8.47/3.68 YES 10.17/4.16 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 10.17/4.16 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 10.17/4.16 10.17/4.16 10.17/4.16 H-Termination with start terms of the given HASKELL could be proven: 10.17/4.16 10.17/4.16 (0) HASKELL 10.17/4.16 (1) BR [EQUIVALENT, 0 ms] 10.17/4.16 (2) HASKELL 10.17/4.16 (3) COR [EQUIVALENT, 0 ms] 10.17/4.16 (4) HASKELL 10.17/4.16 (5) Narrow [SOUND, 0 ms] 10.17/4.16 (6) QDP 10.17/4.16 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 10.17/4.16 (8) YES 10.17/4.16 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (0) 10.17/4.16 Obligation: 10.17/4.16 mainModule Main 10.17/4.16 module Main where { 10.17/4.16 import qualified Prelude; 10.17/4.16 } 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (1) BR (EQUIVALENT) 10.17/4.16 Replaced joker patterns by fresh variables and removed binding patterns. 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (2) 10.17/4.16 Obligation: 10.17/4.16 mainModule Main 10.17/4.16 module Main where { 10.17/4.16 import qualified Prelude; 10.17/4.16 } 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (3) COR (EQUIVALENT) 10.17/4.16 Cond Reductions: 10.17/4.16 The following Function with conditions 10.17/4.16 "max x y|x <= yy|otherwisex; 10.17/4.16 " 10.17/4.16 is transformed to 10.17/4.16 "max x y = max2 x y; 10.17/4.16 " 10.17/4.16 "max0 x y True = x; 10.17/4.16 " 10.17/4.16 "max1 x y True = y; 10.17/4.16 max1 x y False = max0 x y otherwise; 10.17/4.16 " 10.17/4.16 "max2 x y = max1 x y (x <= y); 10.17/4.16 " 10.17/4.16 The following Function with conditions 10.17/4.16 "undefined |Falseundefined; 10.17/4.16 " 10.17/4.16 is transformed to 10.17/4.16 "undefined = undefined1; 10.17/4.16 " 10.17/4.16 "undefined0 True = undefined; 10.17/4.16 " 10.17/4.16 "undefined1 = undefined0 False; 10.17/4.16 " 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (4) 10.17/4.16 Obligation: 10.17/4.16 mainModule Main 10.17/4.16 module Main where { 10.17/4.16 import qualified Prelude; 10.17/4.16 } 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (5) Narrow (SOUND) 10.17/4.16 Haskell To QDPs 10.17/4.16 10.17/4.16 digraph dp_graph { 10.17/4.16 node [outthreshold=100, inthreshold=100];1[label="maximum",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 10.17/4.16 3[label="maximum vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 10.17/4.16 4[label="foldl1 max vx3",fontsize=16,color="burlywood",shape="box"];53[label="vx3/vx30 : vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 53[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 53 -> 5[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 54[label="vx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 54[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 54 -> 6[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 5[label="foldl1 max (vx30 : vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 10.17/4.16 6[label="foldl1 max []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 10.17/4.16 7[label="foldl max vx30 vx31",fontsize=16,color="burlywood",shape="triangle"];55[label="vx31/vx310 : vx311",fontsize=10,color="white",style="solid",shape="box"];7 -> 55[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 55 -> 9[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 56[label="vx31/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 56[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 56 -> 10[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 8[label="error []",fontsize=16,color="red",shape="box"];9[label="foldl max vx30 (vx310 : vx311)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3]; 10.17/4.16 10[label="foldl max vx30 []",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 10.17/4.16 11 -> 7[label="",style="dashed", color="red", weight=0]; 10.17/4.16 11[label="foldl max (max vx30 vx310) vx311",fontsize=16,color="magenta"];11 -> 13[label="",style="dashed", color="magenta", weight=3]; 10.17/4.16 11 -> 14[label="",style="dashed", color="magenta", weight=3]; 10.17/4.16 12[label="vx30",fontsize=16,color="green",shape="box"];13[label="vx311",fontsize=16,color="green",shape="box"];14[label="max vx30 vx310",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3]; 10.17/4.16 15[label="max2 vx30 vx310",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3]; 10.17/4.16 16[label="max1 vx30 vx310 (vx30 <= vx310)",fontsize=16,color="burlywood",shape="box"];57[label="vx30/LT",fontsize=10,color="white",style="solid",shape="box"];16 -> 57[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 57 -> 17[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 58[label="vx30/EQ",fontsize=10,color="white",style="solid",shape="box"];16 -> 58[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 58 -> 18[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 59[label="vx30/GT",fontsize=10,color="white",style="solid",shape="box"];16 -> 59[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 59 -> 19[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 17[label="max1 LT vx310 (LT <= vx310)",fontsize=16,color="burlywood",shape="box"];60[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];17 -> 60[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 60 -> 20[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 61[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];17 -> 61[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 61 -> 21[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 62[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];17 -> 62[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 62 -> 22[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 18[label="max1 EQ vx310 (EQ <= vx310)",fontsize=16,color="burlywood",shape="box"];63[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 63[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 63 -> 23[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 64[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 64[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 64 -> 24[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 65[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 65[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 65 -> 25[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 19[label="max1 GT vx310 (GT <= vx310)",fontsize=16,color="burlywood",shape="box"];66[label="vx310/LT",fontsize=10,color="white",style="solid",shape="box"];19 -> 66[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 66 -> 26[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 67[label="vx310/EQ",fontsize=10,color="white",style="solid",shape="box"];19 -> 67[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 67 -> 27[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 68[label="vx310/GT",fontsize=10,color="white",style="solid",shape="box"];19 -> 68[label="",style="solid", color="burlywood", weight=9]; 10.17/4.16 68 -> 28[label="",style="solid", color="burlywood", weight=3]; 10.17/4.16 20[label="max1 LT LT (LT <= LT)",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3]; 10.17/4.16 21[label="max1 LT EQ (LT <= EQ)",fontsize=16,color="black",shape="box"];21 -> 30[label="",style="solid", color="black", weight=3]; 10.17/4.16 22[label="max1 LT GT (LT <= GT)",fontsize=16,color="black",shape="box"];22 -> 31[label="",style="solid", color="black", weight=3]; 10.17/4.16 23[label="max1 EQ LT (EQ <= LT)",fontsize=16,color="black",shape="box"];23 -> 32[label="",style="solid", color="black", weight=3]; 10.17/4.16 24[label="max1 EQ EQ (EQ <= EQ)",fontsize=16,color="black",shape="box"];24 -> 33[label="",style="solid", color="black", weight=3]; 10.17/4.16 25[label="max1 EQ GT (EQ <= GT)",fontsize=16,color="black",shape="box"];25 -> 34[label="",style="solid", color="black", weight=3]; 10.17/4.16 26[label="max1 GT LT (GT <= LT)",fontsize=16,color="black",shape="box"];26 -> 35[label="",style="solid", color="black", weight=3]; 10.17/4.16 27[label="max1 GT EQ (GT <= EQ)",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3]; 10.17/4.16 28[label="max1 GT GT (GT <= GT)",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3]; 10.17/4.16 29[label="max1 LT LT True",fontsize=16,color="black",shape="box"];29 -> 38[label="",style="solid", color="black", weight=3]; 10.17/4.16 30[label="max1 LT EQ True",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3]; 10.17/4.16 31[label="max1 LT GT True",fontsize=16,color="black",shape="box"];31 -> 40[label="",style="solid", color="black", weight=3]; 10.17/4.16 32[label="max1 EQ LT False",fontsize=16,color="black",shape="box"];32 -> 41[label="",style="solid", color="black", weight=3]; 10.17/4.16 33[label="max1 EQ EQ True",fontsize=16,color="black",shape="box"];33 -> 42[label="",style="solid", color="black", weight=3]; 10.17/4.16 34[label="max1 EQ GT True",fontsize=16,color="black",shape="box"];34 -> 43[label="",style="solid", color="black", weight=3]; 10.17/4.16 35[label="max1 GT LT False",fontsize=16,color="black",shape="box"];35 -> 44[label="",style="solid", color="black", weight=3]; 10.17/4.16 36[label="max1 GT EQ False",fontsize=16,color="black",shape="box"];36 -> 45[label="",style="solid", color="black", weight=3]; 10.17/4.16 37[label="max1 GT GT True",fontsize=16,color="black",shape="box"];37 -> 46[label="",style="solid", color="black", weight=3]; 10.17/4.16 38[label="LT",fontsize=16,color="green",shape="box"];39[label="EQ",fontsize=16,color="green",shape="box"];40[label="GT",fontsize=16,color="green",shape="box"];41[label="max0 EQ LT otherwise",fontsize=16,color="black",shape="box"];41 -> 47[label="",style="solid", color="black", weight=3]; 10.17/4.16 42[label="EQ",fontsize=16,color="green",shape="box"];43[label="GT",fontsize=16,color="green",shape="box"];44[label="max0 GT LT otherwise",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3]; 10.17/4.16 45[label="max0 GT EQ otherwise",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3]; 10.17/4.16 46[label="GT",fontsize=16,color="green",shape="box"];47[label="max0 EQ LT True",fontsize=16,color="black",shape="box"];47 -> 50[label="",style="solid", color="black", weight=3]; 10.17/4.16 48[label="max0 GT LT True",fontsize=16,color="black",shape="box"];48 -> 51[label="",style="solid", color="black", weight=3]; 10.17/4.16 49[label="max0 GT EQ True",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3]; 10.17/4.16 50[label="EQ",fontsize=16,color="green",shape="box"];51[label="GT",fontsize=16,color="green",shape="box"];52[label="GT",fontsize=16,color="green",shape="box"];} 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (6) 10.17/4.16 Obligation: 10.17/4.16 Q DP problem: 10.17/4.16 The TRS P consists of the following rules: 10.17/4.16 10.17/4.16 new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311) 10.17/4.16 10.17/4.16 The TRS R consists of the following rules: 10.17/4.16 10.17/4.16 new_max1(LT, LT) -> LT 10.17/4.16 new_max1(LT, EQ) -> EQ 10.17/4.16 new_max1(EQ, LT) -> EQ 10.17/4.16 new_max1(EQ, GT) -> GT 10.17/4.16 new_max1(GT, EQ) -> GT 10.17/4.16 new_max1(LT, GT) -> GT 10.17/4.16 new_max1(GT, LT) -> GT 10.17/4.16 new_max1(GT, GT) -> GT 10.17/4.16 new_max1(EQ, EQ) -> EQ 10.17/4.16 10.17/4.16 The set Q consists of the following terms: 10.17/4.16 10.17/4.16 new_max1(GT, GT) 10.17/4.16 new_max1(LT, EQ) 10.17/4.16 new_max1(EQ, LT) 10.17/4.16 new_max1(EQ, EQ) 10.17/4.16 new_max1(EQ, GT) 10.17/4.16 new_max1(GT, EQ) 10.17/4.16 new_max1(LT, GT) 10.17/4.16 new_max1(GT, LT) 10.17/4.16 new_max1(LT, LT) 10.17/4.16 10.17/4.16 We have to consider all minimal (P,Q,R)-chains. 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (7) QDPSizeChangeProof (EQUIVALENT) 10.17/4.16 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. 10.17/4.16 10.17/4.16 From the DPs we obtained the following set of size-change graphs: 10.17/4.16 *new_foldl(vx30, :(vx310, vx311)) -> new_foldl(new_max1(vx30, vx310), vx311) 10.17/4.16 The graph contains the following edges 2 > 2 10.17/4.16 10.17/4.16 10.17/4.16 ---------------------------------------- 10.17/4.16 10.17/4.16 (8) 10.17/4.16 YES 10.45/7.53 EOF