/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4)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 (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) 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; liftM4 :: Monad d => (f -> c -> a -> b -> e) -> d f -> d c -> d a -> d b -> d e; liftM4 f m1 m2 m3 m4 = m1 >>= (\x1 ->m2 >>= (\x2 ->m3 >>= (\x3 ->m4 >>= (\x4 ->return (f x1 x2 x3 x4))))); } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\x4->return (f x1 x2 x3 x4)" is transformed to "liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4); " The following Lambda expression "\x3->m4 >>= liftM40 f x1 x2 x3" is transformed to "liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3; " The following Lambda expression "\x2->m3 >>= liftM41 m4 f x1 x2" is transformed to "liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2; " The following Lambda expression "\x1->m2 >>= liftM42 m3 m4 f x1" is transformed to "liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 f x1; " ---------------------------------------- (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; liftM4 :: Monad d => (e -> b -> f -> a -> c) -> d e -> d b -> d f -> d a -> d c; liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f; liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4); liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3; liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2; liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 f x1; } ---------------------------------------- (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; liftM4 :: Monad a => (e -> c -> f -> b -> d) -> a e -> a c -> a f -> a b -> a d; liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f; liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4); liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3; liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2; liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 f x1; } ---------------------------------------- (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; liftM4 :: Monad e => (b -> a -> c -> d -> f) -> e b -> e a -> e c -> e d -> e f; liftM4 f m1 m2 m3 m4 = m1 >>= liftM43 m2 m3 m4 f; liftM40 f x1 x2 x3 x4 = return (f x1 x2 x3 x4); liftM41 m4 f x1 x2 x3 = m4 >>= liftM40 f x1 x2 x3; liftM42 m3 m4 f x1 x2 = m3 >>= liftM41 m4 f x1 x2; liftM43 m2 m3 m4 f x1 = m2 >>= liftM42 m3 m4 f x1; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="Monad.liftM4",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="Monad.liftM4 vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="Monad.liftM4 vy3 vy4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3]; 5[label="Monad.liftM4 vy3 vy4 vy5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3]; 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183 -> 87[label="",style="solid", color="burlywood", weight=3]; 66[label="primbindIO (IO vy60) (Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];66 -> 88[label="",style="solid", color="black", weight=3]; 67[label="primbindIO (AProVE_IO vy60) (Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];67 -> 89[label="",style="solid", color="black", weight=3]; 68[label="primbindIO (AProVE_Exception vy60) (Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];68 -> 90[label="",style="solid", color="black", weight=3]; 69[label="primbindIO (AProVE_Error vy60) (Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="black",shape="box"];69 -> 91[label="",style="solid", color="black", weight=3]; 126 -> 71[label="",style="dashed", color="red", weight=0]; 126[label="Monad.liftM41 vy7 vy3 vy40 vy50 vy60 ++ (vy61 >>= Monad.liftM41 vy7 vy3 vy40 vy50)",fontsize=16,color="magenta"];126 -> 129[label="",style="dashed", color="magenta", weight=3]; 126 -> 130[label="",style="dashed", color="magenta", weight=3]; 127[label="[]",fontsize=16,color="green",shape="box"];86[label="Nothing >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];86 -> 96[label="",style="solid", color="black", weight=3]; 87[label="Just vy70 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];87 -> 97[label="",style="solid", color="black", weight=3]; 88[label="error []",fontsize=16,color="red",shape="box"];89[label="Monad.liftM41 vy7 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];89 -> 98[label="",style="solid", color="black", weight=3]; 90[label="AProVE_Exception vy60",fontsize=16,color="green",shape="box"];91[label="AProVE_Error vy60",fontsize=16,color="green",shape="box"];129 -> 113[label="",style="dashed", color="red", weight=0]; 129[label="vy61 >>= Monad.liftM41 vy7 vy3 vy40 vy50",fontsize=16,color="magenta"];129 -> 132[label="",style="dashed", color="magenta", weight=3]; 130[label="Monad.liftM41 vy7 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];130 -> 133[label="",style="solid", color="black", weight=3]; 96[label="Nothing",fontsize=16,color="green",shape="box"];97[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70",fontsize=16,color="black",shape="box"];97 -> 102[label="",style="solid", color="black", weight=3]; 98[label="vy7 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];98 -> 103[label="",style="solid", color="black", weight=3]; 132[label="vy61",fontsize=16,color="green",shape="box"];133[label="vy7 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="burlywood",shape="triangle"];184[label="vy7/vy70 : vy71",fontsize=10,color="white",style="solid",shape="box"];133 -> 184[label="",style="solid", color="burlywood", weight=9]; 184 -> 135[label="",style="solid", color="burlywood", weight=3]; 185[label="vy7/[]",fontsize=10,color="white",style="solid",shape="box"];133 -> 185[label="",style="solid", color="burlywood", weight=9]; 185 -> 136[label="",style="solid", color="burlywood", weight=3]; 102[label="return (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="black",shape="box"];102 -> 107[label="",style="solid", color="black", weight=3]; 103[label="primbindIO vy7 (Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="burlywood",shape="box"];186[label="vy7/IO vy70",fontsize=10,color="white",style="solid",shape="box"];103 -> 186[label="",style="solid", color="burlywood", weight=9]; 186 -> 108[label="",style="solid", color="burlywood", weight=3]; 187[label="vy7/AProVE_IO vy70",fontsize=10,color="white",style="solid",shape="box"];103 -> 187[label="",style="solid", color="burlywood", weight=9]; 187 -> 109[label="",style="solid", color="burlywood", weight=3]; 188[label="vy7/AProVE_Exception vy70",fontsize=10,color="white",style="solid",shape="box"];103 -> 188[label="",style="solid", color="burlywood", weight=9]; 188 -> 110[label="",style="solid", color="burlywood", weight=3]; 189[label="vy7/AProVE_Error vy70",fontsize=10,color="white",style="solid",shape="box"];103 -> 189[label="",style="solid", color="burlywood", weight=9]; 189 -> 111[label="",style="solid", color="burlywood", weight=3]; 135[label="vy70 : vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];135 -> 141[label="",style="solid", color="black", weight=3]; 136[label="[] >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="black",shape="box"];136 -> 142[label="",style="solid", color="black", weight=3]; 107[label="Just (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="green",shape="box"];107 -> 114[label="",style="dashed", color="green", weight=3]; 108[label="primbindIO (IO vy70) (Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];108 -> 115[label="",style="solid", color="black", weight=3]; 109[label="primbindIO (AProVE_IO vy70) (Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];109 -> 116[label="",style="solid", color="black", weight=3]; 110[label="primbindIO (AProVE_Exception vy70) (Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];110 -> 117[label="",style="solid", color="black", weight=3]; 111[label="primbindIO (AProVE_Error vy70) (Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="black",shape="box"];111 -> 118[label="",style="solid", color="black", weight=3]; 141 -> 71[label="",style="dashed", color="red", weight=0]; 141[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70 ++ (vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60)",fontsize=16,color="magenta"];141 -> 143[label="",style="dashed", color="magenta", weight=3]; 141 -> 144[label="",style="dashed", color="magenta", weight=3]; 142[label="[]",fontsize=16,color="green",shape="box"];114[label="vy3 vy40 vy50 vy60 vy70",fontsize=16,color="green",shape="box"];114 -> 121[label="",style="dashed", color="green", weight=3]; 114 -> 122[label="",style="dashed", color="green", weight=3]; 114 -> 123[label="",style="dashed", color="green", weight=3]; 114 -> 124[label="",style="dashed", color="green", weight=3]; 115[label="error []",fontsize=16,color="red",shape="box"];116[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70",fontsize=16,color="black",shape="box"];116 -> 125[label="",style="solid", color="black", weight=3]; 117[label="AProVE_Exception vy70",fontsize=16,color="green",shape="box"];118[label="AProVE_Error vy70",fontsize=16,color="green",shape="box"];143 -> 133[label="",style="dashed", color="red", weight=0]; 143[label="vy71 >>= Monad.liftM40 vy3 vy40 vy50 vy60",fontsize=16,color="magenta"];143 -> 145[label="",style="dashed", color="magenta", weight=3]; 144[label="Monad.liftM40 vy3 vy40 vy50 vy60 vy70",fontsize=16,color="black",shape="box"];144 -> 146[label="",style="solid", color="black", weight=3]; 121[label="vy40",fontsize=16,color="green",shape="box"];122[label="vy50",fontsize=16,color="green",shape="box"];123[label="vy60",fontsize=16,color="green",shape="box"];124[label="vy70",fontsize=16,color="green",shape="box"];125[label="return (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="black",shape="box"];125 -> 128[label="",style="solid", color="black", weight=3]; 145[label="vy71",fontsize=16,color="green",shape="box"];146[label="return (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="black",shape="box"];146 -> 147[label="",style="solid", color="black", weight=3]; 128[label="primretIO (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="black",shape="box"];128 -> 131[label="",style="solid", color="black", weight=3]; 147[label="vy3 vy40 vy50 vy60 vy70 : []",fontsize=16,color="green",shape="box"];147 -> 148[label="",style="dashed", color="green", weight=3]; 131[label="AProVE_IO (vy3 vy40 vy50 vy60 vy70)",fontsize=16,color="green",shape="box"];131 -> 134[label="",style="dashed", color="green", weight=3]; 148[label="vy3 vy40 vy50 vy60 vy70",fontsize=16,color="green",shape="box"];148 -> 149[label="",style="dashed", color="green", weight=3]; 148 -> 150[label="",style="dashed", color="green", weight=3]; 148 -> 151[label="",style="dashed", color="green", weight=3]; 148 -> 152[label="",style="dashed", color="green", weight=3]; 134[label="vy3 vy40 vy50 vy60 vy70",fontsize=16,color="green",shape="box"];134 -> 137[label="",style="dashed", color="green", weight=3]; 134 -> 138[label="",style="dashed", color="green", weight=3]; 134 -> 139[label="",style="dashed", color="green", weight=3]; 134 -> 140[label="",style="dashed", color="green", weight=3]; 149[label="vy40",fontsize=16,color="green",shape="box"];150[label="vy50",fontsize=16,color="green",shape="box"];151[label="vy60",fontsize=16,color="green",shape="box"];152[label="vy70",fontsize=16,color="green",shape="box"];137[label="vy40",fontsize=16,color="green",shape="box"];138[label="vy50",fontsize=16,color="green",shape="box"];139[label="vy60",fontsize=16,color="green",shape="box"];140[label="vy70",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs0(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) -> new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) 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_gtGtEs0(:(vy60, vy61), vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) -> new_gtGtEs0(vy61, vy7, vy3, vy40, vy50, h, ba, bb, bc, bd) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) -> new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) 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_gtGtEs1(:(vy50, vy51), vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) -> new_gtGtEs1(vy51, vy6, vy7, vy3, vy40, h, ba, bb, bc, bd) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) -> new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) 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_gtGtEs(:(vy70, vy71), vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) -> new_gtGtEs(vy71, vy3, vy40, vy50, vy60, h, ba, bb, bc, bd) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_psPs(:(vy90, vy91), vy8, h) -> new_psPs(vy91, vy8, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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(:(vy90, vy91), vy8, h) -> new_psPs(vy91, vy8, h) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) -> new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) 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_gtGtEs2(:(vy40, vy41), vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) -> new_gtGtEs2(vy41, vy5, vy6, vy7, vy3, h, ba, bb, bc, bd) The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8, 9 >= 9, 10 >= 10 ---------------------------------------- (23) YES