8.08/3.57 YES 9.98/4.04 proof of /export/starexec/sandbox2/benchmark/theBenchmark.hs 9.98/4.04 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.98/4.04 9.98/4.04 9.98/4.04 H-Termination with start terms of the given HASKELL could be proven: 9.98/4.04 9.98/4.04 (0) HASKELL 9.98/4.04 (1) BR [EQUIVALENT, 0 ms] 9.98/4.04 (2) HASKELL 9.98/4.04 (3) COR [EQUIVALENT, 0 ms] 9.98/4.04 (4) HASKELL 9.98/4.04 (5) Narrow [SOUND, 0 ms] 9.98/4.04 (6) QDP 9.98/4.04 (7) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.98/4.04 (8) YES 9.98/4.04 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (0) 9.98/4.04 Obligation: 9.98/4.04 mainModule Main 9.98/4.04 module Main where { 9.98/4.04 import qualified Prelude; 9.98/4.04 } 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (1) BR (EQUIVALENT) 9.98/4.04 Replaced joker patterns by fresh variables and removed binding patterns. 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (2) 9.98/4.04 Obligation: 9.98/4.04 mainModule Main 9.98/4.04 module Main where { 9.98/4.04 import qualified Prelude; 9.98/4.04 } 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (3) COR (EQUIVALENT) 9.98/4.04 Cond Reductions: 9.98/4.04 The following Function with conditions 9.98/4.04 "undefined |Falseundefined; 9.98/4.04 " 9.98/4.04 is transformed to 9.98/4.04 "undefined = undefined1; 9.98/4.04 " 9.98/4.04 "undefined0 True = undefined; 9.98/4.04 " 9.98/4.04 "undefined1 = undefined0 False; 9.98/4.04 " 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (4) 9.98/4.04 Obligation: 9.98/4.04 mainModule Main 9.98/4.04 module Main where { 9.98/4.04 import qualified Prelude; 9.98/4.04 } 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (5) Narrow (SOUND) 9.98/4.04 Haskell To QDPs 9.98/4.04 9.98/4.04 digraph dp_graph { 9.98/4.04 node [outthreshold=100, inthreshold=100];1[label="(/=)",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.98/4.04 3[label="(/=) vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.98/4.04 4[label="(/=) vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.98/4.04 5[label="not (vz3 == vz4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.98/4.04 6[label="not (primEqInt vz3 vz4)",fontsize=16,color="burlywood",shape="box"];63[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];6 -> 63[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 63 -> 7[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 64[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];6 -> 64[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 64 -> 8[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 7[label="not (primEqInt (Pos vz30) vz4)",fontsize=16,color="burlywood",shape="box"];65[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];7 -> 65[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 65 -> 9[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 66[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 66[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 66 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 8[label="not (primEqInt (Neg vz30) vz4)",fontsize=16,color="burlywood",shape="box"];67[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];8 -> 67[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 67 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 68[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];8 -> 68[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 68 -> 12[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 9[label="not (primEqInt (Pos (Succ vz300)) vz4)",fontsize=16,color="burlywood",shape="box"];69[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 69[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 69 -> 13[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 70[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];9 -> 70[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 70 -> 14[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 10[label="not (primEqInt (Pos Zero) vz4)",fontsize=16,color="burlywood",shape="box"];71[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 71[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 71 -> 15[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 72[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];10 -> 72[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 72 -> 16[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 11[label="not (primEqInt (Neg (Succ vz300)) vz4)",fontsize=16,color="burlywood",shape="box"];73[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 73[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 73 -> 17[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 74[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];11 -> 74[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 74 -> 18[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 12[label="not (primEqInt (Neg Zero) vz4)",fontsize=16,color="burlywood",shape="box"];75[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];12 -> 75[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 75 -> 19[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 76[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];12 -> 76[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 76 -> 20[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 13[label="not (primEqInt (Pos (Succ vz300)) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];77[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];13 -> 77[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 77 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 78[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 78[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 78 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 14[label="not (primEqInt (Pos (Succ vz300)) (Neg vz40))",fontsize=16,color="black",shape="box"];14 -> 23[label="",style="solid", color="black", weight=3]; 9.98/4.04 15[label="not (primEqInt (Pos Zero) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];79[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];15 -> 79[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 79 -> 24[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 80[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];15 -> 80[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 80 -> 25[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 16[label="not (primEqInt (Pos Zero) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];81[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];16 -> 81[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 81 -> 26[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 82[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 82[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 82 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 17[label="not (primEqInt (Neg (Succ vz300)) (Pos vz40))",fontsize=16,color="black",shape="box"];17 -> 28[label="",style="solid", color="black", weight=3]; 9.98/4.04 18[label="not (primEqInt (Neg (Succ vz300)) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];83[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];18 -> 83[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 83 -> 29[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 84[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 84[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 84 -> 30[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 19[label="not (primEqInt (Neg Zero) (Pos vz40))",fontsize=16,color="burlywood",shape="box"];85[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];19 -> 85[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 85 -> 31[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 86[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 86[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 86 -> 32[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 20[label="not (primEqInt (Neg Zero) (Neg vz40))",fontsize=16,color="burlywood",shape="box"];87[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];20 -> 87[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 87 -> 33[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 88[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 88[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 88 -> 34[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 21[label="not (primEqInt (Pos (Succ vz300)) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];21 -> 35[label="",style="solid", color="black", weight=3]; 9.98/4.04 22[label="not (primEqInt (Pos (Succ vz300)) (Pos Zero))",fontsize=16,color="black",shape="box"];22 -> 36[label="",style="solid", color="black", weight=3]; 9.98/4.04 23[label="not False",fontsize=16,color="black",shape="triangle"];23 -> 37[label="",style="solid", color="black", weight=3]; 9.98/4.04 24[label="not (primEqInt (Pos Zero) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];24 -> 38[label="",style="solid", color="black", weight=3]; 9.98/4.04 25[label="not (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];25 -> 39[label="",style="solid", color="black", weight=3]; 9.98/4.04 26[label="not (primEqInt (Pos Zero) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];26 -> 40[label="",style="solid", color="black", weight=3]; 9.98/4.04 27[label="not (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3]; 9.98/4.04 28 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 28[label="not False",fontsize=16,color="magenta"];29[label="not (primEqInt (Neg (Succ vz300)) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];29 -> 42[label="",style="solid", color="black", weight=3]; 9.98/4.04 30[label="not (primEqInt (Neg (Succ vz300)) (Neg Zero))",fontsize=16,color="black",shape="box"];30 -> 43[label="",style="solid", color="black", weight=3]; 9.98/4.04 31[label="not (primEqInt (Neg Zero) (Pos (Succ vz400)))",fontsize=16,color="black",shape="box"];31 -> 44[label="",style="solid", color="black", weight=3]; 9.98/4.04 32[label="not (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];32 -> 45[label="",style="solid", color="black", weight=3]; 9.98/4.04 33[label="not (primEqInt (Neg Zero) (Neg (Succ vz400)))",fontsize=16,color="black",shape="box"];33 -> 46[label="",style="solid", color="black", weight=3]; 9.98/4.04 34[label="not (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];34 -> 47[label="",style="solid", color="black", weight=3]; 9.98/4.04 35[label="not (primEqNat vz300 vz400)",fontsize=16,color="burlywood",shape="triangle"];89[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 89[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 89 -> 48[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 90[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 90[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 90 -> 49[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 36 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 36[label="not False",fontsize=16,color="magenta"];37[label="True",fontsize=16,color="green",shape="box"];38 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 38[label="not False",fontsize=16,color="magenta"];39[label="not True",fontsize=16,color="black",shape="triangle"];39 -> 50[label="",style="solid", color="black", weight=3]; 9.98/4.04 40 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 40[label="not False",fontsize=16,color="magenta"];41 -> 39[label="",style="dashed", color="red", weight=0]; 9.98/4.04 41[label="not True",fontsize=16,color="magenta"];42 -> 35[label="",style="dashed", color="red", weight=0]; 9.98/4.04 42[label="not (primEqNat vz300 vz400)",fontsize=16,color="magenta"];42 -> 51[label="",style="dashed", color="magenta", weight=3]; 9.98/4.04 42 -> 52[label="",style="dashed", color="magenta", weight=3]; 9.98/4.04 43 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 43[label="not False",fontsize=16,color="magenta"];44 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 44[label="not False",fontsize=16,color="magenta"];45 -> 39[label="",style="dashed", color="red", weight=0]; 9.98/4.04 45[label="not True",fontsize=16,color="magenta"];46 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 46[label="not False",fontsize=16,color="magenta"];47 -> 39[label="",style="dashed", color="red", weight=0]; 9.98/4.04 47[label="not True",fontsize=16,color="magenta"];48[label="not (primEqNat (Succ vz3000) vz400)",fontsize=16,color="burlywood",shape="box"];91[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];48 -> 91[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 91 -> 53[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 92[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 92[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 92 -> 54[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 49[label="not (primEqNat Zero vz400)",fontsize=16,color="burlywood",shape="box"];93[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];49 -> 93[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 93 -> 55[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 94[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 94[label="",style="solid", color="burlywood", weight=9]; 9.98/4.04 94 -> 56[label="",style="solid", color="burlywood", weight=3]; 9.98/4.04 50[label="False",fontsize=16,color="green",shape="box"];51[label="vz300",fontsize=16,color="green",shape="box"];52[label="vz400",fontsize=16,color="green",shape="box"];53[label="not (primEqNat (Succ vz3000) (Succ vz4000))",fontsize=16,color="black",shape="box"];53 -> 57[label="",style="solid", color="black", weight=3]; 9.98/4.04 54[label="not (primEqNat (Succ vz3000) Zero)",fontsize=16,color="black",shape="box"];54 -> 58[label="",style="solid", color="black", weight=3]; 9.98/4.04 55[label="not (primEqNat Zero (Succ vz4000))",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 9.98/4.04 56[label="not (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 9.98/4.04 57 -> 35[label="",style="dashed", color="red", weight=0]; 9.98/4.04 57[label="not (primEqNat vz3000 vz4000)",fontsize=16,color="magenta"];57 -> 61[label="",style="dashed", color="magenta", weight=3]; 9.98/4.04 57 -> 62[label="",style="dashed", color="magenta", weight=3]; 9.98/4.04 58 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 58[label="not False",fontsize=16,color="magenta"];59 -> 23[label="",style="dashed", color="red", weight=0]; 9.98/4.04 59[label="not False",fontsize=16,color="magenta"];60 -> 39[label="",style="dashed", color="red", weight=0]; 9.98/4.04 60[label="not True",fontsize=16,color="magenta"];61[label="vz3000",fontsize=16,color="green",shape="box"];62[label="vz4000",fontsize=16,color="green",shape="box"];} 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (6) 9.98/4.04 Obligation: 9.98/4.04 Q DP problem: 9.98/4.04 The TRS P consists of the following rules: 9.98/4.04 9.98/4.04 new_not(Succ(vz3000), Succ(vz4000)) -> new_not(vz3000, vz4000) 9.98/4.04 9.98/4.04 R is empty. 9.98/4.04 Q is empty. 9.98/4.04 We have to consider all minimal (P,Q,R)-chains. 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (7) QDPSizeChangeProof (EQUIVALENT) 9.98/4.04 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. 9.98/4.04 9.98/4.04 From the DPs we obtained the following set of size-change graphs: 9.98/4.04 *new_not(Succ(vz3000), Succ(vz4000)) -> new_not(vz3000, vz4000) 9.98/4.04 The graph contains the following edges 1 > 1, 2 > 2 9.98/4.04 9.98/4.04 9.98/4.04 ---------------------------------------- 9.98/4.04 9.98/4.04 (8) 9.98/4.04 YES 10.03/5.01 EOF