/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- YES proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) BR [EQUIVALENT, 0 ms] (2) HASKELL (3) COR [EQUIVALENT, 22 ms] (4) HASKELL (5) LetRed [EQUIVALENT, 0 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) AND (9) QDP (10) DependencyGraphProof [EQUIVALENT, 0 ms] (11) AND (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) QDPSizeChangeProof [EQUIVALENT, 0 ms] (17) YES (18) QDP (19) QDPSizeChangeProof [EQUIVALENT, 0 ms] (20) YES (21) QDP (22) QDPSizeChangeProof [EQUIVALENT, 0 ms] (23) YES ---------------------------------------- (0) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; nub :: Eq a => [a] -> [a]; nub l = nub' l [] where { nub' [] _ = []; nub' (x : xs) ls | x `elem` ls = nub' xs ls | otherwise = x : nub' xs (x : ls); }; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (2) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; nub :: Eq a => [a] -> [a]; nub l = nub' l [] where { nub' [] vy = []; nub' (x : xs) ls | x `elem` ls = nub' xs ls | otherwise = x : nub' xs (x : ls); }; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) 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 "nub' [] vy = []; nub' (x : xs) ls|x `elem` lsnub' xs ls|otherwisex : nub' xs (x : ls); " is transformed to "nub' [] vy = nub'3 [] vy; nub' (x : xs) ls = nub'2 (x : xs) ls; " "nub'0 x xs ls True = x : nub' xs (x : ls); " "nub'1 x xs ls True = nub' xs ls; nub'1 x xs ls False = nub'0 x xs ls otherwise; " "nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls); " "nub'3 [] vy = []; nub'3 wv ww = nub'2 wv ww; " ---------------------------------------- (4) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; nub :: Eq a => [a] -> [a]; nub l = nub' l [] where { nub' [] vy = nub'3 [] vy; nub' (x : xs) ls = nub'2 (x : xs) ls; nub'0 x xs ls True = x : nub' xs (x : ls); nub'1 x xs ls True = nub' xs ls; nub'1 x xs ls False = nub'0 x xs ls otherwise; nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls); nub'3 [] vy = []; nub'3 wv ww = nub'2 wv ww; }; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (5) LetRed (EQUIVALENT) Let/Where Reductions: The bindings of the following Let/Where expression "nub' l [] where { nub' [] vy = nub'3 [] vy; nub' (x : xs) ls = nub'2 (x : xs) ls; ; nub'0 x xs ls True = x : nub' xs (x : ls); ; nub'1 x xs ls True = nub' xs ls; nub'1 x xs ls False = nub'0 x xs ls otherwise; ; nub'2 (x : xs) ls = nub'1 x xs ls (x `elem` ls); ; nub'3 [] vy = []; nub'3 wv ww = nub'2 wv ww; } " are unpacked to the following functions on top level "nubNub'0 x xs ls True = x : nubNub' xs (x : ls); " "nubNub'1 x xs ls True = nubNub' xs ls; nubNub'1 x xs ls False = nubNub'0 x xs ls otherwise; " "nubNub'3 [] vy = []; nubNub'3 wv ww = nubNub'2 wv ww; " "nubNub' [] vy = nubNub'3 [] vy; nubNub' (x : xs) ls = nubNub'2 (x : xs) ls; " "nubNub'2 (x : xs) ls = nubNub'1 x xs ls (x `elem` ls); " ---------------------------------------- (6) Obligation: mainModule Main module Maybe where { import qualified List; import qualified Main; import qualified Prelude; } module List where { import qualified Main; import qualified Maybe; import qualified Prelude; nub :: Eq a => [a] -> [a]; nub l = nubNub' l []; nubNub' [] vy = nubNub'3 [] vy; nubNub' (x : xs) ls = nubNub'2 (x : xs) ls; nubNub'0 x xs ls True = x : nubNub' xs (x : ls); nubNub'1 x xs ls True = nubNub' xs ls; nubNub'1 x xs ls False = nubNub'0 x xs ls otherwise; nubNub'2 (x : xs) ls = nubNub'1 x xs ls (x `elem` ls); nubNub'3 [] vy = []; nubNub'3 wv ww = nubNub'2 wv ww; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (7) Narrow (SOUND) Haskell To QDPs digraph dp_graph { node [outthreshold=100, inthreshold=100];1[label="List.nub",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 3[label="List.nub wx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3]; 4[label="List.nubNub' wx3 []",fontsize=16,color="burlywood",shape="box"];121[label="wx3/wx30 : wx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 121[label="",style="solid", color="burlywood", weight=9]; 121 -> 5[label="",style="solid", color="burlywood", weight=3]; 122[label="wx3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 122[label="",style="solid", color="burlywood", weight=9]; 122 -> 6[label="",style="solid", color="burlywood", weight=3]; 5[label="List.nubNub' (wx30 : wx31) []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3]; 6[label="List.nubNub' [] []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3]; 7[label="List.nubNub'2 (wx30 : wx31) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3]; 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(==))",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3]; 29[label="List.nubNub'1 wx310 wx311 (wx30 : []) (any ((==) wx310) (wx30 : []))",fontsize=16,color="black",shape="box"];29 -> 30[label="",style="solid", color="black", weight=3]; 30[label="List.nubNub'1 wx310 wx311 (wx30 : []) (or . map ((==) wx310))",fontsize=16,color="black",shape="box"];30 -> 31[label="",style="solid", color="black", weight=3]; 31[label="List.nubNub'1 wx310 wx311 (wx30 : []) (or (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];31 -> 32[label="",style="solid", color="black", weight=3]; 32[label="List.nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (map ((==) wx310) (wx30 : [])))",fontsize=16,color="black",shape="box"];32 -> 33[label="",style="solid", color="black", weight=3]; 33[label="List.nubNub'1 wx310 wx311 (wx30 : []) (foldr (||) False (((==) wx310 wx30) : map ((==) wx310) []))",fontsize=16,color="black",shape="box"];33 -> 34[label="",style="solid", color="black", weight=3]; 34[label="List.nubNub'1 wx310 wx311 (wx30 : []) ((||) (==) wx310 wx30 foldr (||) False (map ((==) wx310) []))",fontsize=16,color="burlywood",shape="box"];125[label="wx310/False",fontsize=10,color="white",style="solid",shape="box"];34 -> 125[label="",style="solid", color="burlywood", weight=9]; 125 -> 35[label="",style="solid", color="burlywood", weight=3]; 126[label="wx310/True",fontsize=10,color="white",style="solid",shape="box"];34 -> 126[label="",style="solid", color="burlywood", weight=9]; 126 -> 36[label="",style="solid", color="burlywood", weight=3]; 35[label="List.nubNub'1 False wx311 (wx30 : []) ((||) (==) False wx30 foldr (||) False (map ((==) False) []))",fontsize=16,color="burlywood",shape="box"];127[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];35 -> 127[label="",style="solid", color="burlywood", weight=9]; 127 -> 37[label="",style="solid", color="burlywood", weight=3]; 128[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];35 -> 128[label="",style="solid", color="burlywood", weight=9]; 128 -> 38[label="",style="solid", color="burlywood", weight=3]; 36[label="List.nubNub'1 True wx311 (wx30 : []) ((||) (==) True wx30 foldr (||) False (map ((==) True) []))",fontsize=16,color="burlywood",shape="box"];129[label="wx30/False",fontsize=10,color="white",style="solid",shape="box"];36 -> 129[label="",style="solid", color="burlywood", weight=9]; 129 -> 39[label="",style="solid", color="burlywood", weight=3]; 130[label="wx30/True",fontsize=10,color="white",style="solid",shape="box"];36 -> 130[label="",style="solid", color="burlywood", weight=9]; 130 -> 40[label="",style="solid", color="burlywood", weight=3]; 37[label="List.nubNub'1 False wx311 (False : []) ((||) (==) False False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];37 -> 41[label="",style="solid", color="black", weight=3]; 38[label="List.nubNub'1 False wx311 (True : []) ((||) (==) False True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];38 -> 42[label="",style="solid", color="black", weight=3]; 39[label="List.nubNub'1 True wx311 (False : []) ((||) (==) True False foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];39 -> 43[label="",style="solid", color="black", weight=3]; 40[label="List.nubNub'1 True wx311 (True : []) ((||) (==) True True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];40 -> 44[label="",style="solid", color="black", weight=3]; 41[label="List.nubNub'1 False wx311 (False : []) ((||) True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];41 -> 45[label="",style="solid", color="black", weight=3]; 42[label="List.nubNub'1 False wx311 (True : []) ((||) False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];42 -> 46[label="",style="solid", color="black", weight=3]; 43[label="List.nubNub'1 True wx311 (False : []) ((||) False foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3]; 44[label="List.nubNub'1 True wx311 (True : []) ((||) True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3]; 45[label="List.nubNub'1 False wx311 (False : []) True",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3]; 46[label="List.nubNub'1 False wx311 (True : []) (foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3]; 47[label="List.nubNub'1 True wx311 (False : []) (foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3]; 48[label="List.nubNub'1 True wx311 (True : []) True",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3]; 49 -> 21[label="",style="dashed", color="red", weight=0]; 49[label="List.nubNub' wx311 (False : [])",fontsize=16,color="magenta"];49 -> 53[label="",style="dashed", color="magenta", weight=3]; 49 -> 54[label="",style="dashed", color="magenta", weight=3]; 50[label="List.nubNub'1 False wx311 (True : []) (foldr (||) False [])",fontsize=16,color="black",shape="box"];50 -> 55[label="",style="solid", color="black", weight=3]; 51[label="List.nubNub'1 True wx311 (False : []) (foldr (||) False [])",fontsize=16,color="black",shape="box"];51 -> 56[label="",style="solid", color="black", weight=3]; 52 -> 21[label="",style="dashed", color="red", weight=0]; 52[label="List.nubNub' wx311 (True : [])",fontsize=16,color="magenta"];52 -> 57[label="",style="dashed", color="magenta", weight=3]; 52 -> 58[label="",style="dashed", color="magenta", weight=3]; 53[label="False",fontsize=16,color="green",shape="box"];54[label="wx311",fontsize=16,color="green",shape="box"];55[label="List.nubNub'1 False wx311 (True : []) False",fontsize=16,color="black",shape="box"];55 -> 59[label="",style="solid", color="black", weight=3]; 56[label="List.nubNub'1 True wx311 (False : []) False",fontsize=16,color="black",shape="box"];56 -> 60[label="",style="solid", color="black", weight=3]; 57[label="True",fontsize=16,color="green",shape="box"];58[label="wx311",fontsize=16,color="green",shape="box"];59[label="List.nubNub'0 False wx311 (True : []) otherwise",fontsize=16,color="black",shape="box"];59 -> 61[label="",style="solid", color="black", weight=3]; 60[label="List.nubNub'0 True wx311 (False : []) otherwise",fontsize=16,color="black",shape="box"];60 -> 62[label="",style="solid", color="black", weight=3]; 61[label="List.nubNub'0 False wx311 (True : []) True",fontsize=16,color="black",shape="box"];61 -> 63[label="",style="solid", color="black", weight=3]; 62[label="List.nubNub'0 True wx311 (False : []) True",fontsize=16,color="black",shape="box"];62 -> 64[label="",style="solid", color="black", weight=3]; 63[label="False : List.nubNub' wx311 (False : True : [])",fontsize=16,color="green",shape="box"];63 -> 65[label="",style="dashed", color="green", weight=3]; 64[label="True : List.nubNub' wx311 (True : False : [])",fontsize=16,color="green",shape="box"];64 -> 66[label="",style="dashed", color="green", weight=3]; 65[label="List.nubNub' wx311 (False : True : [])",fontsize=16,color="burlywood",shape="triangle"];131[label="wx311/wx3110 : wx3111",fontsize=10,color="white",style="solid",shape="box"];65 -> 131[label="",style="solid", color="burlywood", weight=9]; 131 -> 67[label="",style="solid", color="burlywood", weight=3]; 132[label="wx311/[]",fontsize=10,color="white",style="solid",shape="box"];65 -> 132[label="",style="solid", color="burlywood", weight=9]; 132 -> 68[label="",style="solid", color="burlywood", weight=3]; 66[label="List.nubNub' wx311 (True : False : [])",fontsize=16,color="burlywood",shape="triangle"];133[label="wx311/wx3110 : wx3111",fontsize=10,color="white",style="solid",shape="box"];66 -> 133[label="",style="solid", color="burlywood", weight=9]; 133 -> 69[label="",style="solid", color="burlywood", weight=3]; 134[label="wx311/[]",fontsize=10,color="white",style="solid",shape="box"];66 -> 134[label="",style="solid", color="burlywood", weight=9]; 134 -> 70[label="",style="solid", color="burlywood", weight=3]; 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74[label="List.nubNub'3 [] (True : False : [])",fontsize=16,color="black",shape="box"];74 -> 78[label="",style="solid", color="black", weight=3]; 75[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (wx3110 `elem` False : True : [])",fontsize=16,color="black",shape="box"];75 -> 79[label="",style="solid", color="black", weight=3]; 76[label="[]",fontsize=16,color="green",shape="box"];77[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (wx3110 `elem` True : False : [])",fontsize=16,color="black",shape="box"];77 -> 80[label="",style="solid", color="black", weight=3]; 78[label="[]",fontsize=16,color="green",shape="box"];79[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (any . (==))",fontsize=16,color="black",shape="box"];79 -> 81[label="",style="solid", color="black", weight=3]; 80[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (any . (==))",fontsize=16,color="black",shape="box"];80 -> 82[label="",style="solid", color="black", weight=3]; 81[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (any ((==) wx3110) (False : True : []))",fontsize=16,color="black",shape="box"];81 -> 83[label="",style="solid", color="black", weight=3]; 82[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (any ((==) wx3110) (True : False : []))",fontsize=16,color="black",shape="box"];82 -> 84[label="",style="solid", color="black", weight=3]; 83[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (or . map ((==) wx3110))",fontsize=16,color="black",shape="box"];83 -> 85[label="",style="solid", color="black", weight=3]; 84[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (or . map ((==) wx3110))",fontsize=16,color="black",shape="box"];84 -> 86[label="",style="solid", color="black", weight=3]; 85[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (or (map ((==) wx3110) (False : True : [])))",fontsize=16,color="black",shape="box"];85 -> 87[label="",style="solid", color="black", weight=3]; 86[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (or (map ((==) wx3110) (True : False : [])))",fontsize=16,color="black",shape="box"];86 -> 88[label="",style="solid", color="black", weight=3]; 87[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (foldr (||) False (map ((==) wx3110) (False : True : [])))",fontsize=16,color="black",shape="box"];87 -> 89[label="",style="solid", color="black", weight=3]; 88[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (foldr (||) False (map ((==) wx3110) (True : False : [])))",fontsize=16,color="black",shape="box"];88 -> 90[label="",style="solid", color="black", weight=3]; 89[label="List.nubNub'1 wx3110 wx3111 (False : True : []) (foldr (||) False (((==) wx3110 False) : map ((==) wx3110) (True : [])))",fontsize=16,color="black",shape="box"];89 -> 91[label="",style="solid", color="black", weight=3]; 90[label="List.nubNub'1 wx3110 wx3111 (True : False : []) (foldr (||) False (((==) wx3110 True) : map ((==) wx3110) (False : [])))",fontsize=16,color="black",shape="box"];90 -> 92[label="",style="solid", color="black", weight=3]; 91[label="List.nubNub'1 wx3110 wx3111 (False : True : []) ((||) (==) wx3110 False foldr (||) False (map ((==) wx3110) (True : [])))",fontsize=16,color="burlywood",shape="box"];135[label="wx3110/False",fontsize=10,color="white",style="solid",shape="box"];91 -> 135[label="",style="solid", color="burlywood", weight=9]; 135 -> 93[label="",style="solid", color="burlywood", weight=3]; 136[label="wx3110/True",fontsize=10,color="white",style="solid",shape="box"];91 -> 136[label="",style="solid", color="burlywood", weight=9]; 136 -> 94[label="",style="solid", color="burlywood", weight=3]; 92[label="List.nubNub'1 wx3110 wx3111 (True : False : []) ((||) (==) wx3110 True foldr (||) False (map ((==) wx3110) (False : [])))",fontsize=16,color="burlywood",shape="box"];137[label="wx3110/False",fontsize=10,color="white",style="solid",shape="box"];92 -> 137[label="",style="solid", color="burlywood", weight=9]; 137 -> 95[label="",style="solid", color="burlywood", weight=3]; 138[label="wx3110/True",fontsize=10,color="white",style="solid",shape="box"];92 -> 138[label="",style="solid", color="burlywood", weight=9]; 138 -> 96[label="",style="solid", color="burlywood", weight=3]; 93[label="List.nubNub'1 False wx3111 (False : True : []) ((||) (==) False False foldr (||) False (map ((==) False) (True : [])))",fontsize=16,color="black",shape="box"];93 -> 97[label="",style="solid", color="black", weight=3]; 94[label="List.nubNub'1 True wx3111 (False : True : []) ((||) (==) True False foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];94 -> 98[label="",style="solid", color="black", weight=3]; 95[label="List.nubNub'1 False wx3111 (True : False : []) ((||) (==) False True foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];95 -> 99[label="",style="solid", color="black", weight=3]; 96[label="List.nubNub'1 True wx3111 (True : False : []) ((||) (==) True True foldr (||) False (map ((==) True) (False : [])))",fontsize=16,color="black",shape="box"];96 -> 100[label="",style="solid", color="black", weight=3]; 97[label="List.nubNub'1 False wx3111 (False : True : []) ((||) True foldr (||) False (map ((==) False) (True : [])))",fontsize=16,color="black",shape="box"];97 -> 101[label="",style="solid", color="black", weight=3]; 98[label="List.nubNub'1 True wx3111 (False : True : []) ((||) False foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];98 -> 102[label="",style="solid", color="black", weight=3]; 99[label="List.nubNub'1 False wx3111 (True : False : []) ((||) False foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];99 -> 103[label="",style="solid", color="black", weight=3]; 100[label="List.nubNub'1 True wx3111 (True : False : []) ((||) True foldr (||) False (map ((==) True) (False : [])))",fontsize=16,color="black",shape="box"];100 -> 104[label="",style="solid", color="black", weight=3]; 101[label="List.nubNub'1 False wx3111 (False : True : []) True",fontsize=16,color="black",shape="box"];101 -> 105[label="",style="solid", color="black", weight=3]; 102[label="List.nubNub'1 True wx3111 (False : True : []) (foldr (||) False (map ((==) True) (True : [])))",fontsize=16,color="black",shape="box"];102 -> 106[label="",style="solid", color="black", weight=3]; 103[label="List.nubNub'1 False wx3111 (True : False : []) (foldr (||) False (map ((==) False) (False : [])))",fontsize=16,color="black",shape="box"];103 -> 107[label="",style="solid", color="black", weight=3]; 104[label="List.nubNub'1 True wx3111 (True : False : []) True",fontsize=16,color="black",shape="box"];104 -> 108[label="",style="solid", color="black", weight=3]; 105 -> 65[label="",style="dashed", color="red", weight=0]; 105[label="List.nubNub' wx3111 (False : True : [])",fontsize=16,color="magenta"];105 -> 109[label="",style="dashed", color="magenta", weight=3]; 106[label="List.nubNub'1 True wx3111 (False : True : []) (foldr (||) False (((==) True True) : map ((==) True) []))",fontsize=16,color="black",shape="box"];106 -> 110[label="",style="solid", color="black", weight=3]; 107[label="List.nubNub'1 False wx3111 (True : False : []) (foldr (||) False (((==) False False) : map ((==) False) []))",fontsize=16,color="black",shape="box"];107 -> 111[label="",style="solid", color="black", weight=3]; 108 -> 66[label="",style="dashed", color="red", weight=0]; 108[label="List.nubNub' wx3111 (True : False : [])",fontsize=16,color="magenta"];108 -> 112[label="",style="dashed", color="magenta", weight=3]; 109[label="wx3111",fontsize=16,color="green",shape="box"];110[label="List.nubNub'1 True wx3111 (False : True : []) ((||) (==) True True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];110 -> 113[label="",style="solid", color="black", weight=3]; 111[label="List.nubNub'1 False wx3111 (True : False : []) ((||) (==) False False foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];111 -> 114[label="",style="solid", color="black", weight=3]; 112[label="wx3111",fontsize=16,color="green",shape="box"];113[label="List.nubNub'1 True wx3111 (False : True : []) ((||) True foldr (||) False (map ((==) True) []))",fontsize=16,color="black",shape="box"];113 -> 115[label="",style="solid", color="black", weight=3]; 114[label="List.nubNub'1 False wx3111 (True : False : []) ((||) True foldr (||) False (map ((==) False) []))",fontsize=16,color="black",shape="box"];114 -> 116[label="",style="solid", color="black", weight=3]; 115[label="List.nubNub'1 True wx3111 (False : True : []) True",fontsize=16,color="black",shape="box"];115 -> 117[label="",style="solid", color="black", weight=3]; 116[label="List.nubNub'1 False wx3111 (True : False : []) True",fontsize=16,color="black",shape="box"];116 -> 118[label="",style="solid", color="black", weight=3]; 117 -> 65[label="",style="dashed", color="red", weight=0]; 117[label="List.nubNub' wx3111 (False : True : [])",fontsize=16,color="magenta"];117 -> 119[label="",style="dashed", color="magenta", weight=3]; 118 -> 66[label="",style="dashed", color="red", weight=0]; 118[label="List.nubNub' wx3111 (True : False : [])",fontsize=16,color="magenta"];118 -> 120[label="",style="dashed", color="magenta", weight=3]; 119[label="wx3111",fontsize=16,color="green",shape="box"];120[label="wx3111",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_nubNub'1(:(False, wx311), False) -> new_nubNub'1(wx311, False) new_nubNub'1(:(True, wx311), True) -> new_nubNub'1(wx311, True) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs. ---------------------------------------- (11) Complex Obligation (AND) ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_nubNub'1(:(True, wx311), True) -> new_nubNub'1(wx311, True) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (13) 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_nubNub'1(:(True, wx311), True) -> new_nubNub'1(wx311, True) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_nubNub'1(:(False, wx311), False) -> new_nubNub'1(wx311, False) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) 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_nubNub'1(:(False, wx311), False) -> new_nubNub'1(wx311, False) The graph contains the following edges 1 > 1, 1 > 2, 2 >= 2 ---------------------------------------- (17) YES ---------------------------------------- (18) Obligation: Q DP problem: The TRS P consists of the following rules: new_nubNub'0(:(False, wx3111)) -> new_nubNub'0(wx3111) new_nubNub'0(:(True, wx3111)) -> new_nubNub'0(wx3111) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (19) 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_nubNub'0(:(False, wx3111)) -> new_nubNub'0(wx3111) The graph contains the following edges 1 > 1 *new_nubNub'0(:(True, wx3111)) -> new_nubNub'0(wx3111) The graph contains the following edges 1 > 1 ---------------------------------------- (20) YES ---------------------------------------- (21) Obligation: Q DP problem: The TRS P consists of the following rules: new_nubNub'(:(False, wx3111)) -> new_nubNub'(wx3111) new_nubNub'(:(True, wx3111)) -> new_nubNub'(wx3111) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (22) 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_nubNub'(:(False, wx3111)) -> new_nubNub'(wx3111) The graph contains the following edges 1 > 1 *new_nubNub'(:(True, wx3111)) -> new_nubNub'(wx3111) The graph contains the following edges 1 > 1 ---------------------------------------- (23) YES