (==) vz4)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 8[label="concat (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz3)",fontsize=16,color="burlywood",shape="triangle"];74[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];9 -> 74[label="",style="solid", color="burlywood", weight=9]; 74 -> 10[label="",style="solid", color="burlywood", weight=3]; 75[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 75[label="",style="solid", color="burlywood", weight=9]; 75 -> 11[label="",style="solid", color="burlywood", weight=3]; 10[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) (vz30 : vz31))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 11[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 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]; 13[label="foldr (++) [] []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3]; 14 -> 16[label="",style="dashed", color="red", weight=0]; 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]; 15[label="[]",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0]; 17[label="foldr (++) [] (map (List.intersectBy0 (==) vz4) vz31)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3]; 16[label="(++) List.intersectBy0 (==) vz4 vz30 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3]; 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]; 20[label="(++) List.intersectBy000 vz30 (any ((==) vz30) vz4) vz5",fontsize=16,color="black",shape="box"];20 -> 21[label="",style="solid", color="black", weight=3]; 21[label="(++) List.intersectBy000 vz30 (or . map ((==) vz30)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3]; 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]; 23[label="(++) List.intersectBy000 vz30 (foldr (||) False (map ((==) vz30) vz4)) vz5",fontsize=16,color="burlywood",shape="triangle"];76[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];23 -> 76[label="",style="solid", color="burlywood", weight=9]; 76 -> 24[label="",style="solid", color="burlywood", weight=3]; 77[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 77[label="",style="solid", color="burlywood", weight=9]; 77 -> 25[label="",style="solid", color="burlywood", weight=3]; 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]; 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]; 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]; 27[label="(++) List.intersectBy000 vz30 (foldr (||) False []) vz5",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3]; 28[label="(++) List.intersectBy000 vz30 ((||) (==) vz30 vz40 foldr (||) False (map ((==) vz30) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];78[label="vz30/LT",fontsize=10,color="white",style="solid",shape="box"];28 -> 78[label="",style="solid", color="burlywood", weight=9]; 78 -> 30[label="",style="solid", color="burlywood", weight=3]; 79[label="vz30/EQ",fontsize=10,color="white",style="solid",shape="box"];28 -> 79[label="",style="solid", color="burlywood", weight=9]; 79 -> 31[label="",style="solid", color="burlywood", weight=3]; 80[label="vz30/GT",fontsize=10,color="white",style="solid",shape="box"];28 -> 80[label="",style="solid", color="burlywood", weight=9]; 80 -> 32[label="",style="solid", color="burlywood", weight=3]; 29[label="(++) List.intersectBy000 vz30 False vz5",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 30[label="(++) List.intersectBy000 LT ((||) (==) LT vz40 foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];81[label="vz40/LT",fontsize=10,color="white",style="solid",shape="box"];30 -> 81[label="",style="solid", color="burlywood", weight=9]; 81 -> 34[label="",style="solid", color="burlywood", weight=3]; 82[label="vz40/EQ",fontsize=10,color="white",style="solid",shape="box"];30 -> 82[label="",style="solid", color="burlywood", weight=9]; 82 -> 35[label="",style="solid", color="burlywood", weight=3]; 83[label="vz40/GT",fontsize=10,color="white",style="solid",shape="box"];30 -> 83[label="",style="solid", color="burlywood", weight=9]; 83 -> 36[label="",style="solid", color="burlywood", weight=3]; 31[label="(++) List.intersectBy000 EQ ((||) (==) EQ vz40 foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];84[label="vz40/LT",fontsize=10,color="white",style="solid",shape="box"];31 -> 84[label="",style="solid", color="burlywood", weight=9]; 84 -> 37[label="",style="solid", color="burlywood", weight=3]; 85[label="vz40/EQ",fontsize=10,color="white",style="solid",shape="box"];31 -> 85[label="",style="solid", color="burlywood", weight=9]; 85 -> 38[label="",style="solid", color="burlywood", weight=3]; 86[label="vz40/GT",fontsize=10,color="white",style="solid",shape="box"];31 -> 86[label="",style="solid", color="burlywood", weight=9]; 86 -> 39[label="",style="solid", color="burlywood", weight=3]; 32[label="(++) List.intersectBy000 GT ((||) (==) GT vz40 foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="burlywood",shape="box"];87[label="vz40/LT",fontsize=10,color="white",style="solid",shape="box"];32 -> 87[label="",style="solid", color="burlywood", weight=9]; 87 -> 40[label="",style="solid", color="burlywood", weight=3]; 88[label="vz40/EQ",fontsize=10,color="white",style="solid",shape="box"];32 -> 88[label="",style="solid", color="burlywood", weight=9]; 88 -> 41[label="",style="solid", color="burlywood", weight=3]; 89[label="vz40/GT",fontsize=10,color="white",style="solid",shape="box"];32 -> 89[label="",style="solid", color="burlywood", weight=9]; 89 -> 42[label="",style="solid", color="burlywood", weight=3]; 33[label="(++) [] vz5",fontsize=16,color="black",shape="triangle"];33 -> 43[label="",style="solid", color="black", weight=3]; 34[label="(++) List.intersectBy000 LT ((||) (==) LT LT foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="black",shape="box"];34 -> 44[label="",style="solid", color="black", weight=3]; 35[label="(++) List.intersectBy000 LT ((||) (==) LT EQ foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="black",shape="box"];35 -> 45[label="",style="solid", color="black", weight=3]; 36[label="(++) List.intersectBy000 LT ((||) (==) LT GT foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="black",shape="box"];36 -> 46[label="",style="solid", color="black", weight=3]; 37[label="(++) List.intersectBy000 EQ ((||) (==) EQ LT foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="black",shape="box"];37 -> 47[label="",style="solid", color="black", weight=3]; 38[label="(++) List.intersectBy000 EQ ((||) (==) EQ EQ foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="black",shape="box"];38 -> 48[label="",style="solid", color="black", weight=3]; 39[label="(++) List.intersectBy000 EQ ((||) (==) EQ GT foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="black",shape="box"];39 -> 49[label="",style="solid", color="black", weight=3]; 40[label="(++) List.intersectBy000 GT ((||) (==) GT LT foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="black",shape="box"];40 -> 50[label="",style="solid", color="black", weight=3]; 41[label="(++) List.intersectBy000 GT ((||) (==) GT EQ foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="black",shape="box"];41 -> 51[label="",style="solid", color="black", weight=3]; 42[label="(++) List.intersectBy000 GT ((||) (==) GT GT foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="black",shape="box"];42 -> 52[label="",style="solid", color="black", weight=3]; 43[label="vz5",fontsize=16,color="green",shape="box"];44[label="(++) List.intersectBy000 LT ((||) True foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="black",shape="box"];44 -> 53[label="",style="solid", color="black", weight=3]; 45[label="(++) List.intersectBy000 LT ((||) False foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="black",shape="triangle"];45 -> 54[label="",style="solid", color="black", weight=3]; 46 -> 45[label="",style="dashed", color="red", weight=0]; 46[label="(++) List.intersectBy000 LT ((||) False foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="magenta"];47[label="(++) List.intersectBy000 EQ ((||) False foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="black",shape="triangle"];47 -> 55[label="",style="solid", color="black", weight=3]; 48[label="(++) List.intersectBy000 EQ ((||) True foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="black",shape="box"];48 -> 56[label="",style="solid", color="black", weight=3]; 49 -> 47[label="",style="dashed", color="red", weight=0]; 49[label="(++) List.intersectBy000 EQ ((||) False foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="magenta"];50[label="(++) List.intersectBy000 GT ((||) False foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="black",shape="triangle"];50 -> 57[label="",style="solid", color="black", weight=3]; 51 -> 50[label="",style="dashed", color="red", weight=0]; 51[label="(++) List.intersectBy000 GT ((||) False foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="magenta"];52[label="(++) List.intersectBy000 GT ((||) True foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="black",shape="box"];52 -> 58[label="",style="solid", color="black", weight=3]; 53[label="(++) List.intersectBy000 LT True vz5",fontsize=16,color="black",shape="box"];53 -> 59[label="",style="solid", color="black", weight=3]; 54 -> 23[label="",style="dashed", color="red", weight=0]; 54[label="(++) List.intersectBy000 LT (foldr (||) False (map ((==) LT) vz41)) vz5",fontsize=16,color="magenta"];54 -> 60[label="",style="dashed", color="magenta", weight=3]; 54 -> 61[label="",style="dashed", color="magenta", weight=3]; 55 -> 23[label="",style="dashed", color="red", weight=0]; 55[label="(++) List.intersectBy000 EQ (foldr (||) False (map ((==) EQ) vz41)) vz5",fontsize=16,color="magenta"];55 -> 62[label="",style="dashed", color="magenta", weight=3]; 55 -> 63[label="",style="dashed", color="magenta", weight=3]; 56[label="(++) List.intersectBy000 EQ True vz5",fontsize=16,color="black",shape="box"];56 -> 64[label="",style="solid", color="black", weight=3]; 57 -> 23[label="",style="dashed", color="red", weight=0]; 57[label="(++) List.intersectBy000 GT (foldr (||) False (map ((==) GT) vz41)) vz5",fontsize=16,color="magenta"];57 -> 65[label="",style="dashed", color="magenta", weight=3]; 57 -> 66[label="",style="dashed", color="magenta", weight=3]; 58[label="(++) List.intersectBy000 GT True vz5",fontsize=16,color="black",shape="box"];58 -> 67[label="",style="solid", color="black", weight=3]; 59[label="(++) (LT : []) vz5",fontsize=16,color="black",shape="box"];59 -> 68[label="",style="solid", color="black", weight=3]; 60[label="vz41",fontsize=16,color="green",shape="box"];61[label="LT",fontsize=16,color="green",shape="box"];62[label="vz41",fontsize=16,color="green",shape="box"];63[label="EQ",fontsize=16,color="green",shape="box"];64[label="(++) (EQ : []) vz5",fontsize=16,color="black",shape="box"];64 -> 69[label="",style="solid", color="black", weight=3]; 65[label="vz41",fontsize=16,color="green",shape="box"];66[label="GT",fontsize=16,color="green",shape="box"];67[label="(++) (GT : []) vz5",fontsize=16,color="black",shape="box"];67 -> 70[label="",style="solid", color="black", weight=3]; 68[label="LT : [] ++ vz5",fontsize=16,color="green",shape="box"];68 -> 71[label="",style="dashed", color="green", weight=3]; 69[label="EQ : [] ++ vz5",fontsize=16,color="green",shape="box"];69 -> 72[label="",style="dashed", color="green", weight=3]; 70[label="GT : [] ++ vz5",fontsize=16,color="green",shape="box"];70 -> 73[label="",style="dashed", color="green", weight=3]; 71 -> 33[label="",style="dashed", color="red", weight=0]; 71[label="[] ++ vz5",fontsize=16,color="magenta"];72 -> 33[label="",style="dashed", color="red", weight=0]; 72[label="[] ++ vz5",fontsize=16,color="magenta"];73 -> 33[label="",style="dashed", color="red", weight=0]; 73[label="[] ++ vz5",fontsize=16,color="magenta"];} ---------------------------------------- (12) Complex Obligation (AND) ---------------------------------------- (13) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs0(EQ, :(LT, vz41), vz5) -> new_psPs0(EQ, vz41, vz5) new_psPs(vz41, vz5) -> new_psPs0(LT, vz41, vz5) new_psPs0(LT, :(GT, vz41), vz5) -> new_psPs(vz41, vz5) new_psPs1(vz41, vz5) -> new_psPs0(EQ, vz41, vz5) new_psPs2(vz41, vz5) -> new_psPs0(GT, vz41, vz5) new_psPs0(GT, :(EQ, vz41), vz5) -> new_psPs2(vz41, vz5) new_psPs0(LT, :(EQ, vz41), vz5) -> new_psPs0(LT, vz41, vz5) new_psPs0(EQ, :(GT, vz41), vz5) -> new_psPs1(vz41, vz5) new_psPs0(GT, :(LT, vz41), vz5) -> new_psPs0(GT, vz41, vz5) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (14) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs. ---------------------------------------- (15) Complex Obligation (AND) ---------------------------------------- (16) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs0(GT, :(EQ, vz41), vz5) -> new_psPs2(vz41, vz5) new_psPs2(vz41, vz5) -> new_psPs0(GT, vz41, vz5) new_psPs0(GT, :(LT, vz41), vz5) -> new_psPs0(GT, vz41, vz5) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (17) 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_psPs2(vz41, vz5) -> new_psPs0(GT, vz41, vz5) The graph contains the following edges 1 >= 2, 2 >= 3 *new_psPs0(GT, :(LT, vz41), vz5) -> new_psPs0(GT, vz41, vz5) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_psPs0(GT, :(EQ, vz41), vz5) -> new_psPs2(vz41, vz5) The graph contains the following edges 2 > 1, 3 >= 2 ---------------------------------------- (18) YES ---------------------------------------- (19) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs0(LT, :(GT, vz41), vz5) -> new_psPs(vz41, vz5) new_psPs(vz41, vz5) -> new_psPs0(LT, vz41, vz5) new_psPs0(LT, :(EQ, vz41), vz5) -> new_psPs0(LT, vz41, vz5) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (20) 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_psPs(vz41, vz5) -> new_psPs0(LT, vz41, vz5) The graph contains the following edges 1 >= 2, 2 >= 3 *new_psPs0(LT, :(EQ, vz41), vz5) -> new_psPs0(LT, vz41, vz5) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_psPs0(LT, :(GT, vz41), vz5) -> new_psPs(vz41, vz5) The graph contains the following edges 2 > 1, 3 >= 2 ---------------------------------------- (21) YES ---------------------------------------- (22) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs0(EQ, :(GT, vz41), vz5) -> new_psPs1(vz41, vz5) new_psPs1(vz41, vz5) -> new_psPs0(EQ, vz41, vz5) new_psPs0(EQ, :(LT, vz41), vz5) -> new_psPs0(EQ, vz41, vz5) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (23) 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_psPs1(vz41, vz5) -> new_psPs0(EQ, vz41, vz5) The graph contains the following edges 1 >= 2, 2 >= 3 *new_psPs0(EQ, :(LT, vz41), vz5) -> new_psPs0(EQ, vz41, vz5) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_psPs0(EQ, :(GT, vz41), vz5) -> new_psPs1(vz41, vz5) The graph contains the following edges 2 > 1, 3 >= 2 ---------------------------------------- (24) YES ---------------------------------------- (25) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (26) 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_foldr(vz4, :(vz30, vz31)) -> new_foldr(vz4, vz31) The graph contains the following edges 1 >= 1, 2 > 2 ---------------------------------------- (27) YES