8.04/3.56 YES 9.52/4.02 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 9.52/4.02 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 9.52/4.02 9.52/4.02 9.52/4.02 H-Termination with start terms of the given HASKELL could be proven: 9.52/4.02 9.52/4.02 (0) HASKELL 9.52/4.02 (1) BR [EQUIVALENT, 0 ms] 9.52/4.02 (2) HASKELL 9.52/4.02 (3) COR [EQUIVALENT, 0 ms] 9.52/4.02 (4) HASKELL 9.52/4.02 (5) Narrow [SOUND, 0 ms] 9.52/4.02 (6) AND 9.52/4.02 (7) QDP 9.52/4.02 (8) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.52/4.02 (9) YES 9.52/4.02 (10) QDP 9.52/4.02 (11) QDPSizeChangeProof [EQUIVALENT, 0 ms] 9.52/4.02 (12) YES 9.52/4.02 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (0) 9.52/4.02 Obligation: 9.52/4.02 mainModule Main 9.52/4.02 module Main where { 9.52/4.02 import qualified Prelude; 9.52/4.02 } 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (1) BR (EQUIVALENT) 9.52/4.02 Replaced joker patterns by fresh variables and removed binding patterns. 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (2) 9.52/4.02 Obligation: 9.52/4.02 mainModule Main 9.52/4.02 module Main where { 9.52/4.02 import qualified Prelude; 9.52/4.02 } 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (3) COR (EQUIVALENT) 9.52/4.02 Cond Reductions: 9.52/4.02 The following Function with conditions 9.52/4.02 "undefined |Falseundefined; 9.52/4.02 " 9.52/4.02 is transformed to 9.52/4.02 "undefined = undefined1; 9.52/4.02 " 9.52/4.02 "undefined0 True = undefined; 9.52/4.02 " 9.52/4.02 "undefined1 = undefined0 False; 9.52/4.02 " 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (4) 9.52/4.02 Obligation: 9.52/4.02 mainModule Main 9.52/4.02 module Main where { 9.52/4.02 import qualified Prelude; 9.52/4.02 } 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (5) Narrow (SOUND) 9.52/4.02 Haskell To QDPs 9.52/4.02 9.52/4.02 digraph dp_graph { 9.52/4.02 node [outthreshold=100, inthreshold=100];1[label="notElem",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 9.52/4.02 3[label="notElem vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 9.52/4.02 4[label="notElem vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 9.52/4.02 5[label="all . (/=)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 9.52/4.02 6[label="all ((/=) vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 9.52/4.02 7[label="and . map ((/=) vz3)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 9.52/4.02 8[label="and (map ((/=) vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 9.52/4.02 9[label="foldr (&&) True (map ((/=) vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];79[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 79[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 79 -> 10[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 80[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 80[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 80 -> 11[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 10[label="foldr (&&) True (map ((/=) vz3) (vz40 : vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3]; 9.52/4.02 11[label="foldr (&&) True (map ((/=) vz3) [])",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3]; 9.52/4.02 12[label="foldr (&&) True (((/=) vz3 vz40) : map ((/=) vz3) vz41)",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3]; 9.52/4.02 13[label="foldr (&&) True []",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3]; 9.52/4.02 14 -> 16[label="",style="dashed", color="red", weight=0]; 9.52/4.02 14[label="(&&) (/=) vz3 vz40 foldr (&&) True (map ((/=) vz3) vz41)",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 15[label="True",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0]; 9.52/4.02 17[label="foldr (&&) True (map ((/=) vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 16[label="(&&) (/=) vz3 vz40 vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3]; 9.52/4.02 18[label="vz41",fontsize=16,color="green",shape="box"];19[label="(&&) not (vz3 == vz40) vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3]; 9.52/4.02 20[label="(&&) not (primEqInt vz3 vz40) vz5",fontsize=16,color="burlywood",shape="box"];81[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 81[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 81 -> 21[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 82[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 82[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 82 -> 22[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 21[label="(&&) not (primEqInt (Pos vz30) vz40) vz5",fontsize=16,color="burlywood",shape="box"];83[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];21 -> 83[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 83 -> 23[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 84[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 84[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 84 -> 24[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 22[label="(&&) not (primEqInt (Neg vz30) vz40) vz5",fontsize=16,color="burlywood",shape="box"];85[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];22 -> 85[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 85 -> 25[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 86[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 86[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 86 -> 26[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 23[label="(&&) not (primEqInt (Pos (Succ vz300)) vz40) vz5",fontsize=16,color="burlywood",shape="box"];87[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 87[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 87 -> 27[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 88[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 88[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 88 -> 28[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 24[label="(&&) not (primEqInt (Pos Zero) vz40) vz5",fontsize=16,color="burlywood",shape="box"];89[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 89 -> 29[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 90[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 90 -> 30[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 25[label="(&&) not (primEqInt (Neg (Succ vz300)) vz40) vz5",fontsize=16,color="burlywood",shape="box"];91[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 91[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 91 -> 31[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 92[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 92[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 92 -> 32[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 26[label="(&&) not (primEqInt (Neg Zero) vz40) vz5",fontsize=16,color="burlywood",shape="box"];93[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];26 -> 93[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 93 -> 33[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 94[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];26 -> 94[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 94 -> 34[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 27[label="(&&) not (primEqInt (Pos (Succ vz300)) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];95[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 95[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 95 -> 35[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 96[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 96[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 96 -> 36[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 28[label="(&&) not (primEqInt (Pos (Succ vz300)) (Neg vz400)) vz5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3]; 9.52/4.02 29[label="(&&) not (primEqInt (Pos Zero) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];97[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 97 -> 38[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 98[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 98 -> 39[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 30[label="(&&) not (primEqInt (Pos Zero) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];99[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];30 -> 99[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 99 -> 40[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 100[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 100[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 100 -> 41[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 31[label="(&&) not (primEqInt (Neg (Succ vz300)) (Pos vz400)) vz5",fontsize=16,color="black",shape="box"];31 -> 42[label="",style="solid", color="black", weight=3]; 9.52/4.02 32[label="(&&) not (primEqInt (Neg (Succ vz300)) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];101[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 101[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 101 -> 43[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 102[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 102[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 102 -> 44[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 33[label="(&&) not (primEqInt (Neg Zero) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];103[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 103 -> 45[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 104[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 104[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 104 -> 46[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 34[label="(&&) not (primEqInt (Neg Zero) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];105[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];34 -> 105[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 105 -> 47[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 106[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 106[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 106 -> 48[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 35[label="(&&) not (primEqInt (Pos (Succ vz300)) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3]; 9.52/4.02 36[label="(&&) not (primEqInt (Pos (Succ vz300)) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3]; 9.52/4.02 37[label="(&&) not False vz5",fontsize=16,color="black",shape="triangle"];37 -> 51[label="",style="solid", color="black", weight=3]; 9.52/4.02 38[label="(&&) not (primEqInt (Pos Zero) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3]; 9.52/4.02 39[label="(&&) not (primEqInt (Pos Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3]; 9.52/4.02 40[label="(&&) not (primEqInt (Pos Zero) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3]; 9.52/4.02 41[label="(&&) not (primEqInt (Pos Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3]; 9.52/4.02 42 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 42[label="(&&) not False vz5",fontsize=16,color="magenta"];43[label="(&&) not (primEqInt (Neg (Succ vz300)) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3]; 9.52/4.02 44[label="(&&) not (primEqInt (Neg (Succ vz300)) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3]; 9.52/4.02 45[label="(&&) not (primEqInt (Neg Zero) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3]; 9.52/4.02 46[label="(&&) not (primEqInt (Neg Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3]; 9.52/4.02 47[label="(&&) not (primEqInt (Neg Zero) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3]; 9.52/4.02 48[label="(&&) not (primEqInt (Neg Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3]; 9.52/4.02 49[label="(&&) not (primEqNat vz300 vz4000) vz5",fontsize=16,color="burlywood",shape="triangle"];107[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];49 -> 107[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 107 -> 62[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 108[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 108[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 108 -> 63[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 50 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 50[label="(&&) not False vz5",fontsize=16,color="magenta"];51[label="(&&) True vz5",fontsize=16,color="black",shape="box"];51 -> 64[label="",style="solid", color="black", weight=3]; 9.52/4.02 52 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 52[label="(&&) not False vz5",fontsize=16,color="magenta"];53[label="(&&) not True vz5",fontsize=16,color="black",shape="triangle"];53 -> 65[label="",style="solid", color="black", weight=3]; 9.52/4.02 54 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 54[label="(&&) not False vz5",fontsize=16,color="magenta"];55 -> 53[label="",style="dashed", color="red", weight=0]; 9.52/4.02 55[label="(&&) not True vz5",fontsize=16,color="magenta"];56 -> 49[label="",style="dashed", color="red", weight=0]; 9.52/4.02 56[label="(&&) not (primEqNat vz300 vz4000) vz5",fontsize=16,color="magenta"];56 -> 66[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 56 -> 67[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 57 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 57[label="(&&) not False vz5",fontsize=16,color="magenta"];58 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 58[label="(&&) not False vz5",fontsize=16,color="magenta"];59 -> 53[label="",style="dashed", color="red", weight=0]; 9.52/4.02 59[label="(&&) not True vz5",fontsize=16,color="magenta"];60 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 60[label="(&&) not False vz5",fontsize=16,color="magenta"];61 -> 53[label="",style="dashed", color="red", weight=0]; 9.52/4.02 61[label="(&&) not True vz5",fontsize=16,color="magenta"];62[label="(&&) not (primEqNat (Succ vz3000) vz4000) vz5",fontsize=16,color="burlywood",shape="box"];109[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 109[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 109 -> 68[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 110[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 110[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 110 -> 69[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 63[label="(&&) not (primEqNat Zero vz4000) vz5",fontsize=16,color="burlywood",shape="box"];111[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];63 -> 111[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 111 -> 70[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 112[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 112[label="",style="solid", color="burlywood", weight=9]; 9.52/4.02 112 -> 71[label="",style="solid", color="burlywood", weight=3]; 9.52/4.02 64[label="vz5",fontsize=16,color="green",shape="box"];65[label="(&&) False vz5",fontsize=16,color="black",shape="box"];65 -> 72[label="",style="solid", color="black", weight=3]; 9.52/4.02 66[label="vz300",fontsize=16,color="green",shape="box"];67[label="vz4000",fontsize=16,color="green",shape="box"];68[label="(&&) not (primEqNat (Succ vz3000) (Succ vz40000)) vz5",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3]; 9.52/4.02 69[label="(&&) not (primEqNat (Succ vz3000) Zero) vz5",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3]; 9.52/4.02 70[label="(&&) not (primEqNat Zero (Succ vz40000)) vz5",fontsize=16,color="black",shape="box"];70 -> 75[label="",style="solid", color="black", weight=3]; 9.52/4.02 71[label="(&&) not (primEqNat Zero Zero) vz5",fontsize=16,color="black",shape="box"];71 -> 76[label="",style="solid", color="black", weight=3]; 9.52/4.02 72[label="False",fontsize=16,color="green",shape="box"];73 -> 49[label="",style="dashed", color="red", weight=0]; 9.52/4.02 73[label="(&&) not (primEqNat vz3000 vz40000) vz5",fontsize=16,color="magenta"];73 -> 77[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 73 -> 78[label="",style="dashed", color="magenta", weight=3]; 9.52/4.02 74 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 74[label="(&&) not False vz5",fontsize=16,color="magenta"];75 -> 37[label="",style="dashed", color="red", weight=0]; 9.52/4.02 75[label="(&&) not False vz5",fontsize=16,color="magenta"];76 -> 53[label="",style="dashed", color="red", weight=0]; 9.52/4.02 76[label="(&&) not True vz5",fontsize=16,color="magenta"];77[label="vz3000",fontsize=16,color="green",shape="box"];78[label="vz40000",fontsize=16,color="green",shape="box"];} 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (6) 9.52/4.02 Complex Obligation (AND) 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (7) 9.52/4.02 Obligation: 9.52/4.02 Q DP problem: 9.52/4.02 The TRS P consists of the following rules: 9.52/4.02 9.52/4.02 new_foldr(vz3, :(vz40, vz41)) -> new_foldr(vz3, vz41) 9.52/4.02 9.52/4.02 R is empty. 9.52/4.02 Q is empty. 9.52/4.02 We have to consider all minimal (P,Q,R)-chains. 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (8) QDPSizeChangeProof (EQUIVALENT) 9.52/4.02 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. 9.52/4.02 9.52/4.02 From the DPs we obtained the following set of size-change graphs: 9.52/4.02 *new_foldr(vz3, :(vz40, vz41)) -> new_foldr(vz3, vz41) 9.52/4.02 The graph contains the following edges 1 >= 1, 2 > 2 9.52/4.02 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (9) 9.52/4.02 YES 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (10) 9.52/4.02 Obligation: 9.52/4.02 Q DP problem: 9.52/4.02 The TRS P consists of the following rules: 9.52/4.02 9.52/4.02 new_asAs(Succ(vz3000), Succ(vz40000), vz5) -> new_asAs(vz3000, vz40000, vz5) 9.52/4.02 9.52/4.02 R is empty. 9.52/4.02 Q is empty. 9.52/4.02 We have to consider all minimal (P,Q,R)-chains. 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (11) QDPSizeChangeProof (EQUIVALENT) 9.52/4.02 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. 9.52/4.02 9.52/4.02 From the DPs we obtained the following set of size-change graphs: 9.52/4.02 *new_asAs(Succ(vz3000), Succ(vz40000), vz5) -> new_asAs(vz3000, vz40000, vz5) 9.52/4.02 The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3 9.52/4.02 9.52/4.02 9.52/4.02 ---------------------------------------- 9.52/4.02 9.52/4.02 (12) 9.52/4.02 YES 9.76/4.06 EOF