9.37/4.12 MAYBE 11.32/4.65 proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs 11.32/4.65 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.32/4.65 11.32/4.65 11.32/4.65 H-Termination with start terms of the given HASKELL could not be shown: 11.32/4.65 11.32/4.65 (0) HASKELL 11.32/4.65 (1) LR [EQUIVALENT, 0 ms] 11.32/4.65 (2) HASKELL 11.32/4.65 (3) BR [EQUIVALENT, 0 ms] 11.32/4.65 (4) HASKELL 11.32/4.65 (5) COR [EQUIVALENT, 0 ms] 11.32/4.65 (6) HASKELL 11.32/4.65 (7) Narrow [SOUND, 0 ms] 11.32/4.65 (8) AND 11.32/4.65 (9) QDP 11.32/4.65 (10) DependencyGraphProof [EQUIVALENT, 0 ms] 11.32/4.65 (11) AND 11.32/4.65 (12) QDP 11.32/4.65 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.32/4.65 (14) YES 11.32/4.65 (15) QDP 11.32/4.65 (16) MRRProof [EQUIVALENT, 0 ms] 11.32/4.65 (17) QDP 11.32/4.65 (18) DependencyGraphProof [EQUIVALENT, 0 ms] 11.32/4.65 (19) QDP 11.32/4.65 (20) NonTerminationLoopProof [COMPLETE, 2 ms] 11.32/4.65 (21) NO 11.32/4.65 (22) QDP 11.32/4.65 (23) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.32/4.65 (24) YES 11.32/4.65 (25) QDP 11.32/4.65 (26) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.32/4.65 (27) YES 11.32/4.65 (28) Narrow [COMPLETE, 0 ms] 11.32/4.65 (29) QDP 11.32/4.65 (30) DependencyGraphProof [EQUIVALENT, 0 ms] 11.32/4.65 (31) TRUE 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (0) 11.32/4.65 Obligation: 11.32/4.65 mainModule Main 11.32/4.65 module Maybe where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Main where { 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Monad where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Prelude; 11.32/4.65 foldM :: Monad c => (b -> a -> c b) -> b -> [a] -> c b; 11.32/4.65 foldM _ a [] = return a; 11.32/4.65 foldM f a (x : xs) = f a x >>= (\fax ->foldM f fax xs); 11.32/4.65 11.32/4.65 } 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (1) LR (EQUIVALENT) 11.32/4.65 Lambda Reductions: 11.32/4.65 The following Lambda expression 11.32/4.65 "\fax->foldM f fax xs" 11.32/4.65 is transformed to 11.32/4.65 "foldM0 f xs fax = foldM f fax xs; 11.32/4.65 " 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (2) 11.32/4.65 Obligation: 11.32/4.65 mainModule Main 11.32/4.65 module Maybe where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Main where { 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Monad where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Prelude; 11.32/4.65 foldM :: Monad c => (a -> b -> c a) -> a -> [b] -> c a; 11.32/4.65 foldM _ a [] = return a; 11.32/4.65 foldM f a (x : xs) = f a x >>= foldM0 f xs; 11.32/4.65 11.32/4.65 foldM0 f xs fax = foldM f fax xs; 11.32/4.65 11.32/4.65 } 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (3) BR (EQUIVALENT) 11.32/4.65 Replaced joker patterns by fresh variables and removed binding patterns. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (4) 11.32/4.65 Obligation: 11.32/4.65 mainModule Main 11.32/4.65 module Maybe where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Main where { 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Monad where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Prelude; 11.32/4.65 foldM :: Monad c => (b -> a -> c b) -> b -> [a] -> c b; 11.32/4.65 foldM vy a [] = return a; 11.32/4.65 foldM f a (x : xs) = f a x >>= foldM0 f xs; 11.32/4.65 11.32/4.65 foldM0 f xs fax = foldM f fax xs; 11.32/4.65 11.32/4.65 } 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (5) COR (EQUIVALENT) 11.32/4.65 Cond Reductions: 11.32/4.65 The following Function with conditions 11.32/4.65 "undefined |Falseundefined; 11.32/4.65 " 11.32/4.65 is transformed to 11.32/4.65 "undefined = undefined1; 11.32/4.65 " 11.32/4.65 "undefined0 True = undefined; 11.32/4.65 " 11.32/4.65 "undefined1 = undefined0 False; 11.32/4.65 " 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (6) 11.32/4.65 Obligation: 11.32/4.65 mainModule Main 11.32/4.65 module Maybe where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Main where { 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Monad; 11.32/4.65 import qualified Prelude; 11.32/4.65 } 11.32/4.65 module Monad where { 11.32/4.65 import qualified Main; 11.32/4.65 import qualified Maybe; 11.32/4.65 import qualified Prelude; 11.32/4.65 foldM :: Monad c => (a -> b -> c a) -> a -> [b] -> c a; 11.32/4.65 foldM vy a [] = return a; 11.32/4.65 foldM f a (x : xs) = f a x >>= foldM0 f xs; 11.32/4.65 11.32/4.65 foldM0 f xs fax = foldM f fax xs; 11.32/4.65 11.32/4.65 } 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (7) Narrow (SOUND) 11.32/4.65 Haskell To QDPs 11.32/4.65 11.32/4.65 digraph dp_graph { 11.32/4.65 node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];76[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 76[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 76 -> 6[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 77[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 77[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 77 -> 7[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 11.32/4.65 7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 11.32/4.65 8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="blue",shape="box"];78[label=">>= :: (Maybe a) -> (a -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 78[label="",style="solid", color="blue", weight=9]; 11.32/4.65 78 -> 10[label="",style="solid", color="blue", weight=3]; 11.32/4.65 79[label=">>= :: (IO a) -> (a -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];8 -> 79[label="",style="solid", color="blue", weight=9]; 11.32/4.65 79 -> 11[label="",style="solid", color="blue", weight=3]; 11.32/4.65 80[label=">>= :: ([] a) -> (a -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 80[label="",style="solid", color="blue", weight=9]; 11.32/4.65 80 -> 12[label="",style="solid", color="blue", weight=3]; 11.32/4.65 9[label="return vz4",fontsize=16,color="blue",shape="box"];81[label="return :: a -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];9 -> 81[label="",style="solid", color="blue", weight=9]; 11.32/4.65 81 -> 13[label="",style="solid", color="blue", weight=3]; 11.32/4.65 82[label="return :: a -> IO a",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="blue", weight=9]; 11.32/4.65 82 -> 14[label="",style="solid", color="blue", weight=3]; 11.32/4.65 83[label="return :: a -> [] a",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="blue", weight=9]; 11.32/4.65 83 -> 15[label="",style="solid", color="blue", weight=3]; 11.32/4.65 10 -> 16[label="",style="dashed", color="red", weight=0]; 11.32/4.65 10[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];10 -> 17[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 11[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];11 -> 18[label="",style="solid", color="black", weight=3]; 11.32/4.65 12 -> 19[label="",style="dashed", color="red", weight=0]; 11.32/4.65 12[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];12 -> 20[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 13[label="return vz4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 11.32/4.65 14[label="return vz4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 11.32/4.65 15[label="return vz4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 11.32/4.65 17[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];17 -> 24[label="",style="dashed", color="green", weight=3]; 11.32/4.65 17 -> 25[label="",style="dashed", color="green", weight=3]; 11.32/4.65 16[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];84[label="vz6/Nothing",fontsize=10,color="white",style="solid",shape="box"];16 -> 84[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 84 -> 26[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 85[label="vz6/Just vz60",fontsize=10,color="white",style="solid",shape="box"];16 -> 85[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 85 -> 27[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 18 -> 28[label="",style="dashed", color="red", weight=0]; 11.32/4.65 18[label="primbindIO (vz3 vz4 vz50) (Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];18 -> 29[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 20[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];20 -> 30[label="",style="dashed", color="green", weight=3]; 11.32/4.65 20 -> 31[label="",style="dashed", color="green", weight=3]; 11.32/4.65 19[label="vz7 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];86[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 86 -> 32[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 87[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 87 -> 33[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 21[label="Just vz4",fontsize=16,color="green",shape="box"];22[label="primretIO vz4",fontsize=16,color="black",shape="box"];22 -> 34[label="",style="solid", color="black", weight=3]; 11.32/4.65 23[label="vz4 : []",fontsize=16,color="green",shape="box"];24[label="vz4",fontsize=16,color="green",shape="box"];25[label="vz50",fontsize=16,color="green",shape="box"];26[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3]; 11.32/4.65 27[label="Just vz60 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3]; 11.32/4.65 29[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];29 -> 43[label="",style="dashed", color="green", weight=3]; 11.32/4.65 29 -> 44[label="",style="dashed", color="green", weight=3]; 11.32/4.65 28[label="primbindIO vz8 (Monad.foldM0 vz3 vz51)",fontsize=16,color="burlywood",shape="triangle"];88[label="vz8/IO vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 88[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 88 -> 39[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 89[label="vz8/AProVE_IO vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 89[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 89 -> 40[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 90[label="vz8/AProVE_Exception vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 90[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 90 -> 41[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 91[label="vz8/AProVE_Error vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 91[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 91 -> 42[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 30[label="vz4",fontsize=16,color="green",shape="box"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="vz70 : vz71 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 11.32/4.65 33[label="[] >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3]; 11.32/4.65 34[label="AProVE_IO vz4",fontsize=16,color="green",shape="box"];35[label="Nothing",fontsize=16,color="green",shape="box"];36[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];36 -> 47[label="",style="solid", color="black", weight=3]; 11.32/4.65 43[label="vz4",fontsize=16,color="green",shape="box"];44[label="vz50",fontsize=16,color="green",shape="box"];39[label="primbindIO (IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3]; 11.32/4.65 40[label="primbindIO (AProVE_IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3]; 11.32/4.65 41[label="primbindIO (AProVE_Exception vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3]; 11.32/4.65 42[label="primbindIO (AProVE_Error vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];42 -> 51[label="",style="solid", color="black", weight=3]; 11.32/4.65 45 -> 61[label="",style="dashed", color="red", weight=0]; 11.32/4.65 45[label="Monad.foldM0 vz3 vz51 vz70 ++ (vz71 >>= Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];45 -> 62[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 45 -> 63[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 46[label="[]",fontsize=16,color="green",shape="box"];47 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 47[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];47 -> 54[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 47 -> 55[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 48[label="error []",fontsize=16,color="red",shape="box"];49[label="Monad.foldM0 vz3 vz51 vz80",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3]; 11.32/4.65 50[label="AProVE_Exception vz80",fontsize=16,color="green",shape="box"];51[label="AProVE_Error vz80",fontsize=16,color="green",shape="box"];62 -> 19[label="",style="dashed", color="red", weight=0]; 11.32/4.65 62[label="vz71 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];62 -> 66[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 63[label="Monad.foldM0 vz3 vz51 vz70",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3]; 11.32/4.65 61[label="vz10 ++ vz9",fontsize=16,color="burlywood",shape="triangle"];92[label="vz10/vz100 : vz101",fontsize=10,color="white",style="solid",shape="box"];61 -> 92[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 92 -> 68[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 93[label="vz10/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 93[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 93 -> 69[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 54[label="vz51",fontsize=16,color="green",shape="box"];55[label="vz60",fontsize=16,color="green",shape="box"];56 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 56[label="Monad.foldM vz3 vz80 vz51",fontsize=16,color="magenta"];56 -> 59[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 56 -> 60[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 66[label="vz71",fontsize=16,color="green",shape="box"];67 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 67[label="Monad.foldM vz3 vz70 vz51",fontsize=16,color="magenta"];67 -> 70[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 67 -> 71[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 68[label="(vz100 : vz101) ++ vz9",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3]; 11.32/4.65 69[label="[] ++ vz9",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3]; 11.32/4.65 59[label="vz51",fontsize=16,color="green",shape="box"];60[label="vz80",fontsize=16,color="green",shape="box"];70[label="vz51",fontsize=16,color="green",shape="box"];71[label="vz70",fontsize=16,color="green",shape="box"];72[label="vz100 : vz101 ++ vz9",fontsize=16,color="green",shape="box"];72 -> 74[label="",style="dashed", color="green", weight=3]; 11.32/4.65 73[label="vz9",fontsize=16,color="green",shape="box"];74 -> 61[label="",style="dashed", color="red", weight=0]; 11.32/4.65 74[label="vz101 ++ vz9",fontsize=16,color="magenta"];74 -> 75[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 75[label="vz101",fontsize=16,color="green",shape="box"];} 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (8) 11.32/4.65 Complex Obligation (AND) 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (9) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs(vz3, vz51, h, ba) 11.32/4.65 new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba) 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba) 11.32/4.65 new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba) 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (10) DependencyGraphProof (EQUIVALENT) 11.32/4.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (11) 11.32/4.65 Complex Obligation (AND) 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (12) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba) 11.32/4.65 new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (13) QDPSizeChangeProof (EQUIVALENT) 11.32/4.65 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. 11.32/4.65 11.32/4.65 From the DPs we obtained the following set of size-change graphs: 11.32/4.65 *new_primbindIO(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_IO, h, ba) 11.32/4.65 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5 11.32/4.65 11.32/4.65 11.32/4.65 *new_foldM(vz3, :(vz50, vz51), ty_IO, h, ba) -> new_primbindIO(vz3, vz51, h, ba) 11.32/4.65 The graph contains the following edges 1 >= 1, 2 > 2, 4 >= 3, 5 >= 4 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (14) 11.32/4.65 YES 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (15) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba) 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (16) MRRProof (EQUIVALENT) 11.32/4.65 By using the rule removal processor [LPAR04] with the following ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented. 11.32/4.65 11.32/4.65 Strictly oriented dependency pairs: 11.32/4.65 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_[], h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 11.32/4.65 Used ordering: Polynomial interpretation [POLO]: 11.32/4.65 11.32/4.65 POL(:(x_1, x_2)) = 1 + x_1 + x_2 11.32/4.65 POL(new_foldM(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + 2*x_3 + x_4 + x_5 11.32/4.65 POL(new_gtGtEs0(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4 11.32/4.65 POL(ty_[]) = 0 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (17) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_[], h, ba) 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (18) DependencyGraphProof (EQUIVALENT) 11.32/4.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (19) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_gtGtEs0(vz3, vz51, h, ba) -> new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (20) NonTerminationLoopProof (COMPLETE) 11.32/4.65 We used the non-termination processor [FROCOS05] to show that the DP problem is infinite. 11.32/4.65 Found a loop by semiunifying a rule from P directly. 11.32/4.65 11.32/4.65 s = new_gtGtEs0(vz3, vz51, h, ba) evaluates to t =new_gtGtEs0(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 Thus s starts an infinite chain as s semiunifies with t with the following substitutions: 11.32/4.65 * Matcher: [ ] 11.32/4.65 * Semiunifier: [ ] 11.32/4.65 11.32/4.65 -------------------------------------------------------------------------------- 11.32/4.65 Rewriting sequence 11.32/4.65 11.32/4.65 The DP semiunifies directly so there is only one rewrite step from new_gtGtEs0(vz3, vz51, h, ba) to new_gtGtEs0(vz3, vz51, h, ba). 11.32/4.65 11.32/4.65 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (21) 11.32/4.65 NO 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (22) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba) 11.32/4.65 new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs(vz3, vz51, h, ba) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (23) QDPSizeChangeProof (EQUIVALENT) 11.32/4.65 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. 11.32/4.65 11.32/4.65 From the DPs we obtained the following set of size-change graphs: 11.32/4.65 *new_foldM(vz3, :(vz50, vz51), ty_Maybe, h, ba) -> new_gtGtEs(vz3, vz51, h, ba) 11.32/4.65 The graph contains the following edges 1 >= 1, 2 > 2, 4 >= 3, 5 >= 4 11.32/4.65 11.32/4.65 11.32/4.65 *new_gtGtEs(vz3, vz51, h, ba) -> new_foldM(vz3, vz51, ty_Maybe, h, ba) 11.32/4.65 The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 4, 4 >= 5 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (24) 11.32/4.65 YES 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (25) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_psPs(:(vz100, vz101), vz9, h) -> new_psPs(vz101, vz9, h) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all minimal (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (26) QDPSizeChangeProof (EQUIVALENT) 11.32/4.65 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. 11.32/4.65 11.32/4.65 From the DPs we obtained the following set of size-change graphs: 11.32/4.65 *new_psPs(:(vz100, vz101), vz9, h) -> new_psPs(vz101, vz9, h) 11.32/4.65 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 11.32/4.65 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (27) 11.32/4.65 YES 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (28) Narrow (COMPLETE) 11.32/4.65 Haskell To QDPs 11.32/4.65 11.32/4.65 digraph dp_graph { 11.32/4.65 node [outthreshold=100, inthreshold=100];1[label="Monad.foldM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 3[label="Monad.foldM vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 4[label="Monad.foldM vz3 vz4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 11.32/4.65 5[label="Monad.foldM vz3 vz4 vz5",fontsize=16,color="burlywood",shape="triangle"];76[label="vz5/vz50 : vz51",fontsize=10,color="white",style="solid",shape="box"];5 -> 76[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 76 -> 6[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 77[label="vz5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 77[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 77 -> 7[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 6[label="Monad.foldM vz3 vz4 (vz50 : vz51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 11.32/4.65 7[label="Monad.foldM vz3 vz4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 11.32/4.65 8[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="blue",shape="box"];78[label=">>= :: (Maybe a) -> (a -> Maybe a) -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];8 -> 78[label="",style="solid", color="blue", weight=9]; 11.32/4.65 78 -> 10[label="",style="solid", color="blue", weight=3]; 11.32/4.65 79[label=">>= :: (IO a) -> (a -> IO a) -> IO a",fontsize=10,color="white",style="solid",shape="box"];8 -> 79[label="",style="solid", color="blue", weight=9]; 11.32/4.65 79 -> 11[label="",style="solid", color="blue", weight=3]; 11.32/4.65 80[label=">>= :: ([] a) -> (a -> [] a) -> [] a",fontsize=10,color="white",style="solid",shape="box"];8 -> 80[label="",style="solid", color="blue", weight=9]; 11.32/4.65 80 -> 12[label="",style="solid", color="blue", weight=3]; 11.32/4.65 9[label="return vz4",fontsize=16,color="blue",shape="box"];81[label="return :: a -> Maybe a",fontsize=10,color="white",style="solid",shape="box"];9 -> 81[label="",style="solid", color="blue", weight=9]; 11.32/4.65 81 -> 13[label="",style="solid", color="blue", weight=3]; 11.32/4.65 82[label="return :: a -> IO a",fontsize=10,color="white",style="solid",shape="box"];9 -> 82[label="",style="solid", color="blue", weight=9]; 11.32/4.65 82 -> 14[label="",style="solid", color="blue", weight=3]; 11.32/4.65 83[label="return :: a -> [] a",fontsize=10,color="white",style="solid",shape="box"];9 -> 83[label="",style="solid", color="blue", weight=9]; 11.32/4.65 83 -> 15[label="",style="solid", color="blue", weight=3]; 11.32/4.65 10 -> 16[label="",style="dashed", color="red", weight=0]; 11.32/4.65 10[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];10 -> 17[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 11[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];11 -> 18[label="",style="solid", color="black", weight=3]; 11.32/4.65 12 -> 19[label="",style="dashed", color="red", weight=0]; 11.32/4.65 12[label="vz3 vz4 vz50 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];12 -> 20[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 13[label="return vz4",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3]; 11.32/4.65 14[label="return vz4",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3]; 11.32/4.65 15[label="return vz4",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3]; 11.32/4.65 17[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];17 -> 24[label="",style="dashed", color="green", weight=3]; 11.32/4.65 17 -> 25[label="",style="dashed", color="green", weight=3]; 11.32/4.65 16[label="vz6 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];84[label="vz6/Nothing",fontsize=10,color="white",style="solid",shape="box"];16 -> 84[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 84 -> 26[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 85[label="vz6/Just vz60",fontsize=10,color="white",style="solid",shape="box"];16 -> 85[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 85 -> 27[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 18 -> 28[label="",style="dashed", color="red", weight=0]; 11.32/4.65 18[label="primbindIO (vz3 vz4 vz50) (Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];18 -> 29[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 20[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];20 -> 30[label="",style="dashed", color="green", weight=3]; 11.32/4.65 20 -> 31[label="",style="dashed", color="green", weight=3]; 11.32/4.65 19[label="vz7 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="burlywood",shape="triangle"];86[label="vz7/vz70 : vz71",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 86 -> 32[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 87[label="vz7/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 87[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 87 -> 33[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 21[label="Just vz4",fontsize=16,color="green",shape="box"];22[label="primretIO vz4",fontsize=16,color="black",shape="box"];22 -> 34[label="",style="solid", color="black", weight=3]; 11.32/4.65 23[label="vz4 : []",fontsize=16,color="green",shape="box"];24[label="vz4",fontsize=16,color="green",shape="box"];25[label="vz50",fontsize=16,color="green",shape="box"];26[label="Nothing >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3]; 11.32/4.65 27[label="Just vz60 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3]; 11.32/4.65 29[label="vz3 vz4 vz50",fontsize=16,color="green",shape="box"];29 -> 43[label="",style="dashed", color="green", weight=3]; 11.32/4.65 29 -> 44[label="",style="dashed", color="green", weight=3]; 11.32/4.65 28[label="primbindIO vz8 (Monad.foldM0 vz3 vz51)",fontsize=16,color="burlywood",shape="triangle"];88[label="vz8/IO vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 88[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 88 -> 39[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 89[label="vz8/AProVE_IO vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 89[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 89 -> 40[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 90[label="vz8/AProVE_Exception vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 90[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 90 -> 41[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 91[label="vz8/AProVE_Error vz80",fontsize=10,color="white",style="solid",shape="box"];28 -> 91[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 91 -> 42[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 30[label="vz4",fontsize=16,color="green",shape="box"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="vz70 : vz71 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 11.32/4.65 33[label="[] >>= Monad.foldM0 vz3 vz51",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3]; 11.32/4.65 34[label="AProVE_IO vz4",fontsize=16,color="green",shape="box"];35[label="Nothing",fontsize=16,color="green",shape="box"];36[label="Monad.foldM0 vz3 vz51 vz60",fontsize=16,color="black",shape="box"];36 -> 47[label="",style="solid", color="black", weight=3]; 11.32/4.65 43[label="vz4",fontsize=16,color="green",shape="box"];44[label="vz50",fontsize=16,color="green",shape="box"];39[label="primbindIO (IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];39 -> 48[label="",style="solid", color="black", weight=3]; 11.32/4.65 40[label="primbindIO (AProVE_IO vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];40 -> 49[label="",style="solid", color="black", weight=3]; 11.32/4.65 41[label="primbindIO (AProVE_Exception vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];41 -> 50[label="",style="solid", color="black", weight=3]; 11.32/4.65 42[label="primbindIO (AProVE_Error vz80) (Monad.foldM0 vz3 vz51)",fontsize=16,color="black",shape="box"];42 -> 51[label="",style="solid", color="black", weight=3]; 11.32/4.65 45 -> 61[label="",style="dashed", color="red", weight=0]; 11.32/4.65 45[label="Monad.foldM0 vz3 vz51 vz70 ++ (vz71 >>= Monad.foldM0 vz3 vz51)",fontsize=16,color="magenta"];45 -> 62[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 45 -> 63[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 46[label="[]",fontsize=16,color="green",shape="box"];47 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 47[label="Monad.foldM vz3 vz60 vz51",fontsize=16,color="magenta"];47 -> 54[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 47 -> 55[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 48[label="error []",fontsize=16,color="red",shape="box"];49[label="Monad.foldM0 vz3 vz51 vz80",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3]; 11.32/4.65 50[label="AProVE_Exception vz80",fontsize=16,color="green",shape="box"];51[label="AProVE_Error vz80",fontsize=16,color="green",shape="box"];62 -> 19[label="",style="dashed", color="red", weight=0]; 11.32/4.65 62[label="vz71 >>= Monad.foldM0 vz3 vz51",fontsize=16,color="magenta"];62 -> 66[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 63[label="Monad.foldM0 vz3 vz51 vz70",fontsize=16,color="black",shape="box"];63 -> 67[label="",style="solid", color="black", weight=3]; 11.32/4.65 61[label="vz10 ++ vz9",fontsize=16,color="burlywood",shape="triangle"];92[label="vz10/vz100 : vz101",fontsize=10,color="white",style="solid",shape="box"];61 -> 92[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 92 -> 68[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 93[label="vz10/[]",fontsize=10,color="white",style="solid",shape="box"];61 -> 93[label="",style="solid", color="burlywood", weight=9]; 11.32/4.65 93 -> 69[label="",style="solid", color="burlywood", weight=3]; 11.32/4.65 54[label="vz51",fontsize=16,color="green",shape="box"];55[label="vz60",fontsize=16,color="green",shape="box"];56 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 56[label="Monad.foldM vz3 vz80 vz51",fontsize=16,color="magenta"];56 -> 59[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 56 -> 60[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 66[label="vz71",fontsize=16,color="green",shape="box"];67 -> 5[label="",style="dashed", color="red", weight=0]; 11.32/4.65 67[label="Monad.foldM vz3 vz70 vz51",fontsize=16,color="magenta"];67 -> 70[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 67 -> 71[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 68[label="(vz100 : vz101) ++ vz9",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3]; 11.32/4.65 69[label="[] ++ vz9",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3]; 11.32/4.65 59[label="vz51",fontsize=16,color="green",shape="box"];60[label="vz80",fontsize=16,color="green",shape="box"];70[label="vz51",fontsize=16,color="green",shape="box"];71[label="vz70",fontsize=16,color="green",shape="box"];72[label="vz100 : vz101 ++ vz9",fontsize=16,color="green",shape="box"];72 -> 74[label="",style="dashed", color="green", weight=3]; 11.32/4.65 73[label="vz9",fontsize=16,color="green",shape="box"];74 -> 61[label="",style="dashed", color="red", weight=0]; 11.32/4.65 74[label="vz101 ++ vz9",fontsize=16,color="magenta"];74 -> 75[label="",style="dashed", color="magenta", weight=3]; 11.32/4.65 75[label="vz101",fontsize=16,color="green",shape="box"];} 11.32/4.65 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (29) 11.32/4.65 Obligation: 11.32/4.65 Q DP problem: 11.32/4.65 The TRS P consists of the following rules: 11.32/4.65 11.32/4.65 new_primbindIO(AProVE_IO(vz80), vz3, vz51, h, ba, []) -> new_foldM(vz3, vz80, vz51, ty_IO, h, ba, []) 11.32/4.65 new_gtGtEs(Just(vz60), vz3, vz51, h, ba, []) -> new_foldM(vz3, vz60, vz51, ty_Maybe, h, ba, []) 11.32/4.65 11.32/4.65 R is empty. 11.32/4.65 Q is empty. 11.32/4.65 We have to consider all (P,Q,R)-chains. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (30) DependencyGraphProof (EQUIVALENT) 11.32/4.65 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes. 11.32/4.65 ---------------------------------------- 11.32/4.65 11.32/4.65 (31) 11.32/4.65 TRUE 11.46/4.68 EOF