/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4)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) LetRed [EQUIVALENT, 0 ms] (8) HASKELL (9) Narrow [SOUND, 0 ms] (10) QDP (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] (12) YES ---------------------------------------- (0) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (1) LR (EQUIVALENT) Lambda Reductions: The following Lambda expression "\(_,zs)->zs" is transformed to "zs0 (_,zs) = zs; " The following Lambda expression "\(ys,_)->ys" is transformed to "ys0 (ys,_) = ys; " ---------------------------------------- (2) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. Binding Reductions: The bind variable of the following binding Pattern "xs@(vw : vx)" is replaced by the following term "vw : vx" ---------------------------------------- (4) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (5) COR (EQUIVALENT) Cond Reductions: The following Function with conditions "undefined |Falseundefined; " is transformed to "undefined = undefined1; " "undefined0 True = undefined; " "undefined1 = undefined0 False; " The following Function with conditions "span p [] = ([],[]); span p (vw : vx)|p vw(vw : ys,zs)|otherwise([],vw : vx) where { vu43 = span p vx; ; ys = ys0 vu43; ; ys0 (ys,vz) = ys; ; zs = zs0 vu43; ; zs0 (vy,zs) = zs; } ; " is transformed to "span p [] = span3 p []; span p (vw : vx) = span2 p (vw : vx); " "span2 p (vw : vx) = span1 p vw vx (p vw) where { span0 p vw vx True = ([],vw : vx); ; span1 p vw vx True = (vw : ys,zs); span1 p vw vx False = span0 p vw vx otherwise; ; vu43 = span p vx; ; ys = ys0 vu43; ; ys0 (ys,vz) = ys; ; zs = zs0 vu43; ; zs0 (vy,zs) = zs; } ; " "span3 p [] = ([],[]); span3 wx wy = span2 wx wy; " ---------------------------------------- (6) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (7) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "span1 p vw vx (p vw) where { span0 p vw vx True = ([],vw : vx); ; span1 p vw vx True = (vw : ys,zs); span1 p vw vx False = span0 p vw vx otherwise; ; vu43 = span p vx; ; ys = ys0 vu43; ; ys0 (ys,vz) = ys; ; zs = zs0 vu43; ; zs0 (vy,zs) = zs; } " are unpacked to the following functions on top level "span2Zs wz xu = span2Zs0 wz xu (span2Vu43 wz xu); " "span2Zs0 wz xu (vy,zs) = zs; " "span2Vu43 wz xu = span wz xu; " "span2Span1 wz xu p vw vx True = (vw : span2Ys wz xu,span2Zs wz xu); span2Span1 wz xu p vw vx False = span2Span0 wz xu p vw vx otherwise; " "span2Ys0 wz xu (ys,vz) = ys; " "span2Ys wz xu = span2Ys0 wz xu (span2Vu43 wz xu); " "span2Span0 wz xu p vw vx True = ([],vw : vx); " ---------------------------------------- (8) Obligation: mainModule Main module Main where { import qualified Prelude; } ---------------------------------------- (9) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="break",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="break xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="break xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 5[label="span (not . xv3) xv4",fontsize=16,color="burlywood",shape="triangle"];43[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];5 -> 43[label="",style="solid", color="burlywood", weight=9]; 43 -> 6[label="",style="solid", color="burlywood", weight=3]; 44[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 44[label="",style="solid", color="burlywood", weight=9]; 44 -> 7[label="",style="solid", color="burlywood", weight=3]; 6[label="span (not . xv3) (xv40 : xv41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="span (not . xv3) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8[label="span2 (not . xv3) (xv40 : xv41)",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9[label="span3 (not . xv3) []",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not . xv3)",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 11[label="([],[])",fontsize=16,color="green",shape="box"];12 -> 13[label="",style="dashed", color="red", weight=0]; 12[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not (xv3 xv40))",fontsize=16,color="magenta"];12 -> 14[label="",style="dashed", color="magenta", weight=3]; 14[label="xv3 xv40",fontsize=16,color="green",shape="box"];14 -> 18[label="",style="dashed", color="green", weight=3]; 13[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not xv5)",fontsize=16,color="burlywood",shape="triangle"];45[label="xv5/False",fontsize=10,color="white",style="solid",shape="box"];13 -> 45[label="",style="solid", color="burlywood", weight=9]; 45 -> 16[label="",style="solid", color="burlywood", weight=3]; 46[label="xv5/True",fontsize=10,color="white",style="solid",shape="box"];13 -> 46[label="",style="solid", color="burlywood", weight=9]; 46 -> 17[label="",style="solid", color="burlywood", weight=3]; 18[label="xv40",fontsize=16,color="green",shape="box"];16[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not False)",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3]; 17[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 (not True)",fontsize=16,color="black",shape="box"];17 -> 20[label="",style="solid", color="black", weight=3]; 19[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 True",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3]; 20[label="span2Span1 (not . xv3) xv41 (not . xv3) xv40 xv41 False",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3]; 21[label="(xv40 : span2Ys (not . xv3) xv41,span2Zs (not . xv3) xv41)",fontsize=16,color="green",shape="box"];21 -> 23[label="",style="dashed", color="green", weight=3]; 21 -> 24[label="",style="dashed", color="green", weight=3]; 22[label="span2Span0 (not . xv3) xv41 (not . xv3) xv40 xv41 otherwise",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3]; 23[label="span2Ys (not . xv3) xv41",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3]; 24[label="span2Zs (not . xv3) xv41",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3]; 25[label="span2Span0 (not . xv3) xv41 (not . xv3) xv40 xv41 True",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3]; 26 -> 31[label="",style="dashed", color="red", weight=0]; 26[label="span2Ys0 (not . xv3) xv41 (span2Vu43 (not . xv3) xv41)",fontsize=16,color="magenta"];26 -> 32[label="",style="dashed", color="magenta", weight=3]; 27 -> 36[label="",style="dashed", color="red", weight=0]; 27[label="span2Zs0 (not . xv3) xv41 (span2Vu43 (not . xv3) xv41)",fontsize=16,color="magenta"];27 -> 37[label="",style="dashed", color="magenta", weight=3]; 28[label="([],xv40 : xv41)",fontsize=16,color="green",shape="box"];32[label="span2Vu43 (not . xv3) xv41",fontsize=16,color="black",shape="triangle"];32 -> 34[label="",style="solid", color="black", weight=3]; 31[label="span2Ys0 (not . xv3) xv41 xv6",fontsize=16,color="burlywood",shape="triangle"];47[label="xv6/(xv60,xv61)",fontsize=10,color="white",style="solid",shape="box"];31 -> 47[label="",style="solid", color="burlywood", weight=9]; 47 -> 35[label="",style="solid", color="burlywood", weight=3]; 37 -> 32[label="",style="dashed", color="red", weight=0]; 37[label="span2Vu43 (not . xv3) xv41",fontsize=16,color="magenta"];36[label="span2Zs0 (not . xv3) xv41 xv7",fontsize=16,color="burlywood",shape="triangle"];48[label="xv7/(xv70,xv71)",fontsize=10,color="white",style="solid",shape="box"];36 -> 48[label="",style="solid", color="burlywood", weight=9]; 48 -> 39[label="",style="solid", color="burlywood", weight=3]; 34 -> 5[label="",style="dashed", color="red", weight=0]; 34[label="span (not . xv3) xv41",fontsize=16,color="magenta"];34 -> 40[label="",style="dashed", color="magenta", weight=3]; 35[label="span2Ys0 (not . xv3) xv41 (xv60,xv61)",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3]; 39[label="span2Zs0 (not . xv3) xv41 (xv70,xv71)",fontsize=16,color="black",shape="box"];39 -> 42[label="",style="solid", color="black", weight=3]; 40[label="xv41",fontsize=16,color="green",shape="box"];41[label="xv60",fontsize=16,color="green",shape="box"];42[label="xv71",fontsize=16,color="green",shape="box"];} ---------------------------------------- (10) Obligation: Q DP problem: The TRS P consists of the following rules: new_span(xv3, :(xv40, xv41), h) -> new_span2Span1(xv3, xv41, xv40, h) new_span2Vu43(xv3, xv41, h) -> new_span(xv3, xv41, h) new_span2Span1(xv3, xv41, xv40, h) -> new_span(xv3, xv41, h) new_span2Span1(xv3, xv41, xv40, h) -> new_span2Vu43(xv3, xv41, h) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (11) 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_span2Span1(xv3, xv41, xv40, h) -> new_span(xv3, xv41, h) The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3 *new_span2Span1(xv3, xv41, xv40, h) -> new_span2Vu43(xv3, xv41, h) The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3 *new_span2Vu43(xv3, xv41, h) -> new_span(xv3, xv41, h) The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3 *new_span(xv3, :(xv40, xv41), h) -> new_span2Span1(xv3, xv41, xv40, h) The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4 ---------------------------------------- (12) YES