/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) LR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4)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 "span2Ys wz xu = span2Ys0 wz xu (span2Vu43 wz xu); " "span2Span0 wz xu p vw vx True = ([],vw : vx); " "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; " "span2Vu43 wz xu = span wz xu; " "span2Zs0 wz xu (vy,zs) = zs; " "span2Zs wz xu = span2Zs0 wz xu (span2Vu43 wz xu); " "span2Ys0 wz xu (ys,vz) = ys; " ---------------------------------------- (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="span",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="span xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 4[label="span xv3 xv4",fontsize=16,color="burlywood",shape="triangle"];39[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9]; 39 -> 5[label="",style="solid", color="burlywood", weight=3]; 40[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9]; 40 -> 6[label="",style="solid", color="burlywood", weight=3]; 5[label="span xv3 (xv40 : xv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 6[label="span xv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="span2 xv3 (xv40 : xv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 8[label="span3 xv3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3]; 9 -> 11[label="",style="dashed", color="red", weight=0]; 9[label="span2Span1 xv3 xv41 xv3 xv40 xv41 (xv3 xv40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3]; 10[label="([],[])",fontsize=16,color="green",shape="box"];12[label="xv3 xv40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3]; 11[label="span2Span1 xv3 xv41 xv3 xv40 xv41 xv5",fontsize=16,color="burlywood",shape="triangle"];41[label="xv5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 41[label="",style="solid", color="burlywood", weight=9]; 41 -> 14[label="",style="solid", color="burlywood", weight=3]; 42[label="xv5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 42[label="",style="solid", color="burlywood", weight=9]; 42 -> 15[label="",style="solid", color="burlywood", weight=3]; 16[label="xv40",fontsize=16,color="green",shape="box"];14[label="span2Span1 xv3 xv41 xv3 xv40 xv41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3]; 15[label="span2Span1 xv3 xv41 xv3 xv40 xv41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3]; 17[label="span2Span0 xv3 xv41 xv3 xv40 xv41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3]; 18[label="(xv40 : span2Ys xv3 xv41,span2Zs xv3 xv41)",fontsize=16,color="green",shape="box"];18 -> 20[label="",style="dashed", color="green", weight=3]; 18 -> 21[label="",style="dashed", color="green", weight=3]; 19[label="span2Span0 xv3 xv41 xv3 xv40 xv41 True",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3]; 20[label="span2Ys xv3 xv41",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3]; 21[label="span2Zs xv3 xv41",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3]; 22[label="([],xv40 : xv41)",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="red", weight=0]; 23[label="span2Ys0 xv3 xv41 (span2Vu43 xv3 xv41)",fontsize=16,color="magenta"];23 -> 28[label="",style="dashed", color="magenta", weight=3]; 24 -> 32[label="",style="dashed", color="red", weight=0]; 24[label="span2Zs0 xv3 xv41 (span2Vu43 xv3 xv41)",fontsize=16,color="magenta"];24 -> 33[label="",style="dashed", color="magenta", weight=3]; 28[label="span2Vu43 xv3 xv41",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3]; 27[label="span2Ys0 xv3 xv41 xv6",fontsize=16,color="burlywood",shape="triangle"];43[label="xv6/(xv60,xv61)",fontsize=10,color="white",style="solid",shape="box"];27 -> 43[label="",style="solid", color="burlywood", weight=9]; 43 -> 31[label="",style="solid", color="burlywood", weight=3]; 33 -> 28[label="",style="dashed", color="red", weight=0]; 33[label="span2Vu43 xv3 xv41",fontsize=16,color="magenta"];32[label="span2Zs0 xv3 xv41 xv7",fontsize=16,color="burlywood",shape="triangle"];44[label="xv7/(xv70,xv71)",fontsize=10,color="white",style="solid",shape="box"];32 -> 44[label="",style="solid", color="burlywood", weight=9]; 44 -> 35[label="",style="solid", color="burlywood", weight=3]; 30 -> 4[label="",style="dashed", color="red", weight=0]; 30[label="span xv3 xv41",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3]; 31[label="span2Ys0 xv3 xv41 (xv60,xv61)",fontsize=16,color="black",shape="box"];31 -> 37[label="",style="solid", color="black", weight=3]; 35[label="span2Zs0 xv3 xv41 (xv70,xv71)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3]; 36[label="xv41",fontsize=16,color="green",shape="box"];37[label="xv60",fontsize=16,color="green",shape="box"];38[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