/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) AND (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES ---------------------------------------- (0) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; zipWithM_ :: Monad c => (d -> a -> c b) -> [d] -> [a] -> c (); zipWithM_ f xs ys = sequence_ (zipWith f xs ys); } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\_->q" is transformed to "gtGt0 q _ = q; " ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; zipWithM_ :: Monad c => (d -> b -> c a) -> [d] -> [b] -> c (); zipWithM_ f xs ys = sequence_ (zipWith f xs ys); } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (4) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; zipWithM_ :: Monad c => (b -> a -> c d) -> [b] -> [a] -> c (); zipWithM_ f xs ys = sequence_ (zipWith f xs ys); } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (6) Obligation: mainModule Main module Maybe where { import qualified Main; import qualified Monad; import qualified Prelude; } module Main where { import qualified Maybe; import qualified Monad; import qualified Prelude; } module Monad where { import qualified Main; import qualified Maybe; import qualified Prelude; zipWithM_ :: Monad b => (a -> d -> b c) -> [a] -> [d] -> b (); zipWithM_ f xs ys = sequence_ (zipWith f xs ys); } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="Monad.zipWithM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="Monad.zipWithM_ ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="Monad.zipWithM_ ww3 ww4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 5[label="Monad.zipWithM_ ww3 ww4 ww5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3]; 6[label="sequence_ (zipWith ww3 ww4 ww5)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 7[label="foldr (>>) (return ()) (zipWith ww3 ww4 ww5)",fontsize=16,color="burlywood",shape="triangle"];43[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];7 -> 43[label="",style="solid", color="burlywood", weight=9]; 43 -> 8[label="",style="solid", color="burlywood", weight=3]; 44[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 44[label="",style="solid", color="burlywood", weight=9]; 44 -> 9[label="",style="solid", color="burlywood", weight=3]; 8[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) ww5)",fontsize=16,color="burlywood",shape="box"];45[label="ww5/ww50 : ww51",fontsize=10,color="white",style="solid",shape="box"];8 -> 45[label="",style="solid", color="burlywood", weight=9]; 45 -> 10[label="",style="solid", color="burlywood", weight=3]; 46[label="ww5/[]",fontsize=10,color="white",style="solid",shape="box"];8 -> 46[label="",style="solid", color="burlywood", weight=9]; 46 -> 11[label="",style="solid", color="burlywood", weight=3]; 9[label="foldr (>>) (return ()) (zipWith ww3 [] ww5)",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3]; 10[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) (ww50 : ww51))",fontsize=16,color="black",shape="box"];10 -> 13[label="",style="solid", color="black", weight=3]; 11[label="foldr (>>) (return ()) (zipWith ww3 (ww40 : ww41) [])",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3]; 12[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];12 -> 15[label="",style="solid", color="black", weight=3]; 13[label="foldr (>>) (return ()) (ww3 ww40 ww50 : zipWith ww3 ww41 ww51)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3]; 14 -> 12[label="",style="dashed", color="red", weight=0]; 14[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];15[label="return ()",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3]; 16 -> 18[label="",style="dashed", color="red", weight=0]; 16[label="(>>) ww3 ww40 ww50 foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];16 -> 19[label="",style="dashed", color="magenta", weight=3]; 17[label="() : []",fontsize=16,color="green",shape="box"];19 -> 7[label="",style="dashed", color="red", weight=0]; 19[label="foldr (>>) (return ()) (zipWith ww3 ww41 ww51)",fontsize=16,color="magenta"];19 -> 20[label="",style="dashed", color="magenta", weight=3]; 19 -> 21[label="",style="dashed", color="magenta", weight=3]; 18[label="(>>) ww3 ww40 ww50 ww6",fontsize=16,color="black",shape="triangle"];18 -> 22[label="",style="solid", color="black", weight=3]; 20[label="ww51",fontsize=16,color="green",shape="box"];21[label="ww41",fontsize=16,color="green",shape="box"];22 -> 23[label="",style="dashed", color="red", weight=0]; 22[label="ww3 ww40 ww50 >>= gtGt0 ww6",fontsize=16,color="magenta"];22 -> 24[label="",style="dashed", color="magenta", weight=3]; 24[label="ww3 ww40 ww50",fontsize=16,color="green",shape="box"];24 -> 29[label="",style="dashed", color="green", weight=3]; 24 -> 30[label="",style="dashed", color="green", weight=3]; 23[label="ww8 >>= gtGt0 ww6",fontsize=16,color="burlywood",shape="triangle"];47[label="ww8/ww80 : ww81",fontsize=10,color="white",style="solid",shape="box"];23 -> 47[label="",style="solid", color="burlywood", weight=9]; 47 -> 27[label="",style="solid", color="burlywood", weight=3]; 48[label="ww8/[]",fontsize=10,color="white",style="solid",shape="box"];23 -> 48[label="",style="solid", color="burlywood", weight=9]; 48 -> 28[label="",style="solid", color="burlywood", weight=3]; 29[label="ww40",fontsize=16,color="green",shape="box"];30[label="ww50",fontsize=16,color="green",shape="box"];27[label="ww80 : ww81 >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3]; 28[label="[] >>= gtGt0 ww6",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 31 -> 33[label="",style="dashed", color="red", weight=0]; 31[label="gtGt0 ww6 ww80 ++ (ww81 >>= gtGt0 ww6)",fontsize=16,color="magenta"];31 -> 34[label="",style="dashed", color="magenta", weight=3]; 32[label="[]",fontsize=16,color="green",shape="box"];34 -> 23[label="",style="dashed", color="red", weight=0]; 34[label="ww81 >>= gtGt0 ww6",fontsize=16,color="magenta"];34 -> 35[label="",style="dashed", color="magenta", weight=3]; 33[label="gtGt0 ww6 ww80 ++ ww9",fontsize=16,color="black",shape="triangle"];33 -> 36[label="",style="solid", color="black", weight=3]; 35[label="ww81",fontsize=16,color="green",shape="box"];36[label="ww6 ++ ww9",fontsize=16,color="burlywood",shape="triangle"];49[label="ww6/ww60 : ww61",fontsize=10,color="white",style="solid",shape="box"];36 -> 49[label="",style="solid", color="burlywood", weight=9]; 49 -> 37[label="",style="solid", color="burlywood", weight=3]; 50[label="ww6/[]",fontsize=10,color="white",style="solid",shape="box"];36 -> 50[label="",style="solid", color="burlywood", weight=9]; 50 -> 38[label="",style="solid", color="burlywood", weight=3]; 37[label="(ww60 : ww61) ++ ww9",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3]; 38[label="[] ++ ww9",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3]; 39[label="ww60 : ww61 ++ ww9",fontsize=16,color="green",shape="box"];39 -> 41[label="",style="dashed", color="green", weight=3]; 40[label="ww9",fontsize=16,color="green",shape="box"];41 -> 36[label="",style="dashed", color="red", weight=0]; 41[label="ww61 ++ ww9",fontsize=16,color="magenta"];41 -> 42[label="",style="dashed", color="magenta", weight=3]; 42[label="ww61",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) 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_gtGtEs(:(ww80, ww81), ww6, h) -> new_gtGtEs(ww81, ww6, h) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_foldr(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) 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(ww3, :(ww40, ww41), :(ww50, ww51), h, ba, bb) -> new_foldr(ww3, ww41, ww51, h, ba, bb) The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs(:(ww60, ww61), ww9) -> new_psPs(ww61, ww9) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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(:(ww60, ww61), ww9) -> new_psPs(ww61, ww9) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (17) YES