8.37/3.59 YES 10.20/4.11 proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs 10.20/4.11 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.20/4.11 10.20/4.11 10.20/4.11 H-Termination with start terms of the given HASKELL could be proven: 10.20/4.11 10.20/4.11 (0) HASKELL 10.20/4.11 (1) LR [EQUIVALENT, 0 ms] 10.20/4.11 (2) HASKELL 10.20/4.11 (3) BR [EQUIVALENT, 0 ms] 10.20/4.11 (4) HASKELL 10.20/4.11 (5) COR [EQUIVALENT, 0 ms] 10.20/4.11 (6) HASKELL 10.20/4.11 (7) Narrow [SOUND, 0 ms] 10.20/4.11 (8) AND 10.20/4.11 (9) QDP 10.20/4.11 (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.20/4.11 (11) YES 10.20/4.11 (12) QDP 10.20/4.11 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.20/4.11 (14) YES 10.20/4.11 (15) QDP 10.20/4.11 (16) DependencyGraphProof [EQUIVALENT, 0 ms] 10.20/4.11 (17) AND 10.20/4.11 (18) QDP 10.20/4.11 (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.20/4.11 (20) YES 10.20/4.11 (21) QDP 10.20/4.11 (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.20/4.11 (23) YES 10.20/4.11 (24) QDP 10.20/4.11 (25) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.20/4.11 (26) YES 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (0) 10.20/4.11 Obligation: 10.20/4.11 mainModule Main 10.20/4.11 module Main where { 10.20/4.11 import qualified Prelude; 10.20/4.11 } 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (1) LR (EQUIVALENT) 10.20/4.11 Lambda Reductions: 10.20/4.11 The following Lambda expression 10.20/4.11 "\_->q" 10.20/4.11 is transformed to 10.20/4.11 "gtGt0 q _ = q; 10.20/4.11 " 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (2) 10.20/4.11 Obligation: 10.20/4.11 mainModule Main 10.20/4.11 module Main where { 10.20/4.11 import qualified Prelude; 10.20/4.11 } 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (3) BR (EQUIVALENT) 10.20/4.11 Replaced joker patterns by fresh variables and removed binding patterns. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (4) 10.20/4.11 Obligation: 10.20/4.11 mainModule Main 10.20/4.11 module Main where { 10.20/4.11 import qualified Prelude; 10.20/4.11 } 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (5) COR (EQUIVALENT) 10.20/4.11 Cond Reductions: 10.20/4.11 The following Function with conditions 10.20/4.11 "undefined |Falseundefined; 10.20/4.11 " 10.20/4.11 is transformed to 10.20/4.11 "undefined = undefined1; 10.20/4.11 " 10.20/4.11 "undefined0 True = undefined; 10.20/4.11 " 10.20/4.11 "undefined1 = undefined0 False; 10.20/4.11 " 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (6) 10.20/4.11 Obligation: 10.20/4.11 mainModule Main 10.20/4.11 module Main where { 10.20/4.11 import qualified Prelude; 10.20/4.11 } 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (7) Narrow (SOUND) 10.20/4.11 Haskell To QDPs 10.20/4.11 10.20/4.11 digraph dp_graph { 10.20/4.11 node [outthreshold=100, inthreshold=100];1[label="mapM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.20/4.11 3[label="mapM_ vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 10.20/4.11 4[label="mapM_ vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 10.20/4.11 5[label="sequence_ . map vy3",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 10.20/4.11 6[label="sequence_ (map vy3 vy4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 10.20/4.11 7[label="foldr (>>) (return ()) (map vy3 vy4)",fontsize=16,color="burlywood",shape="triangle"];79[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];7 -> 79[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 79 -> 8[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 80[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 80[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 80 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 8[label="foldr (>>) (return ()) (map vy3 (vy40 : vy41))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 10.20/4.11 9[label="foldr (>>) (return ()) (map vy3 [])",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10.20/4.11 10[label="foldr (>>) (return ()) (vy3 vy40 : map vy3 vy41)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 10.20/4.11 11[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 10.20/4.11 12[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="blue",shape="box"];81[label=">> :: (Maybe a) -> (Maybe ()) -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 81[label="",style="solid", color="blue", weight=9]; 10.20/4.11 81 -> 26[label="",style="solid", color="blue", weight=3]; 10.20/4.11 82[label=">> :: ([] a) -> ([] ()) -> [] ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="blue", weight=9]; 10.20/4.11 82 -> 27[label="",style="solid", color="blue", weight=3]; 10.20/4.11 83[label=">> :: (IO a) -> (IO ()) -> IO ()",fontsize=10,color="white",style="solid",shape="box"];12 -> 83[label="",style="solid", color="blue", weight=9]; 10.20/4.11 83 -> 28[label="",style="solid", color="blue", weight=3]; 10.20/4.11 13[label="return ()",fontsize=16,color="blue",shape="box"];84[label="return :: () -> Maybe ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 84[label="",style="solid", color="blue", weight=9]; 10.20/4.11 84 -> 16[label="",style="solid", color="blue", weight=3]; 10.20/4.11 85[label="return :: () -> [] ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 85[label="",style="solid", color="blue", weight=9]; 10.20/4.11 85 -> 17[label="",style="solid", color="blue", weight=3]; 10.20/4.11 86[label="return :: () -> IO ()",fontsize=10,color="white",style="solid",shape="box"];13 -> 86[label="",style="solid", color="blue", weight=9]; 10.20/4.11 86 -> 18[label="",style="solid", color="blue", weight=3]; 10.20/4.11 26 -> 20[label="",style="dashed", color="red", weight=0]; 10.20/4.11 26[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];26 -> 33[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 27 -> 21[label="",style="dashed", color="red", weight=0]; 10.20/4.11 27[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];27 -> 34[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 28 -> 22[label="",style="dashed", color="red", weight=0]; 10.20/4.11 28[label="(>>) vy3 vy40 foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];28 -> 35[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 16[label="return ()",fontsize=16,color="black",shape="box"];16 -> 23[label="",style="solid", color="black", weight=3]; 10.20/4.11 17[label="return ()",fontsize=16,color="black",shape="box"];17 -> 24[label="",style="solid", color="black", weight=3]; 10.20/4.11 18[label="return ()",fontsize=16,color="black",shape="box"];18 -> 25[label="",style="solid", color="black", weight=3]; 10.20/4.11 33 -> 7[label="",style="dashed", color="red", weight=0]; 10.20/4.11 33[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];33 -> 38[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 20[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];20 -> 29[label="",style="solid", color="black", weight=3]; 10.20/4.11 34 -> 7[label="",style="dashed", color="red", weight=0]; 10.20/4.11 34[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];34 -> 39[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 21[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];21 -> 30[label="",style="solid", color="black", weight=3]; 10.20/4.11 35 -> 7[label="",style="dashed", color="red", weight=0]; 10.20/4.11 35[label="foldr (>>) (return ()) (map vy3 vy41)",fontsize=16,color="magenta"];35 -> 40[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 22[label="(>>) vy3 vy40 vy5",fontsize=16,color="black",shape="triangle"];22 -> 31[label="",style="solid", color="black", weight=3]; 10.20/4.11 23[label="Just ()",fontsize=16,color="green",shape="box"];24[label="() : []",fontsize=16,color="green",shape="box"];25[label="primretIO ()",fontsize=16,color="black",shape="box"];25 -> 32[label="",style="solid", color="black", weight=3]; 10.20/4.11 38[label="vy41",fontsize=16,color="green",shape="box"];29 -> 36[label="",style="dashed", color="red", weight=0]; 10.20/4.11 29[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];29 -> 37[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 39[label="vy41",fontsize=16,color="green",shape="box"];30 -> 41[label="",style="dashed", color="red", weight=0]; 10.20/4.11 30[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="magenta"];30 -> 42[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 40[label="vy41",fontsize=16,color="green",shape="box"];31[label="vy3 vy40 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];31 -> 43[label="",style="solid", color="black", weight=3]; 10.20/4.11 32[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];37[label="vy3 vy40",fontsize=16,color="green",shape="box"];37 -> 44[label="",style="dashed", color="green", weight=3]; 10.20/4.11 36[label="vy6 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];87[label="vy6/Nothing",fontsize=10,color="white",style="solid",shape="box"];36 -> 87[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 87 -> 45[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 88[label="vy6/Just vy60",fontsize=10,color="white",style="solid",shape="box"];36 -> 88[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 88 -> 46[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 42[label="vy3 vy40",fontsize=16,color="green",shape="box"];42 -> 50[label="",style="dashed", color="green", weight=3]; 10.20/4.11 41[label="vy7 >>= gtGt0 vy5",fontsize=16,color="burlywood",shape="triangle"];89[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];41 -> 89[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 89 -> 48[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 90[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 90[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 90 -> 49[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 43 -> 51[label="",style="dashed", color="red", weight=0]; 10.20/4.11 43[label="primbindIO (vy3 vy40) (gtGt0 vy5)",fontsize=16,color="magenta"];43 -> 52[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 44[label="vy40",fontsize=16,color="green",shape="box"];45[label="Nothing >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3]; 10.20/4.11 46[label="Just vy60 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];46 -> 54[label="",style="solid", color="black", weight=3]; 10.20/4.11 50[label="vy40",fontsize=16,color="green",shape="box"];48[label="vy70 : vy71 >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];48 -> 55[label="",style="solid", color="black", weight=3]; 10.20/4.11 49[label="[] >>= gtGt0 vy5",fontsize=16,color="black",shape="box"];49 -> 56[label="",style="solid", color="black", weight=3]; 10.20/4.11 52[label="vy3 vy40",fontsize=16,color="green",shape="box"];52 -> 62[label="",style="dashed", color="green", weight=3]; 10.20/4.11 51[label="primbindIO vy8 (gtGt0 vy5)",fontsize=16,color="burlywood",shape="triangle"];91[label="vy8/IO vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 91[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 91 -> 58[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 92[label="vy8/AProVE_IO vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 92[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 92 -> 59[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 93[label="vy8/AProVE_Exception vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 93[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 93 -> 60[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 94[label="vy8/AProVE_Error vy80",fontsize=10,color="white",style="solid",shape="box"];51 -> 94[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 94 -> 61[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 53[label="Nothing",fontsize=16,color="green",shape="box"];54[label="gtGt0 vy5 vy60",fontsize=16,color="black",shape="box"];54 -> 63[label="",style="solid", color="black", weight=3]; 10.20/4.11 55 -> 64[label="",style="dashed", color="red", weight=0]; 10.20/4.11 55[label="gtGt0 vy5 vy70 ++ (vy71 >>= gtGt0 vy5)",fontsize=16,color="magenta"];55 -> 65[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 56[label="[]",fontsize=16,color="green",shape="box"];62[label="vy40",fontsize=16,color="green",shape="box"];58[label="primbindIO (IO vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];58 -> 66[label="",style="solid", color="black", weight=3]; 10.20/4.11 59[label="primbindIO (AProVE_IO vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];59 -> 67[label="",style="solid", color="black", weight=3]; 10.20/4.11 60[label="primbindIO (AProVE_Exception vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];60 -> 68[label="",style="solid", color="black", weight=3]; 10.20/4.11 61[label="primbindIO (AProVE_Error vy80) (gtGt0 vy5)",fontsize=16,color="black",shape="box"];61 -> 69[label="",style="solid", color="black", weight=3]; 10.20/4.11 63[label="vy5",fontsize=16,color="green",shape="box"];65 -> 41[label="",style="dashed", color="red", weight=0]; 10.20/4.11 65[label="vy71 >>= gtGt0 vy5",fontsize=16,color="magenta"];65 -> 70[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 64[label="gtGt0 vy5 vy70 ++ vy9",fontsize=16,color="black",shape="triangle"];64 -> 71[label="",style="solid", color="black", weight=3]; 10.20/4.11 66[label="error []",fontsize=16,color="red",shape="box"];67[label="gtGt0 vy5 vy80",fontsize=16,color="black",shape="box"];67 -> 72[label="",style="solid", color="black", weight=3]; 10.20/4.11 68[label="AProVE_Exception vy80",fontsize=16,color="green",shape="box"];69[label="AProVE_Error vy80",fontsize=16,color="green",shape="box"];70[label="vy71",fontsize=16,color="green",shape="box"];71[label="vy5 ++ vy9",fontsize=16,color="burlywood",shape="triangle"];95[label="vy5/vy50 : vy51",fontsize=10,color="white",style="solid",shape="box"];71 -> 95[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 95 -> 73[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 96[label="vy5/[]",fontsize=10,color="white",style="solid",shape="box"];71 -> 96[label="",style="solid", color="burlywood", weight=9]; 10.20/4.11 96 -> 74[label="",style="solid", color="burlywood", weight=3]; 10.20/4.11 72[label="vy5",fontsize=16,color="green",shape="box"];73[label="(vy50 : vy51) ++ vy9",fontsize=16,color="black",shape="box"];73 -> 75[label="",style="solid", color="black", weight=3]; 10.20/4.11 74[label="[] ++ vy9",fontsize=16,color="black",shape="box"];74 -> 76[label="",style="solid", color="black", weight=3]; 10.20/4.11 75[label="vy50 : vy51 ++ vy9",fontsize=16,color="green",shape="box"];75 -> 77[label="",style="dashed", color="green", weight=3]; 10.20/4.11 76[label="vy9",fontsize=16,color="green",shape="box"];77 -> 71[label="",style="dashed", color="red", weight=0]; 10.20/4.11 77[label="vy51 ++ vy9",fontsize=16,color="magenta"];77 -> 78[label="",style="dashed", color="magenta", weight=3]; 10.20/4.11 78[label="vy51",fontsize=16,color="green",shape="box"];} 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (8) 10.20/4.11 Complex Obligation (AND) 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (9) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_gtGtEs(:(vy70, vy71), vy5, h) -> new_gtGtEs(vy71, vy5, h) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (10) QDPSizeChangeProof (EQUIVALENT) 10.20/4.11 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. 10.20/4.11 10.20/4.11 From the DPs we obtained the following set of size-change graphs: 10.20/4.11 *new_gtGtEs(:(vy70, vy71), vy5, h) -> new_gtGtEs(vy71, vy5, h) 10.20/4.11 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (11) 10.20/4.11 YES 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (12) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_psPs(:(vy50, vy51), vy9) -> new_psPs(vy51, vy9) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (13) QDPSizeChangeProof (EQUIVALENT) 10.20/4.11 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. 10.20/4.11 10.20/4.11 From the DPs we obtained the following set of size-change graphs: 10.20/4.11 *new_psPs(:(vy50, vy51), vy9) -> new_psPs(vy51, vy9) 10.20/4.11 The graph contains the following edges 1 > 1, 2 >= 2 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (14) 10.20/4.11 YES 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (15) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba) 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba) 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (16) DependencyGraphProof (EQUIVALENT) 10.20/4.11 The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (17) 10.20/4.11 Complex Obligation (AND) 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (18) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (19) QDPSizeChangeProof (EQUIVALENT) 10.20/4.11 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. 10.20/4.11 10.20/4.11 From the DPs we obtained the following set of size-change graphs: 10.20/4.11 *new_foldr(vy3, :(vy40, vy41), ty_IO, h, ba) -> new_foldr(vy3, vy41, ty_IO, h, ba) 10.20/4.11 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (20) 10.20/4.11 YES 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (21) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (22) QDPSizeChangeProof (EQUIVALENT) 10.20/4.11 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. 10.20/4.11 10.20/4.11 From the DPs we obtained the following set of size-change graphs: 10.20/4.11 *new_foldr(vy3, :(vy40, vy41), ty_Maybe, h, ba) -> new_foldr(vy3, vy41, ty_Maybe, h, ba) 10.20/4.11 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (23) 10.20/4.11 YES 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (24) 10.20/4.11 Obligation: 10.20/4.11 Q DP problem: 10.20/4.11 The TRS P consists of the following rules: 10.20/4.11 10.20/4.11 new_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba) 10.20/4.11 10.20/4.11 R is empty. 10.20/4.11 Q is empty. 10.20/4.11 We have to consider all minimal (P,Q,R)-chains. 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (25) QDPSizeChangeProof (EQUIVALENT) 10.20/4.11 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. 10.20/4.11 10.20/4.11 From the DPs we obtained the following set of size-change graphs: 10.20/4.11 *new_foldr(vy3, :(vy40, vy41), ty_[], h, ba) -> new_foldr(vy3, vy41, ty_[], h, ba) 10.20/4.11 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4, 5 >= 5 10.20/4.11 10.20/4.11 10.20/4.11 ---------------------------------------- 10.20/4.11 10.20/4.11 (26) 10.20/4.11 YES 10.20/4.15 EOF