/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 0 ms] (4) HASKELL (5) Narrow [SOUND, 0 ms] (6) QDP (7) DependencyGraphProof [EQUIVALENT, 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 Main where { import qualified Prelude; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "lookup k [] = Nothing; lookup k ((x,y) : xys)|k == xJust y|otherwiselookup k xys; " is transformed to "lookup k [] = lookup3 k []; lookup k ((x,y) : xys) = lookup2 k ((x,y) : xys); " "lookup1 k x y xys True = Just y; lookup1 k x y xys False = lookup0 k x y xys otherwise; " "lookup0 k x y xys True = lookup k xys; " "lookup2 k ((x,y) : xys) = lookup1 k x y xys (k == x); " "lookup3 k [] = Nothing; lookup3 wu wv = lookup2 wu wv; " The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="lookup ww3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="lookup ww3 ww4",fontsize=16,color="burlywood",shape="triangle"];66[label="ww4/ww40 : ww41",fontsize=10,color="white",style="solid",shape="box"];4 -> 66[label="",style="solid", color="burlywood", weight=9]; 66 -> 5[label="",style="solid", color="burlywood", weight=3]; 67[label="ww4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 67[label="",style="solid", color="burlywood", weight=9]; 67 -> 6[label="",style="solid", color="burlywood", weight=3]; 5[label="lookup ww3 (ww40 : ww41)",fontsize=16,color="burlywood",shape="box"];68[label="ww40/(ww400,ww401)",fontsize=10,color="white",style="solid",shape="box"];5 -> 68[label="",style="solid", color="burlywood", weight=9]; 68 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="lookup ww3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="lookup ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8[label="lookup3 ww3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9[label="lookup2 ww3 ((ww400,ww401) : ww41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 ww3 ww400 ww401 ww41 (ww3 == ww400)",fontsize=16,color="burlywood",shape="box"];69[label="ww3/LT",fontsize=10,color="white",style="solid",shape="box"];11 -> 69[label="",style="solid", color="burlywood", weight=9]; 69 -> 12[label="",style="solid", color="burlywood", weight=3]; 70[label="ww3/EQ",fontsize=10,color="white",style="solid",shape="box"];11 -> 70[label="",style="solid", color="burlywood", weight=9]; 70 -> 13[label="",style="solid", color="burlywood", weight=3]; 71[label="ww3/GT",fontsize=10,color="white",style="solid",shape="box"];11 -> 71[label="",style="solid", color="burlywood", weight=9]; 71 -> 14[label="",style="solid", color="burlywood", weight=3]; 12[label="lookup1 LT ww400 ww401 ww41 (LT == ww400)",fontsize=16,color="burlywood",shape="box"];72[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];12 -> 72[label="",style="solid", color="burlywood", weight=9]; 72 -> 15[label="",style="solid", color="burlywood", weight=3]; 73[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];12 -> 73[label="",style="solid", color="burlywood", weight=9]; 73 -> 16[label="",style="solid", color="burlywood", weight=3]; 74[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];12 -> 74[label="",style="solid", color="burlywood", weight=9]; 74 -> 17[label="",style="solid", color="burlywood", weight=3]; 13[label="lookup1 EQ ww400 ww401 ww41 (EQ == ww400)",fontsize=16,color="burlywood",shape="box"];75[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];13 -> 75[label="",style="solid", color="burlywood", weight=9]; 75 -> 18[label="",style="solid", color="burlywood", weight=3]; 76[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];13 -> 76[label="",style="solid", color="burlywood", weight=9]; 76 -> 19[label="",style="solid", color="burlywood", weight=3]; 77[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9]; 77 -> 20[label="",style="solid", color="burlywood", weight=3]; 14[label="lookup1 GT ww400 ww401 ww41 (GT == ww400)",fontsize=16,color="burlywood",shape="box"];78[label="ww400/LT",fontsize=10,color="white",style="solid",shape="box"];14 -> 78[label="",style="solid", color="burlywood", weight=9]; 78 -> 21[label="",style="solid", color="burlywood", weight=3]; 79[label="ww400/EQ",fontsize=10,color="white",style="solid",shape="box"];14 -> 79[label="",style="solid", color="burlywood", weight=9]; 79 -> 22[label="",style="solid", color="burlywood", weight=3]; 80[label="ww400/GT",fontsize=10,color="white",style="solid",shape="box"];14 -> 80[label="",style="solid", color="burlywood", weight=9]; 80 -> 23[label="",style="solid", color="burlywood", weight=3]; 15[label="lookup1 LT LT ww401 ww41 (LT == LT)",fontsize=16,color="black",shape="box"];15 -> 24[label="",style="solid", color="black", weight=3]; 16[label="lookup1 LT EQ ww401 ww41 (LT == EQ)",fontsize=16,color="black",shape="box"];16 -> 25[label="",style="solid", color="black", weight=3]; 17[label="lookup1 LT GT ww401 ww41 (LT == GT)",fontsize=16,color="black",shape="box"];17 -> 26[label="",style="solid", color="black", weight=3]; 18[label="lookup1 EQ LT ww401 ww41 (EQ == LT)",fontsize=16,color="black",shape="box"];18 -> 27[label="",style="solid", color="black", weight=3]; 19[label="lookup1 EQ EQ ww401 ww41 (EQ == EQ)",fontsize=16,color="black",shape="box"];19 -> 28[label="",style="solid", color="black", weight=3]; 20[label="lookup1 EQ GT ww401 ww41 (EQ == GT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3]; 21[label="lookup1 GT LT ww401 ww41 (GT == LT)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3]; 22[label="lookup1 GT EQ ww401 ww41 (GT == EQ)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3]; 23[label="lookup1 GT GT ww401 ww41 (GT == GT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3]; 24[label="lookup1 LT LT ww401 ww41 True",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3]; 25[label="lookup1 LT EQ ww401 ww41 False",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3]; 26[label="lookup1 LT GT ww401 ww41 False",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3]; 27[label="lookup1 EQ LT ww401 ww41 False",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3]; 28[label="lookup1 EQ EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3]; 29[label="lookup1 EQ GT ww401 ww41 False",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3]; 30[label="lookup1 GT LT ww401 ww41 False",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3]; 31[label="lookup1 GT EQ ww401 ww41 False",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3]; 32[label="lookup1 GT GT ww401 ww41 True",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3]; 33[label="Just ww401",fontsize=16,color="green",shape="box"];34[label="lookup0 LT EQ ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];34 -> 42[label="",style="solid", color="black", weight=3]; 35[label="lookup0 LT GT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];35 -> 43[label="",style="solid", color="black", weight=3]; 36[label="lookup0 EQ LT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];36 -> 44[label="",style="solid", color="black", weight=3]; 37[label="Just ww401",fontsize=16,color="green",shape="box"];38[label="lookup0 EQ GT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3]; 39[label="lookup0 GT LT ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3]; 40[label="lookup0 GT EQ ww401 ww41 otherwise",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3]; 41[label="Just ww401",fontsize=16,color="green",shape="box"];42[label="lookup0 LT EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];42 -> 48[label="",style="solid", color="black", weight=3]; 43[label="lookup0 LT GT ww401 ww41 True",fontsize=16,color="black",shape="box"];43 -> 49[label="",style="solid", color="black", weight=3]; 44[label="lookup0 EQ LT ww401 ww41 True",fontsize=16,color="black",shape="box"];44 -> 50[label="",style="solid", color="black", weight=3]; 45[label="lookup0 EQ GT ww401 ww41 True",fontsize=16,color="black",shape="box"];45 -> 51[label="",style="solid", color="black", weight=3]; 46[label="lookup0 GT LT ww401 ww41 True",fontsize=16,color="black",shape="box"];46 -> 52[label="",style="solid", color="black", weight=3]; 47[label="lookup0 GT EQ ww401 ww41 True",fontsize=16,color="black",shape="box"];47 -> 53[label="",style="solid", color="black", weight=3]; 48 -> 4[label="",style="dashed", color="red", weight=0]; 48[label="lookup LT ww41",fontsize=16,color="magenta"];48 -> 54[label="",style="dashed", color="magenta", weight=3]; 48 -> 55[label="",style="dashed", color="magenta", weight=3]; 49 -> 4[label="",style="dashed", color="red", weight=0]; 49[label="lookup LT ww41",fontsize=16,color="magenta"];49 -> 56[label="",style="dashed", color="magenta", weight=3]; 49 -> 57[label="",style="dashed", color="magenta", weight=3]; 50 -> 4[label="",style="dashed", color="red", weight=0]; 50[label="lookup EQ ww41",fontsize=16,color="magenta"];50 -> 58[label="",style="dashed", color="magenta", weight=3]; 50 -> 59[label="",style="dashed", color="magenta", weight=3]; 51 -> 4[label="",style="dashed", color="red", weight=0]; 51[label="lookup EQ ww41",fontsize=16,color="magenta"];51 -> 60[label="",style="dashed", color="magenta", weight=3]; 51 -> 61[label="",style="dashed", color="magenta", weight=3]; 52 -> 4[label="",style="dashed", color="red", weight=0]; 52[label="lookup GT ww41",fontsize=16,color="magenta"];52 -> 62[label="",style="dashed", color="magenta", weight=3]; 52 -> 63[label="",style="dashed", color="magenta", weight=3]; 53 -> 4[label="",style="dashed", color="red", weight=0]; 53[label="lookup GT ww41",fontsize=16,color="magenta"];53 -> 64[label="",style="dashed", color="magenta", weight=3]; 53 -> 65[label="",style="dashed", color="magenta", weight=3]; 54[label="ww41",fontsize=16,color="green",shape="box"];55[label="LT",fontsize=16,color="green",shape="box"];56[label="ww41",fontsize=16,color="green",shape="box"];57[label="LT",fontsize=16,color="green",shape="box"];58[label="ww41",fontsize=16,color="green",shape="box"];59[label="EQ",fontsize=16,color="green",shape="box"];60[label="ww41",fontsize=16,color="green",shape="box"];61[label="EQ",fontsize=16,color="green",shape="box"];62[label="ww41",fontsize=16,color="green",shape="box"];63[label="GT",fontsize=16,color="green",shape="box"];64[label="ww41",fontsize=16,color="green",shape="box"];65[label="GT",fontsize=16,color="green",shape="box"];} ---------------------------------------- (6) Obligation: Q DP problem: The TRS P consists of the following rules: new_lookup(LT, :(@2(GT, ww401), ww41), h) -> new_lookup(LT, ww41, h) new_lookup(LT, :(@2(EQ, ww401), ww41), h) -> new_lookup(LT, ww41, h) new_lookup(EQ, :(@2(LT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) new_lookup(GT, :(@2(LT, ww401), ww41), h) -> new_lookup(GT, ww41, h) new_lookup(GT, :(@2(EQ, ww401), ww41), h) -> new_lookup(GT, ww41, h) new_lookup(EQ, :(@2(GT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (7) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs. ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_lookup(GT, :(@2(EQ, ww401), ww41), h) -> new_lookup(GT, ww41, h) new_lookup(GT, :(@2(LT, ww401), ww41), h) -> new_lookup(GT, ww41, 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_lookup(GT, :(@2(EQ, ww401), ww41), h) -> new_lookup(GT, ww41, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_lookup(GT, :(@2(LT, ww401), ww41), h) -> new_lookup(GT, ww41, 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_lookup(EQ, :(@2(GT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) new_lookup(EQ, :(@2(LT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) 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_lookup(EQ, :(@2(GT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_lookup(EQ, :(@2(LT, ww401), ww41), h) -> new_lookup(EQ, ww41, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_lookup(LT, :(@2(EQ, ww401), ww41), h) -> new_lookup(LT, ww41, h) new_lookup(LT, :(@2(GT, ww401), ww41), h) -> new_lookup(LT, ww41, h) 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_lookup(LT, :(@2(EQ, ww401), ww41), h) -> new_lookup(LT, ww41, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 *new_lookup(LT, :(@2(GT, ww401), ww41), h) -> new_lookup(LT, ww41, h) The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3 ---------------------------------------- (17) YES