/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty H-Termination with start terms of the given HASKELL could be proven: (0) HASKELL (1) CR [EQUIVALENT, 0 ms] (2) HASKELL (3) BR [EQUIVALENT, 0 ms] (4) HASKELL (5) COR [EQUIVALENT, 20 ms] (6) HASKELL (7) Narrow [SOUND, 0 ms] (8) AND (9) QDP (10) QDPSizeChangeProof [EQUIVALENT, 0 ms] (11) YES (12) QDP (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] (14) YES (15) QDP (16) DependencyGraphProof [EQUIVALENT, 0 ms] (17) QDP (18) QDPSizeChangeProof [EQUIVALENT, 92 ms] (19) YES (20) QDP (21) QDPSizeChangeProof [EQUIVALENT, 0 ms] (22) YES (23) QDP (24) QDPSizeChangeProof [EQUIVALENT, 0 ms] (25) YES (26) QDP (27) QDPOrderProof [EQUIVALENT, 66 ms] (28) QDP (29) DependencyGraphProof [EQUIVALENT, 0 ms] (30) TRUE (31) QDP (32) QDPSizeChangeProof [EQUIVALENT, 0 ms] (33) 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; merge :: (a -> a -> Ordering) -> [a] -> [a] -> [a]; merge cmp xs [] = xs; merge cmp [] ys = ys; merge cmp (x : xs) (y : ys) = case x `cmp` y of { GT-> y : merge cmp (x : xs) ys; _-> x : merge cmp xs (y : ys); } ; merge_pairs :: (a -> a -> Ordering) -> [[a]] -> [[a]]; merge_pairs cmp [] = []; merge_pairs cmp (xs : []) = xs : []; merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss; mergesort :: (a -> a -> Ordering) -> [a] -> [a]; mergesort cmp = mergesort' cmp . map wrap; mergesort' :: (a -> a -> Ordering) -> [[a]] -> [a]; mergesort' cmp [] = []; mergesort' cmp (xs : []) = xs; mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss); sort :: Ord a => [a] -> [a]; sort l = mergesort compare l; wrap :: a -> [a]; wrap x = x : []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (1) CR (EQUIVALENT) Case Reductions: The following Case expression "case cmp x y of { GT -> y : merge cmp (x : xs) ys; _ -> x : merge cmp xs (y : ys)} " is transformed to "merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys; merge0 y cmp x xs ys _ = x : merge cmp xs (y : ys); " ---------------------------------------- (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; merge :: (a -> a -> Ordering) -> [a] -> [a] -> [a]; merge cmp xs [] = xs; merge cmp [] ys = ys; merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y); merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys; merge0 y cmp x xs ys _ = x : merge cmp xs (y : ys); merge_pairs :: (a -> a -> Ordering) -> [[a]] -> [[a]]; merge_pairs cmp [] = []; merge_pairs cmp (xs : []) = xs : []; merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss; mergesort :: (a -> a -> Ordering) -> [a] -> [a]; mergesort cmp = mergesort' cmp . map wrap; mergesort' :: (a -> a -> Ordering) -> [[a]] -> [a]; mergesort' cmp [] = []; mergesort' cmp (xs : []) = xs; mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss); sort :: Ord a => [a] -> [a]; sort l = mergesort compare l; wrap :: a -> [a]; wrap x = x : []; } module Main where { import qualified List; import qualified Maybe; import qualified Prelude; } ---------------------------------------- (3) BR (EQUIVALENT) Replaced joker patterns by fresh variables and removed binding patterns. ---------------------------------------- (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; merge :: (a -> a -> Ordering) -> [a] -> [a] -> [a]; merge cmp xs [] = xs; merge cmp [] ys = ys; merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y); merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys; merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys); merge_pairs :: (a -> a -> Ordering) -> [[a]] -> [[a]]; merge_pairs cmp [] = []; merge_pairs cmp (xs : []) = xs : []; merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss; mergesort :: (a -> a -> Ordering) -> [a] -> [a]; mergesort cmp = mergesort' cmp . map wrap; mergesort' :: (a -> a -> Ordering) -> [[a]] -> [a]; mergesort' cmp [] = []; mergesort' cmp (xs : []) = xs; mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss); sort :: Ord a => [a] -> [a]; sort l = mergesort compare l; wrap :: a -> [a]; wrap x = x : []; } module Main where { import qualified List; import qualified Maybe; 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; " ---------------------------------------- (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; merge :: (a -> a -> Ordering) -> [a] -> [a] -> [a]; merge cmp xs [] = xs; merge cmp [] ys = ys; merge cmp (x : xs) (y : ys) = merge0 y cmp x xs ys (x `cmp` y); merge0 y cmp x xs ys GT = y : merge cmp (x : xs) ys; merge0 y cmp x xs ys vy = x : merge cmp xs (y : ys); merge_pairs :: (a -> a -> Ordering) -> [[a]] -> [[a]]; merge_pairs cmp [] = []; merge_pairs cmp (xs : []) = xs : []; merge_pairs cmp (xs : ys : xss) = merge cmp xs ys : merge_pairs cmp xss; mergesort :: (a -> a -> Ordering) -> [a] -> [a]; mergesort cmp = mergesort' cmp . map wrap; mergesort' :: (a -> a -> Ordering) -> [[a]] -> [a]; mergesort' cmp [] = []; mergesort' cmp (xs : []) = xs; mergesort' cmp xss = mergesort' cmp (merge_pairs cmp xss); sort :: Ord a => [a] -> [a]; sort l = mergesort compare l; wrap :: a -> [a]; wrap x = x : []; } 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.sort",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 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16238[label="vz25610/[]",fontsize=10,color="white",style="solid",shape="box"];15916 -> 16238[label="",style="solid", color="burlywood", weight=9]; 16238 -> 15924[label="",style="solid", color="burlywood", weight=3]; 15917[label="[]",fontsize=16,color="green",shape="box"];15918[label="List.merge compare vz255 vz2560",fontsize=16,color="burlywood",shape="triangle"];16239[label="vz2560/vz25600 : vz25601",fontsize=10,color="white",style="solid",shape="box"];15918 -> 16239[label="",style="solid", color="burlywood", weight=9]; 16239 -> 15925[label="",style="solid", color="burlywood", weight=3]; 16240[label="vz2560/[]",fontsize=10,color="white",style="solid",shape="box"];15918 -> 16240[label="",style="solid", color="burlywood", weight=9]; 16240 -> 15926[label="",style="solid", color="burlywood", weight=3]; 16007[label="primCmpFloat (Float vz25800 vz25801) vz2570",fontsize=16,color="burlywood",shape="box"];16241[label="vz25801/Pos vz258010",fontsize=10,color="white",style="solid",shape="box"];16007 -> 16241[label="",style="solid", color="burlywood", weight=9]; 16241 -> 16054[label="",style="solid", color="burlywood", weight=3]; 16242[label="vz25801/Neg vz258010",fontsize=10,color="white",style="solid",shape="box"];16007 -> 16242[label="",style="solid", color="burlywood", weight=9]; 16242 -> 16055[label="",style="solid", color="burlywood", weight=3]; 16008[label="vz266 : List.merge compare vz267 (vz265 : vz268)",fontsize=16,color="green",shape="box"];16008 -> 16056[label="",style="dashed", color="green", weight=3]; 16009[label="vz266 : List.merge compare vz267 (vz265 : vz268)",fontsize=16,color="green",shape="box"];16009 -> 16057[label="",style="dashed", color="green", weight=3]; 16010[label="vz265 : List.merge compare (vz266 : vz267) vz268",fontsize=16,color="green",shape="box"];16010 -> 16058[label="",style="dashed", color="green", weight=3]; 15921[label="List.merge_pairs compare (vz256110 : vz256111)",fontsize=16,color="burlywood",shape="box"];16243[label="vz256111/vz2561110 : vz2561111",fontsize=10,color="white",style="solid",shape="box"];15921 -> 16243[label="",style="solid", color="burlywood", weight=9]; 16243 -> 15929[label="",style="solid", color="burlywood", weight=3]; 16244[label="vz256111/[]",fontsize=10,color="white",style="solid",shape="box"];15921 -> 16244[label="",style="solid", color="burlywood", weight=9]; 16244 -> 15930[label="",style="solid", color="burlywood", weight=3]; 15922[label="List.merge_pairs compare []",fontsize=16,color="black",shape="box"];15922 -> 15931[label="",style="solid", color="black", weight=3]; 15923[label="List.merge compare vz255 (List.merge compare vz2560 (vz256100 : vz256101))",fontsize=16,color="burlywood",shape="box"];16245[label="vz2560/vz25600 : vz25601",fontsize=10,color="white",style="solid",shape="box"];15923 -> 16245[label="",style="solid", color="burlywood", weight=9]; 16245 -> 15932[label="",style="solid", color="burlywood", weight=3]; 16246[label="vz2560/[]",fontsize=10,color="white",style="solid",shape="box"];15923 -> 16246[label="",style="solid", color="burlywood", weight=9]; 16246 -> 15933[label="",style="solid", color="burlywood", weight=3]; 15924[label="List.merge compare vz255 (List.merge compare vz2560 [])",fontsize=16,color="black",shape="box"];15924 -> 15934[label="",style="solid", color="black", weight=3]; 15925[label="List.merge compare vz255 (vz25600 : vz25601)",fontsize=16,color="burlywood",shape="box"];16247[label="vz255/vz2550 : vz2551",fontsize=10,color="white",style="solid",shape="box"];15925 -> 16247[label="",style="solid", color="burlywood", weight=9]; 16247 -> 15935[label="",style="solid", color="burlywood", weight=3]; 16248[label="vz255/[]",fontsize=10,color="white",style="solid",shape="box"];15925 -> 16248[label="",style="solid", color="burlywood", weight=9]; 16248 -> 15936[label="",style="solid", color="burlywood", weight=3]; 15926[label="List.merge compare vz255 []",fontsize=16,color="black",shape="box"];15926 -> 15937[label="",style="solid", color="black", weight=3]; 16054[label="primCmpFloat (Float vz25800 (Pos vz258010)) vz2570",fontsize=16,color="burlywood",shape="box"];16249[label="vz2570/Float vz25700 vz25701",fontsize=10,color="white",style="solid",shape="box"];16054 -> 16249[label="",style="solid", color="burlywood", weight=9]; 16249 -> 16061[label="",style="solid", color="burlywood", weight=3]; 16055[label="primCmpFloat (Float vz25800 (Neg vz258010)) vz2570",fontsize=16,color="burlywood",shape="box"];16250[label="vz2570/Float vz25700 vz25701",fontsize=10,color="white",style="solid",shape="box"];16055 -> 16250[label="",style="solid", color="burlywood", weight=9]; 16250 -> 16062[label="",style="solid", color="burlywood", weight=3]; 16056 -> 15918[label="",style="dashed", color="red", weight=0]; 16056[label="List.merge compare vz267 (vz265 : vz268)",fontsize=16,color="magenta"];16056 -> 16063[label="",style="dashed", color="magenta", weight=3]; 16056 -> 16064[label="",style="dashed", color="magenta", weight=3]; 16057 -> 15918[label="",style="dashed", color="red", weight=0]; 16057[label="List.merge compare vz267 (vz265 : vz268)",fontsize=16,color="magenta"];16057 -> 16065[label="",style="dashed", color="magenta", weight=3]; 16057 -> 16066[label="",style="dashed", color="magenta", weight=3]; 16058 -> 15918[label="",style="dashed", color="red", weight=0]; 16058[label="List.merge compare (vz266 : vz267) vz268",fontsize=16,color="magenta"];16058 -> 16067[label="",style="dashed", color="magenta", weight=3]; 16058 -> 16068[label="",style="dashed", color="magenta", weight=3]; 15929[label="List.merge_pairs compare (vz256110 : vz2561110 : vz2561111)",fontsize=16,color="black",shape="box"];15929 -> 15942[label="",style="solid", color="black", weight=3]; 15930[label="List.merge_pairs compare (vz256110 : [])",fontsize=16,color="black",shape="box"];15930 -> 15943[label="",style="solid", color="black", weight=3]; 15931[label="[]",fontsize=16,color="green",shape="box"];15932[label="List.merge compare vz255 (List.merge compare (vz25600 : vz25601) (vz256100 : vz256101))",fontsize=16,color="black",shape="box"];15932 -> 15944[label="",style="solid", color="black", weight=3]; 15933[label="List.merge compare vz255 (List.merge compare [] (vz256100 : vz256101))",fontsize=16,color="black",shape="box"];15933 -> 15945[label="",style="solid", color="black", weight=3]; 15934 -> 15918[label="",style="dashed", color="red", weight=0]; 15934[label="List.merge compare vz255 vz2560",fontsize=16,color="magenta"];15935[label="List.merge compare (vz2550 : vz2551) (vz25600 : vz25601)",fontsize=16,color="black",shape="box"];15935 -> 15946[label="",style="solid", color="black", weight=3]; 15936[label="List.merge compare [] (vz25600 : vz25601)",fontsize=16,color="black",shape="box"];15936 -> 15947[label="",style="solid", color="black", weight=3]; 15937[label="vz255",fontsize=16,color="green",shape="box"];16061[label="primCmpFloat (Float vz25800 (Pos vz258010)) (Float vz25700 vz25701)",fontsize=16,color="burlywood",shape="box"];16251[label="vz25701/Pos vz257010",fontsize=10,color="white",style="solid",shape="box"];16061 -> 16251[label="",style="solid", color="burlywood", weight=9]; 16251 -> 16073[label="",style="solid", color="burlywood", weight=3]; 16252[label="vz25701/Neg vz257010",fontsize=10,color="white",style="solid",shape="box"];16061 -> 16252[label="",style="solid", color="burlywood", weight=9]; 16252 -> 16074[label="",style="solid", color="burlywood", weight=3]; 16062[label="primCmpFloat (Float vz25800 (Neg vz258010)) (Float vz25700 vz25701)",fontsize=16,color="burlywood",shape="box"];16253[label="vz25701/Pos vz257010",fontsize=10,color="white",style="solid",shape="box"];16062 -> 16253[label="",style="solid", color="burlywood", weight=9]; 16253 -> 16075[label="",style="solid", color="burlywood", weight=3]; 16254[label="vz25701/Neg vz257010",fontsize=10,color="white",style="solid",shape="box"];16062 -> 16254[label="",style="solid", color="burlywood", weight=9]; 16254 -> 16076[label="",style="solid", color="burlywood", weight=3]; 16063[label="vz267",fontsize=16,color="green",shape="box"];16064[label="vz265 : vz268",fontsize=16,color="green",shape="box"];16065[label="vz267",fontsize=16,color="green",shape="box"];16066[label="vz265 : vz268",fontsize=16,color="green",shape="box"];16067[label="vz266 : vz267",fontsize=16,color="green",shape="box"];16068[label="vz268",fontsize=16,color="green",shape="box"];15942[label="List.merge compare vz256110 vz2561110 : List.merge_pairs compare vz2561111",fontsize=16,color="green",shape="box"];15942 -> 15952[label="",style="dashed", color="green", weight=3]; 15942 -> 15953[label="",style="dashed", color="green", weight=3]; 15943[label="vz256110 : []",fontsize=16,color="green",shape="box"];15944 -> 15918[label="",style="dashed", color="red", weight=0]; 15944[label="List.merge compare vz255 (List.merge0 vz256100 compare vz25600 vz25601 vz256101 (compare vz25600 vz256100))",fontsize=16,color="magenta"];15944 -> 15954[label="",style="dashed", color="magenta", weight=3]; 15945 -> 15918[label="",style="dashed", color="red", weight=0]; 15945[label="List.merge compare vz255 (vz256100 : vz256101)",fontsize=16,color="magenta"];15945 -> 15955[label="",style="dashed", color="magenta", weight=3]; 15946 -> 15956[label="",style="dashed", color="red", weight=0]; 15946[label="List.merge0 vz25600 compare vz2550 vz2551 vz25601 (compare vz2550 vz25600)",fontsize=16,color="magenta"];15946 -> 15962[label="",style="dashed", color="magenta", weight=3]; 15946 -> 15963[label="",style="dashed", color="magenta", weight=3]; 15946 -> 15964[label="",style="dashed", color="magenta", weight=3]; 15946 -> 15965[label="",style="dashed", color="magenta", weight=3]; 15946 -> 15966[label="",style="dashed", color="magenta", weight=3]; 15947[label="vz25600 : vz25601",fontsize=16,color="green",shape="box"];16073[label="primCmpFloat (Float vz25800 (Pos vz258010)) (Float vz25700 (Pos vz257010))",fontsize=16,color="black",shape="box"];16073 -> 16085[label="",style="solid", color="black", weight=3]; 16074[label="primCmpFloat (Float vz25800 (Pos vz258010)) (Float vz25700 (Neg vz257010))",fontsize=16,color="black",shape="box"];16074 -> 16086[label="",style="solid", color="black", weight=3]; 16075[label="primCmpFloat (Float vz25800 (Neg vz258010)) (Float vz25700 (Pos vz257010))",fontsize=16,color="black",shape="box"];16075 -> 16087[label="",style="solid", color="black", weight=3]; 16076[label="primCmpFloat (Float vz25800 (Neg vz258010)) (Float vz25700 (Neg vz257010))",fontsize=16,color="black",shape="box"];16076 -> 16088[label="",style="solid", color="black", weight=3]; 15952 -> 15918[label="",style="dashed", color="red", weight=0]; 15952[label="List.merge compare vz256110 vz2561110",fontsize=16,color="magenta"];15952 -> 15976[label="",style="dashed", color="magenta", weight=3]; 15952 -> 15977[label="",style="dashed", color="magenta", weight=3]; 15953 -> 15915[label="",style="dashed", color="red", weight=0]; 15953[label="List.merge_pairs compare vz2561111",fontsize=16,color="magenta"];15953 -> 15978[label="",style="dashed", color="magenta", weight=3]; 15954 -> 15956[label="",style="dashed", color="red", weight=0]; 15954[label="List.merge0 vz256100 compare vz25600 vz25601 vz256101 (compare vz25600 vz256100)",fontsize=16,color="magenta"];15954 -> 15967[label="",style="dashed", color="magenta", weight=3]; 15954 -> 15968[label="",style="dashed", color="magenta", weight=3]; 15954 -> 15969[label="",style="dashed", color="magenta", weight=3]; 15954 -> 15970[label="",style="dashed", color="magenta", weight=3]; 15954 -> 15971[label="",style="dashed", color="magenta", weight=3]; 15955[label="vz256100 : vz256101",fontsize=16,color="green",shape="box"];15962[label="vz25601",fontsize=16,color="green",shape="box"];15963[label="vz25600",fontsize=16,color="green",shape="box"];15964[label="vz2550",fontsize=16,color="green",shape="box"];15965[label="vz2551",fontsize=16,color="green",shape="box"];15966[label="compare vz2550 vz25600",fontsize=16,color="blue",shape="box"];16255[label="compare :: () -> () -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16255[label="",style="solid", color="blue", weight=9]; 16255 -> 15979[label="",style="solid", color="blue", weight=3]; 16256[label="compare :: ((@3) a b c) -> ((@3) a b c) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16256[label="",style="solid", color="blue", weight=9]; 16256 -> 15980[label="",style="solid", color="blue", weight=3]; 16257[label="compare :: Int -> Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16257[label="",style="solid", color="blue", weight=9]; 16257 -> 15981[label="",style="solid", color="blue", weight=3]; 16258[label="compare :: Float -> Float -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16258[label="",style="solid", color="blue", weight=9]; 16258 -> 15982[label="",style="solid", color="blue", weight=3]; 16259[label="compare :: (Either a b) -> (Either a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16259[label="",style="solid", color="blue", weight=9]; 16259 -> 15983[label="",style="solid", color="blue", weight=3]; 16260[label="compare :: Integer -> Integer -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16260[label="",style="solid", color="blue", weight=9]; 16260 -> 15984[label="",style="solid", color="blue", weight=3]; 16261[label="compare :: ([] a) -> ([] a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16261[label="",style="solid", color="blue", weight=9]; 16261 -> 15985[label="",style="solid", color="blue", weight=3]; 16262[label="compare :: Bool -> Bool -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16262[label="",style="solid", color="blue", weight=9]; 16262 -> 15986[label="",style="solid", color="blue", weight=3]; 16263[label="compare :: Char -> Char -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16263[label="",style="solid", color="blue", weight=9]; 16263 -> 15987[label="",style="solid", color="blue", weight=3]; 16264[label="compare :: Ordering -> Ordering -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16264[label="",style="solid", color="blue", weight=9]; 16264 -> 15988[label="",style="solid", color="blue", weight=3]; 16265[label="compare :: (Ratio a) -> (Ratio a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16265[label="",style="solid", color="blue", weight=9]; 16265 -> 15989[label="",style="solid", color="blue", weight=3]; 16266[label="compare :: Double -> Double -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16266[label="",style="solid", color="blue", weight=9]; 16266 -> 15990[label="",style="solid", color="blue", weight=3]; 16267[label="compare :: ((@2) a b) -> ((@2) a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16267[label="",style="solid", color="blue", weight=9]; 16267 -> 15991[label="",style="solid", color="blue", weight=3]; 16268[label="compare :: (Maybe a) -> (Maybe a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15966 -> 16268[label="",style="solid", color="blue", weight=9]; 16268 -> 15992[label="",style="solid", color="blue", weight=3]; 16085 -> 15981[label="",style="dashed", color="red", weight=0]; 16085[label="compare (vz25800 * Pos vz257010) (Pos vz258010 * vz25700)",fontsize=16,color="magenta"];16085 -> 16101[label="",style="dashed", color="magenta", weight=3]; 16085 -> 16102[label="",style="dashed", color="magenta", weight=3]; 16086 -> 15981[label="",style="dashed", color="red", weight=0]; 16086[label="compare (vz25800 * Pos vz257010) (Neg vz258010 * vz25700)",fontsize=16,color="magenta"];16086 -> 16103[label="",style="dashed", color="magenta", weight=3]; 16086 -> 16104[label="",style="dashed", color="magenta", weight=3]; 16087 -> 15981[label="",style="dashed", color="red", weight=0]; 16087[label="compare (vz25800 * Neg vz257010) (Pos vz258010 * vz25700)",fontsize=16,color="magenta"];16087 -> 16105[label="",style="dashed", color="magenta", weight=3]; 16087 -> 16106[label="",style="dashed", color="magenta", weight=3]; 16088 -> 15981[label="",style="dashed", color="red", weight=0]; 16088[label="compare (vz25800 * Neg vz257010) (Neg vz258010 * vz25700)",fontsize=16,color="magenta"];16088 -> 16107[label="",style="dashed", color="magenta", weight=3]; 16088 -> 16108[label="",style="dashed", color="magenta", weight=3]; 15976[label="vz256110",fontsize=16,color="green",shape="box"];15977[label="vz2561110",fontsize=16,color="green",shape="box"];15978[label="vz2561111",fontsize=16,color="green",shape="box"];15967[label="vz256101",fontsize=16,color="green",shape="box"];15968[label="vz256100",fontsize=16,color="green",shape="box"];15969[label="vz25600",fontsize=16,color="green",shape="box"];15970[label="vz25601",fontsize=16,color="green",shape="box"];15971[label="compare vz25600 vz256100",fontsize=16,color="blue",shape="box"];16269[label="compare :: () -> () -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16269[label="",style="solid", color="blue", weight=9]; 16269 -> 15993[label="",style="solid", color="blue", weight=3]; 16270[label="compare :: ((@3) a b c) -> ((@3) a b c) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16270[label="",style="solid", color="blue", weight=9]; 16270 -> 15994[label="",style="solid", color="blue", weight=3]; 16271[label="compare :: Int -> Int -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16271[label="",style="solid", color="blue", weight=9]; 16271 -> 15995[label="",style="solid", color="blue", weight=3]; 16272[label="compare :: Float -> Float -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16272[label="",style="solid", color="blue", weight=9]; 16272 -> 15996[label="",style="solid", color="blue", weight=3]; 16273[label="compare :: (Either a b) -> (Either a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16273[label="",style="solid", color="blue", weight=9]; 16273 -> 15997[label="",style="solid", color="blue", weight=3]; 16274[label="compare :: Integer -> Integer -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16274[label="",style="solid", color="blue", weight=9]; 16274 -> 15998[label="",style="solid", color="blue", weight=3]; 16275[label="compare :: ([] a) -> ([] a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16275[label="",style="solid", color="blue", weight=9]; 16275 -> 15999[label="",style="solid", color="blue", weight=3]; 16276[label="compare :: Bool -> Bool -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16276[label="",style="solid", color="blue", weight=9]; 16276 -> 16000[label="",style="solid", color="blue", weight=3]; 16277[label="compare :: Char -> Char -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16277[label="",style="solid", color="blue", weight=9]; 16277 -> 16001[label="",style="solid", color="blue", weight=3]; 16278[label="compare :: Ordering -> Ordering -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16278[label="",style="solid", color="blue", weight=9]; 16278 -> 16002[label="",style="solid", color="blue", weight=3]; 16279[label="compare :: (Ratio a) -> (Ratio a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16279[label="",style="solid", color="blue", weight=9]; 16279 -> 16003[label="",style="solid", color="blue", weight=3]; 16280[label="compare :: Double -> Double -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16280[label="",style="solid", color="blue", weight=9]; 16280 -> 16004[label="",style="solid", color="blue", weight=3]; 16281[label="compare :: ((@2) a b) -> ((@2) a b) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16281[label="",style="solid", color="blue", weight=9]; 16281 -> 16005[label="",style="solid", color="blue", weight=3]; 16282[label="compare :: (Maybe a) -> (Maybe a) -> Ordering",fontsize=10,color="white",style="solid",shape="box"];15971 -> 16282[label="",style="solid", color="blue", weight=9]; 16282 -> 16006[label="",style="solid", color="blue", weight=3]; 15979[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15979 -> 16011[label="",style="solid", color="black", weight=3]; 15980[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15980 -> 16012[label="",style="solid", color="black", weight=3]; 15981[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15981 -> 16013[label="",style="solid", color="black", weight=3]; 15982 -> 15961[label="",style="dashed", color="red", weight=0]; 15982[label="compare vz2550 vz25600",fontsize=16,color="magenta"];15982 -> 16014[label="",style="dashed", color="magenta", weight=3]; 15982 -> 16015[label="",style="dashed", color="magenta", weight=3]; 15983[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15983 -> 16016[label="",style="solid", color="black", weight=3]; 15984[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15984 -> 16017[label="",style="solid", color="black", weight=3]; 15985[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15985 -> 16018[label="",style="solid", color="black", weight=3]; 15986[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15986 -> 16019[label="",style="solid", color="black", weight=3]; 15987[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15987 -> 16020[label="",style="solid", color="black", weight=3]; 15988[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15988 -> 16021[label="",style="solid", color="black", weight=3]; 15989[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15989 -> 16022[label="",style="solid", color="black", weight=3]; 15990[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15990 -> 16023[label="",style="solid", color="black", weight=3]; 15991[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15991 -> 16024[label="",style="solid", color="black", weight=3]; 15992[label="compare vz2550 vz25600",fontsize=16,color="black",shape="triangle"];15992 -> 16025[label="",style="solid", color="black", weight=3]; 16101[label="vz25800 * Pos vz257010",fontsize=16,color="black",shape="triangle"];16101 -> 16121[label="",style="solid", color="black", weight=3]; 16102[label="Pos vz258010 * vz25700",fontsize=16,color="black",shape="triangle"];16102 -> 16122[label="",style="solid", color="black", weight=3]; 16103 -> 16101[label="",style="dashed", color="red", weight=0]; 16103[label="vz25800 * Pos vz257010",fontsize=16,color="magenta"];16103 -> 16123[label="",style="dashed", color="magenta", weight=3]; 16104[label="Neg vz258010 * vz25700",fontsize=16,color="black",shape="triangle"];16104 -> 16124[label="",style="solid", color="black", weight=3]; 16105[label="vz25800 * Neg vz257010",fontsize=16,color="black",shape="triangle"];16105 -> 16125[label="",style="solid", color="black", weight=3]; 16106 -> 16102[label="",style="dashed", color="red", weight=0]; 16106[label="Pos vz258010 * vz25700",fontsize=16,color="magenta"];16106 -> 16126[label="",style="dashed", color="magenta", weight=3]; 16107 -> 16105[label="",style="dashed", color="red", weight=0]; 16107[label="vz25800 * Neg vz257010",fontsize=16,color="magenta"];16107 -> 16127[label="",style="dashed", color="magenta", weight=3]; 16108 -> 16104[label="",style="dashed", color="red", weight=0]; 16108[label="Neg vz258010 * vz25700",fontsize=16,color="magenta"];16108 -> 16128[label="",style="dashed", color="magenta", weight=3]; 15993 -> 15979[label="",style="dashed", color="red", weight=0]; 15993[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15993 -> 16026[label="",style="dashed", color="magenta", weight=3]; 15993 -> 16027[label="",style="dashed", color="magenta", weight=3]; 15994 -> 15980[label="",style="dashed", color="red", weight=0]; 15994[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15994 -> 16028[label="",style="dashed", color="magenta", weight=3]; 15994 -> 16029[label="",style="dashed", color="magenta", weight=3]; 15995 -> 15981[label="",style="dashed", color="red", weight=0]; 15995[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15995 -> 16030[label="",style="dashed", color="magenta", weight=3]; 15995 -> 16031[label="",style="dashed", color="magenta", weight=3]; 15996 -> 15961[label="",style="dashed", color="red", weight=0]; 15996[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15996 -> 16032[label="",style="dashed", color="magenta", weight=3]; 15996 -> 16033[label="",style="dashed", color="magenta", weight=3]; 15997 -> 15983[label="",style="dashed", color="red", weight=0]; 15997[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15997 -> 16034[label="",style="dashed", color="magenta", weight=3]; 15997 -> 16035[label="",style="dashed", color="magenta", weight=3]; 15998 -> 15984[label="",style="dashed", color="red", weight=0]; 15998[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15998 -> 16036[label="",style="dashed", color="magenta", weight=3]; 15998 -> 16037[label="",style="dashed", color="magenta", weight=3]; 15999 -> 15985[label="",style="dashed", color="red", weight=0]; 15999[label="compare vz25600 vz256100",fontsize=16,color="magenta"];15999 -> 16038[label="",style="dashed", color="magenta", weight=3]; 15999 -> 16039[label="",style="dashed", color="magenta", weight=3]; 16000 -> 15986[label="",style="dashed", color="red", weight=0]; 16000[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16000 -> 16040[label="",style="dashed", color="magenta", weight=3]; 16000 -> 16041[label="",style="dashed", color="magenta", weight=3]; 16001 -> 15987[label="",style="dashed", color="red", weight=0]; 16001[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16001 -> 16042[label="",style="dashed", color="magenta", weight=3]; 16001 -> 16043[label="",style="dashed", color="magenta", weight=3]; 16002 -> 15988[label="",style="dashed", color="red", weight=0]; 16002[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16002 -> 16044[label="",style="dashed", color="magenta", weight=3]; 16002 -> 16045[label="",style="dashed", color="magenta", weight=3]; 16003 -> 15989[label="",style="dashed", color="red", weight=0]; 16003[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16003 -> 16046[label="",style="dashed", color="magenta", weight=3]; 16003 -> 16047[label="",style="dashed", color="magenta", weight=3]; 16004 -> 15990[label="",style="dashed", color="red", weight=0]; 16004[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16004 -> 16048[label="",style="dashed", color="magenta", weight=3]; 16004 -> 16049[label="",style="dashed", color="magenta", weight=3]; 16005 -> 15991[label="",style="dashed", color="red", weight=0]; 16005[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16005 -> 16050[label="",style="dashed", color="magenta", weight=3]; 16005 -> 16051[label="",style="dashed", color="magenta", weight=3]; 16006 -> 15992[label="",style="dashed", color="red", weight=0]; 16006[label="compare vz25600 vz256100",fontsize=16,color="magenta"];16006 -> 16052[label="",style="dashed", color="magenta", weight=3]; 16006 -> 16053[label="",style="dashed", color="magenta", weight=3]; 16011[label="error []",fontsize=16,color="red",shape="box"];16012[label="error []",fontsize=16,color="red",shape="box"];16013[label="primCmpInt vz2550 vz25600",fontsize=16,color="burlywood",shape="box"];16283[label="vz2550/Pos vz25500",fontsize=10,color="white",style="solid",shape="box"];16013 -> 16283[label="",style="solid", color="burlywood", weight=9]; 16283 -> 16059[label="",style="solid", color="burlywood", weight=3]; 16284[label="vz2550/Neg vz25500",fontsize=10,color="white",style="solid",shape="box"];16013 -> 16284[label="",style="solid", color="burlywood", weight=9]; 16284 -> 16060[label="",style="solid", color="burlywood", weight=3]; 16014[label="vz2550",fontsize=16,color="green",shape="box"];16015[label="vz25600",fontsize=16,color="green",shape="box"];16016[label="error []",fontsize=16,color="red",shape="box"];16017[label="error []",fontsize=16,color="red",shape="box"];16018[label="error []",fontsize=16,color="red",shape="box"];16019[label="error []",fontsize=16,color="red",shape="box"];16020[label="error []",fontsize=16,color="red",shape="box"];16021[label="error []",fontsize=16,color="red",shape="box"];16022[label="error []",fontsize=16,color="red",shape="box"];16023[label="error []",fontsize=16,color="red",shape="box"];16024[label="error []",fontsize=16,color="red",shape="box"];16025[label="error []",fontsize=16,color="red",shape="box"];16121[label="primMulInt vz25800 (Pos vz257010)",fontsize=16,color="burlywood",shape="box"];16285[label="vz25800/Pos vz258000",fontsize=10,color="white",style="solid",shape="box"];16121 -> 16285[label="",style="solid", color="burlywood", weight=9]; 16285 -> 16137[label="",style="solid", color="burlywood", weight=3]; 16286[label="vz25800/Neg vz258000",fontsize=10,color="white",style="solid",shape="box"];16121 -> 16286[label="",style="solid", color="burlywood", weight=9]; 16286 -> 16138[label="",style="solid", color="burlywood", weight=3]; 16122[label="primMulInt (Pos vz258010) vz25700",fontsize=16,color="burlywood",shape="box"];16287[label="vz25700/Pos vz257000",fontsize=10,color="white",style="solid",shape="box"];16122 -> 16287[label="",style="solid", color="burlywood", weight=9]; 16287 -> 16139[label="",style="solid", color="burlywood", weight=3]; 16288[label="vz25700/Neg vz257000",fontsize=10,color="white",style="solid",shape="box"];16122 -> 16288[label="",style="solid", color="burlywood", weight=9]; 16288 -> 16140[label="",style="solid", color="burlywood", weight=3]; 16123[label="vz257010",fontsize=16,color="green",shape="box"];16124[label="primMulInt (Neg vz258010) vz25700",fontsize=16,color="burlywood",shape="box"];16289[label="vz25700/Pos vz257000",fontsize=10,color="white",style="solid",shape="box"];16124 -> 16289[label="",style="solid", color="burlywood", weight=9]; 16289 -> 16141[label="",style="solid", color="burlywood", weight=3]; 16290[label="vz25700/Neg vz257000",fontsize=10,color="white",style="solid",shape="box"];16124 -> 16290[label="",style="solid", color="burlywood", weight=9]; 16290 -> 16142[label="",style="solid", color="burlywood", weight=3]; 16125[label="primMulInt vz25800 (Neg vz257010)",fontsize=16,color="burlywood",shape="box"];16291[label="vz25800/Pos vz258000",fontsize=10,color="white",style="solid",shape="box"];16125 -> 16291[label="",style="solid", color="burlywood", weight=9]; 16291 -> 16143[label="",style="solid", color="burlywood", weight=3]; 16292[label="vz25800/Neg vz258000",fontsize=10,color="white",style="solid",shape="box"];16125 -> 16292[label="",style="solid", color="burlywood", weight=9]; 16292 -> 16144[label="",style="solid", color="burlywood", weight=3]; 16126[label="vz258010",fontsize=16,color="green",shape="box"];16127[label="vz257010",fontsize=16,color="green",shape="box"];16128[label="vz258010",fontsize=16,color="green",shape="box"];16026[label="vz25600",fontsize=16,color="green",shape="box"];16027[label="vz256100",fontsize=16,color="green",shape="box"];16028[label="vz25600",fontsize=16,color="green",shape="box"];16029[label="vz256100",fontsize=16,color="green",shape="box"];16030[label="vz25600",fontsize=16,color="green",shape="box"];16031[label="vz256100",fontsize=16,color="green",shape="box"];16032[label="vz25600",fontsize=16,color="green",shape="box"];16033[label="vz256100",fontsize=16,color="green",shape="box"];16034[label="vz25600",fontsize=16,color="green",shape="box"];16035[label="vz256100",fontsize=16,color="green",shape="box"];16036[label="vz25600",fontsize=16,color="green",shape="box"];16037[label="vz256100",fontsize=16,color="green",shape="box"];16038[label="vz25600",fontsize=16,color="green",shape="box"];16039[label="vz256100",fontsize=16,color="green",shape="box"];16040[label="vz25600",fontsize=16,color="green",shape="box"];16041[label="vz256100",fontsize=16,color="green",shape="box"];16042[label="vz25600",fontsize=16,color="green",shape="box"];16043[label="vz256100",fontsize=16,color="green",shape="box"];16044[label="vz25600",fontsize=16,color="green",shape="box"];16045[label="vz256100",fontsize=16,color="green",shape="box"];16046[label="vz25600",fontsize=16,color="green",shape="box"];16047[label="vz256100",fontsize=16,color="green",shape="box"];16048[label="vz25600",fontsize=16,color="green",shape="box"];16049[label="vz256100",fontsize=16,color="green",shape="box"];16050[label="vz25600",fontsize=16,color="green",shape="box"];16051[label="vz256100",fontsize=16,color="green",shape="box"];16052[label="vz25600",fontsize=16,color="green",shape="box"];16053[label="vz256100",fontsize=16,color="green",shape="box"];16059[label="primCmpInt (Pos vz25500) vz25600",fontsize=16,color="burlywood",shape="box"];16293[label="vz25500/Succ vz255000",fontsize=10,color="white",style="solid",shape="box"];16059 -> 16293[label="",style="solid", color="burlywood", weight=9]; 16293 -> 16069[label="",style="solid", color="burlywood", weight=3]; 16294[label="vz25500/Zero",fontsize=10,color="white",style="solid",shape="box"];16059 -> 16294[label="",style="solid", color="burlywood", weight=9]; 16294 -> 16070[label="",style="solid", color="burlywood", weight=3]; 16060[label="primCmpInt (Neg vz25500) vz25600",fontsize=16,color="burlywood",shape="box"];16295[label="vz25500/Succ vz255000",fontsize=10,color="white",style="solid",shape="box"];16060 -> 16295[label="",style="solid", color="burlywood", weight=9]; 16295 -> 16071[label="",style="solid", color="burlywood", weight=3]; 16296[label="vz25500/Zero",fontsize=10,color="white",style="solid",shape="box"];16060 -> 16296[label="",style="solid", color="burlywood", weight=9]; 16296 -> 16072[label="",style="solid", color="burlywood", weight=3]; 16137[label="primMulInt (Pos vz258000) (Pos vz257010)",fontsize=16,color="black",shape="box"];16137 -> 16149[label="",style="solid", color="black", weight=3]; 16138[label="primMulInt (Neg vz258000) (Pos vz257010)",fontsize=16,color="black",shape="box"];16138 -> 16150[label="",style="solid", color="black", weight=3]; 16139[label="primMulInt (Pos vz258010) (Pos vz257000)",fontsize=16,color="black",shape="box"];16139 -> 16151[label="",style="solid", color="black", weight=3]; 16140[label="primMulInt (Pos vz258010) (Neg vz257000)",fontsize=16,color="black",shape="box"];16140 -> 16152[label="",style="solid", color="black", weight=3]; 16141[label="primMulInt (Neg vz258010) (Pos vz257000)",fontsize=16,color="black",shape="box"];16141 -> 16153[label="",style="solid", color="black", weight=3]; 16142[label="primMulInt (Neg vz258010) (Neg vz257000)",fontsize=16,color="black",shape="box"];16142 -> 16154[label="",style="solid", color="black", weight=3]; 16143[label="primMulInt (Pos vz258000) (Neg vz257010)",fontsize=16,color="black",shape="box"];16143 -> 16155[label="",style="solid", color="black", weight=3]; 16144[label="primMulInt (Neg vz258000) (Neg vz257010)",fontsize=16,color="black",shape="box"];16144 -> 16156[label="",style="solid", color="black", weight=3]; 16069[label="primCmpInt (Pos (Succ vz255000)) vz25600",fontsize=16,color="burlywood",shape="box"];16297[label="vz25600/Pos vz256000",fontsize=10,color="white",style="solid",shape="box"];16069 -> 16297[label="",style="solid", color="burlywood", weight=9]; 16297 -> 16077[label="",style="solid", color="burlywood", weight=3]; 16298[label="vz25600/Neg vz256000",fontsize=10,color="white",style="solid",shape="box"];16069 -> 16298[label="",style="solid", color="burlywood", weight=9]; 16298 -> 16078[label="",style="solid", color="burlywood", weight=3]; 16070[label="primCmpInt (Pos Zero) vz25600",fontsize=16,color="burlywood",shape="box"];16299[label="vz25600/Pos vz256000",fontsize=10,color="white",style="solid",shape="box"];16070 -> 16299[label="",style="solid", color="burlywood", weight=9]; 16299 -> 16079[label="",style="solid", color="burlywood", weight=3]; 16300[label="vz25600/Neg vz256000",fontsize=10,color="white",style="solid",shape="box"];16070 -> 16300[label="",style="solid", color="burlywood", weight=9]; 16300 -> 16080[label="",style="solid", color="burlywood", weight=3]; 16071[label="primCmpInt (Neg (Succ vz255000)) vz25600",fontsize=16,color="burlywood",shape="box"];16301[label="vz25600/Pos vz256000",fontsize=10,color="white",style="solid",shape="box"];16071 -> 16301[label="",style="solid", color="burlywood", weight=9]; 16301 -> 16081[label="",style="solid", color="burlywood", weight=3]; 16302[label="vz25600/Neg vz256000",fontsize=10,color="white",style="solid",shape="box"];16071 -> 16302[label="",style="solid", color="burlywood", weight=9]; 16302 -> 16082[label="",style="solid", color="burlywood", weight=3]; 16072[label="primCmpInt (Neg Zero) vz25600",fontsize=16,color="burlywood",shape="box"];16303[label="vz25600/Pos vz256000",fontsize=10,color="white",style="solid",shape="box"];16072 -> 16303[label="",style="solid", color="burlywood", weight=9]; 16303 -> 16083[label="",style="solid", color="burlywood", weight=3]; 16304[label="vz25600/Neg vz256000",fontsize=10,color="white",style="solid",shape="box"];16072 -> 16304[label="",style="solid", color="burlywood", weight=9]; 16304 -> 16084[label="",style="solid", color="burlywood", weight=3]; 16149[label="Pos (primMulNat vz258000 vz257010)",fontsize=16,color="green",shape="box"];16149 -> 16161[label="",style="dashed", color="green", weight=3]; 16150[label="Neg (primMulNat vz258000 vz257010)",fontsize=16,color="green",shape="box"];16150 -> 16162[label="",style="dashed", color="green", weight=3]; 16151[label="Pos (primMulNat vz258010 vz257000)",fontsize=16,color="green",shape="box"];16151 -> 16163[label="",style="dashed", color="green", weight=3]; 16152[label="Neg (primMulNat vz258010 vz257000)",fontsize=16,color="green",shape="box"];16152 -> 16164[label="",style="dashed", color="green", weight=3]; 16153[label="Neg (primMulNat vz258010 vz257000)",fontsize=16,color="green",shape="box"];16153 -> 16165[label="",style="dashed", color="green", weight=3]; 16154[label="Pos (primMulNat vz258010 vz257000)",fontsize=16,color="green",shape="box"];16154 -> 16166[label="",style="dashed", color="green", weight=3]; 16155[label="Neg (primMulNat vz258000 vz257010)",fontsize=16,color="green",shape="box"];16155 -> 16167[label="",style="dashed", color="green", weight=3]; 16156[label="Pos (primMulNat vz258000 vz257010)",fontsize=16,color="green",shape="box"];16156 -> 16168[label="",style="dashed", color="green", weight=3]; 16077[label="primCmpInt (Pos (Succ vz255000)) (Pos vz256000)",fontsize=16,color="black",shape="box"];16077 -> 16089[label="",style="solid", color="black", weight=3]; 16078[label="primCmpInt (Pos (Succ vz255000)) (Neg vz256000)",fontsize=16,color="black",shape="box"];16078 -> 16090[label="",style="solid", color="black", weight=3]; 16079[label="primCmpInt (Pos Zero) (Pos vz256000)",fontsize=16,color="burlywood",shape="box"];16305[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16079 -> 16305[label="",style="solid", color="burlywood", weight=9]; 16305 -> 16091[label="",style="solid", color="burlywood", weight=3]; 16306[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16079 -> 16306[label="",style="solid", color="burlywood", weight=9]; 16306 -> 16092[label="",style="solid", color="burlywood", weight=3]; 16080[label="primCmpInt (Pos Zero) (Neg vz256000)",fontsize=16,color="burlywood",shape="box"];16307[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16080 -> 16307[label="",style="solid", color="burlywood", weight=9]; 16307 -> 16093[label="",style="solid", color="burlywood", weight=3]; 16308[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16080 -> 16308[label="",style="solid", color="burlywood", weight=9]; 16308 -> 16094[label="",style="solid", color="burlywood", weight=3]; 16081[label="primCmpInt (Neg (Succ vz255000)) (Pos vz256000)",fontsize=16,color="black",shape="box"];16081 -> 16095[label="",style="solid", color="black", weight=3]; 16082[label="primCmpInt (Neg (Succ vz255000)) (Neg vz256000)",fontsize=16,color="black",shape="box"];16082 -> 16096[label="",style="solid", color="black", weight=3]; 16083[label="primCmpInt (Neg Zero) (Pos vz256000)",fontsize=16,color="burlywood",shape="box"];16309[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16083 -> 16309[label="",style="solid", color="burlywood", weight=9]; 16309 -> 16097[label="",style="solid", color="burlywood", weight=3]; 16310[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16083 -> 16310[label="",style="solid", color="burlywood", weight=9]; 16310 -> 16098[label="",style="solid", color="burlywood", weight=3]; 16084[label="primCmpInt (Neg Zero) (Neg vz256000)",fontsize=16,color="burlywood",shape="box"];16311[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16084 -> 16311[label="",style="solid", color="burlywood", weight=9]; 16311 -> 16099[label="",style="solid", color="burlywood", weight=3]; 16312[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16084 -> 16312[label="",style="solid", color="burlywood", weight=9]; 16312 -> 16100[label="",style="solid", color="burlywood", weight=3]; 16161[label="primMulNat vz258000 vz257010",fontsize=16,color="burlywood",shape="triangle"];16313[label="vz258000/Succ vz2580000",fontsize=10,color="white",style="solid",shape="box"];16161 -> 16313[label="",style="solid", color="burlywood", weight=9]; 16313 -> 16173[label="",style="solid", color="burlywood", weight=3]; 16314[label="vz258000/Zero",fontsize=10,color="white",style="solid",shape="box"];16161 -> 16314[label="",style="solid", color="burlywood", weight=9]; 16314 -> 16174[label="",style="solid", color="burlywood", weight=3]; 16162 -> 16161[label="",style="dashed", color="red", weight=0]; 16162[label="primMulNat vz258000 vz257010",fontsize=16,color="magenta"];16162 -> 16175[label="",style="dashed", color="magenta", weight=3]; 16163 -> 16161[label="",style="dashed", color="red", weight=0]; 16163[label="primMulNat vz258010 vz257000",fontsize=16,color="magenta"];16163 -> 16176[label="",style="dashed", color="magenta", weight=3]; 16163 -> 16177[label="",style="dashed", color="magenta", weight=3]; 16164 -> 16161[label="",style="dashed", color="red", weight=0]; 16164[label="primMulNat vz258010 vz257000",fontsize=16,color="magenta"];16164 -> 16178[label="",style="dashed", color="magenta", weight=3]; 16164 -> 16179[label="",style="dashed", color="magenta", weight=3]; 16165 -> 16161[label="",style="dashed", color="red", weight=0]; 16165[label="primMulNat vz258010 vz257000",fontsize=16,color="magenta"];16165 -> 16180[label="",style="dashed", color="magenta", weight=3]; 16165 -> 16181[label="",style="dashed", color="magenta", weight=3]; 16166 -> 16161[label="",style="dashed", color="red", weight=0]; 16166[label="primMulNat vz258010 vz257000",fontsize=16,color="magenta"];16166 -> 16182[label="",style="dashed", color="magenta", weight=3]; 16166 -> 16183[label="",style="dashed", color="magenta", weight=3]; 16167 -> 16161[label="",style="dashed", color="red", weight=0]; 16167[label="primMulNat vz258000 vz257010",fontsize=16,color="magenta"];16168 -> 16161[label="",style="dashed", color="red", weight=0]; 16168[label="primMulNat vz258000 vz257010",fontsize=16,color="magenta"];16168 -> 16184[label="",style="dashed", color="magenta", weight=3]; 16089[label="primCmpNat (Succ vz255000) vz256000",fontsize=16,color="burlywood",shape="triangle"];16315[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16089 -> 16315[label="",style="solid", color="burlywood", weight=9]; 16315 -> 16109[label="",style="solid", color="burlywood", weight=3]; 16316[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16089 -> 16316[label="",style="solid", color="burlywood", weight=9]; 16316 -> 16110[label="",style="solid", color="burlywood", weight=3]; 16090[label="GT",fontsize=16,color="green",shape="box"];16091[label="primCmpInt (Pos Zero) (Pos (Succ vz2560000))",fontsize=16,color="black",shape="box"];16091 -> 16111[label="",style="solid", color="black", weight=3]; 16092[label="primCmpInt (Pos Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];16092 -> 16112[label="",style="solid", color="black", weight=3]; 16093[label="primCmpInt (Pos Zero) (Neg (Succ vz2560000))",fontsize=16,color="black",shape="box"];16093 -> 16113[label="",style="solid", color="black", weight=3]; 16094[label="primCmpInt (Pos Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];16094 -> 16114[label="",style="solid", color="black", weight=3]; 16095[label="LT",fontsize=16,color="green",shape="box"];16096[label="primCmpNat vz256000 (Succ vz255000)",fontsize=16,color="burlywood",shape="triangle"];16317[label="vz256000/Succ vz2560000",fontsize=10,color="white",style="solid",shape="box"];16096 -> 16317[label="",style="solid", color="burlywood", weight=9]; 16317 -> 16115[label="",style="solid", color="burlywood", weight=3]; 16318[label="vz256000/Zero",fontsize=10,color="white",style="solid",shape="box"];16096 -> 16318[label="",style="solid", color="burlywood", weight=9]; 16318 -> 16116[label="",style="solid", color="burlywood", weight=3]; 16097[label="primCmpInt (Neg Zero) (Pos (Succ vz2560000))",fontsize=16,color="black",shape="box"];16097 -> 16117[label="",style="solid", color="black", weight=3]; 16098[label="primCmpInt (Neg Zero) (Pos Zero)",fontsize=16,color="black",shape="box"];16098 -> 16118[label="",style="solid", color="black", weight=3]; 16099[label="primCmpInt (Neg Zero) (Neg (Succ vz2560000))",fontsize=16,color="black",shape="box"];16099 -> 16119[label="",style="solid", color="black", weight=3]; 16100[label="primCmpInt (Neg Zero) (Neg Zero)",fontsize=16,color="black",shape="box"];16100 -> 16120[label="",style="solid", color="black", weight=3]; 16173[label="primMulNat (Succ vz2580000) vz257010",fontsize=16,color="burlywood",shape="box"];16319[label="vz257010/Succ vz2570100",fontsize=10,color="white",style="solid",shape="box"];16173 -> 16319[label="",style="solid", color="burlywood", weight=9]; 16319 -> 16187[label="",style="solid", color="burlywood", weight=3]; 16320[label="vz257010/Zero",fontsize=10,color="white",style="solid",shape="box"];16173 -> 16320[label="",style="solid", color="burlywood", weight=9]; 16320 -> 16188[label="",style="solid", color="burlywood", weight=3]; 16174[label="primMulNat Zero vz257010",fontsize=16,color="burlywood",shape="box"];16321[label="vz257010/Succ vz2570100",fontsize=10,color="white",style="solid",shape="box"];16174 -> 16321[label="",style="solid", color="burlywood", weight=9]; 16321 -> 16189[label="",style="solid", color="burlywood", weight=3]; 16322[label="vz257010/Zero",fontsize=10,color="white",style="solid",shape="box"];16174 -> 16322[label="",style="solid", color="burlywood", weight=9]; 16322 -> 16190[label="",style="solid", color="burlywood", weight=3]; 16175[label="vz258000",fontsize=16,color="green",shape="box"];16176[label="vz258010",fontsize=16,color="green",shape="box"];16177[label="vz257000",fontsize=16,color="green",shape="box"];16178[label="vz258010",fontsize=16,color="green",shape="box"];16179[label="vz257000",fontsize=16,color="green",shape="box"];16180[label="vz258010",fontsize=16,color="green",shape="box"];16181[label="vz257000",fontsize=16,color="green",shape="box"];16182[label="vz258010",fontsize=16,color="green",shape="box"];16183[label="vz257000",fontsize=16,color="green",shape="box"];16184[label="vz258000",fontsize=16,color="green",shape="box"];16109[label="primCmpNat (Succ vz255000) (Succ vz2560000)",fontsize=16,color="black",shape="box"];16109 -> 16129[label="",style="solid", color="black", weight=3]; 16110[label="primCmpNat (Succ vz255000) Zero",fontsize=16,color="black",shape="box"];16110 -> 16130[label="",style="solid", color="black", weight=3]; 16111 -> 16096[label="",style="dashed", color="red", weight=0]; 16111[label="primCmpNat Zero (Succ vz2560000)",fontsize=16,color="magenta"];16111 -> 16131[label="",style="dashed", color="magenta", weight=3]; 16111 -> 16132[label="",style="dashed", color="magenta", weight=3]; 16112[label="EQ",fontsize=16,color="green",shape="box"];16113[label="GT",fontsize=16,color="green",shape="box"];16114[label="EQ",fontsize=16,color="green",shape="box"];16115[label="primCmpNat (Succ vz2560000) (Succ vz255000)",fontsize=16,color="black",shape="box"];16115 -> 16133[label="",style="solid", color="black", weight=3]; 16116[label="primCmpNat Zero (Succ vz255000)",fontsize=16,color="black",shape="box"];16116 -> 16134[label="",style="solid", color="black", weight=3]; 16117[label="LT",fontsize=16,color="green",shape="box"];16118[label="EQ",fontsize=16,color="green",shape="box"];16119 -> 16089[label="",style="dashed", color="red", weight=0]; 16119[label="primCmpNat (Succ vz2560000) Zero",fontsize=16,color="magenta"];16119 -> 16135[label="",style="dashed", color="magenta", weight=3]; 16119 -> 16136[label="",style="dashed", color="magenta", weight=3]; 16120[label="EQ",fontsize=16,color="green",shape="box"];16187[label="primMulNat (Succ vz2580000) (Succ vz2570100)",fontsize=16,color="black",shape="box"];16187 -> 16191[label="",style="solid", color="black", weight=3]; 16188[label="primMulNat (Succ vz2580000) Zero",fontsize=16,color="black",shape="box"];16188 -> 16192[label="",style="solid", color="black", weight=3]; 16189[label="primMulNat Zero (Succ vz2570100)",fontsize=16,color="black",shape="box"];16189 -> 16193[label="",style="solid", color="black", weight=3]; 16190[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];16190 -> 16194[label="",style="solid", color="black", weight=3]; 16129[label="primCmpNat vz255000 vz2560000",fontsize=16,color="burlywood",shape="triangle"];16323[label="vz255000/Succ vz2550000",fontsize=10,color="white",style="solid",shape="box"];16129 -> 16323[label="",style="solid", color="burlywood", weight=9]; 16323 -> 16145[label="",style="solid", color="burlywood", weight=3]; 16324[label="vz255000/Zero",fontsize=10,color="white",style="solid",shape="box"];16129 -> 16324[label="",style="solid", color="burlywood", weight=9]; 16324 -> 16146[label="",style="solid", color="burlywood", weight=3]; 16130[label="GT",fontsize=16,color="green",shape="box"];16131[label="Zero",fontsize=16,color="green",shape="box"];16132[label="vz2560000",fontsize=16,color="green",shape="box"];16133 -> 16129[label="",style="dashed", color="red", weight=0]; 16133[label="primCmpNat vz2560000 vz255000",fontsize=16,color="magenta"];16133 -> 16147[label="",style="dashed", color="magenta", weight=3]; 16133 -> 16148[label="",style="dashed", color="magenta", weight=3]; 16134[label="LT",fontsize=16,color="green",shape="box"];16135[label="Zero",fontsize=16,color="green",shape="box"];16136[label="vz2560000",fontsize=16,color="green",shape="box"];16191 -> 16195[label="",style="dashed", color="red", weight=0]; 16191[label="primPlusNat (primMulNat vz2580000 (Succ vz2570100)) (Succ vz2570100)",fontsize=16,color="magenta"];16191 -> 16196[label="",style="dashed", color="magenta", weight=3]; 16192[label="Zero",fontsize=16,color="green",shape="box"];16193[label="Zero",fontsize=16,color="green",shape="box"];16194[label="Zero",fontsize=16,color="green",shape="box"];16145[label="primCmpNat (Succ vz2550000) vz2560000",fontsize=16,color="burlywood",shape="box"];16325[label="vz2560000/Succ vz25600000",fontsize=10,color="white",style="solid",shape="box"];16145 -> 16325[label="",style="solid", color="burlywood", weight=9]; 16325 -> 16157[label="",style="solid", color="burlywood", weight=3]; 16326[label="vz2560000/Zero",fontsize=10,color="white",style="solid",shape="box"];16145 -> 16326[label="",style="solid", color="burlywood", weight=9]; 16326 -> 16158[label="",style="solid", color="burlywood", weight=3]; 16146[label="primCmpNat Zero vz2560000",fontsize=16,color="burlywood",shape="box"];16327[label="vz2560000/Succ vz25600000",fontsize=10,color="white",style="solid",shape="box"];16146 -> 16327[label="",style="solid", color="burlywood", weight=9]; 16327 -> 16159[label="",style="solid", color="burlywood", weight=3]; 16328[label="vz2560000/Zero",fontsize=10,color="white",style="solid",shape="box"];16146 -> 16328[label="",style="solid", color="burlywood", weight=9]; 16328 -> 16160[label="",style="solid", color="burlywood", weight=3]; 16147[label="vz255000",fontsize=16,color="green",shape="box"];16148[label="vz2560000",fontsize=16,color="green",shape="box"];16196 -> 16161[label="",style="dashed", color="red", weight=0]; 16196[label="primMulNat vz2580000 (Succ vz2570100)",fontsize=16,color="magenta"];16196 -> 16197[label="",style="dashed", color="magenta", weight=3]; 16196 -> 16198[label="",style="dashed", color="magenta", weight=3]; 16195[label="primPlusNat vz270 (Succ vz2570100)",fontsize=16,color="burlywood",shape="triangle"];16329[label="vz270/Succ vz2700",fontsize=10,color="white",style="solid",shape="box"];16195 -> 16329[label="",style="solid", color="burlywood", weight=9]; 16329 -> 16199[label="",style="solid", color="burlywood", weight=3]; 16330[label="vz270/Zero",fontsize=10,color="white",style="solid",shape="box"];16195 -> 16330[label="",style="solid", color="burlywood", weight=9]; 16330 -> 16200[label="",style="solid", color="burlywood", weight=3]; 16157[label="primCmpNat (Succ vz2550000) (Succ vz25600000)",fontsize=16,color="black",shape="box"];16157 -> 16169[label="",style="solid", color="black", weight=3]; 16158[label="primCmpNat (Succ vz2550000) Zero",fontsize=16,color="black",shape="box"];16158 -> 16170[label="",style="solid", color="black", weight=3]; 16159[label="primCmpNat Zero (Succ vz25600000)",fontsize=16,color="black",shape="box"];16159 -> 16171[label="",style="solid", color="black", weight=3]; 16160[label="primCmpNat Zero Zero",fontsize=16,color="black",shape="box"];16160 -> 16172[label="",style="solid", color="black", weight=3]; 16197[label="vz2580000",fontsize=16,color="green",shape="box"];16198[label="Succ vz2570100",fontsize=16,color="green",shape="box"];16199[label="primPlusNat (Succ vz2700) (Succ vz2570100)",fontsize=16,color="black",shape="box"];16199 -> 16201[label="",style="solid", color="black", weight=3]; 16200[label="primPlusNat Zero (Succ vz2570100)",fontsize=16,color="black",shape="box"];16200 -> 16202[label="",style="solid", color="black", weight=3]; 16169 -> 16129[label="",style="dashed", color="red", weight=0]; 16169[label="primCmpNat vz2550000 vz25600000",fontsize=16,color="magenta"];16169 -> 16185[label="",style="dashed", color="magenta", weight=3]; 16169 -> 16186[label="",style="dashed", color="magenta", weight=3]; 16170[label="GT",fontsize=16,color="green",shape="box"];16171[label="LT",fontsize=16,color="green",shape="box"];16172[label="EQ",fontsize=16,color="green",shape="box"];16201[label="Succ (Succ (primPlusNat vz2700 vz2570100))",fontsize=16,color="green",shape="box"];16201 -> 16203[label="",style="dashed", color="green", weight=3]; 16202[label="Succ vz2570100",fontsize=16,color="green",shape="box"];16185[label="vz25600000",fontsize=16,color="green",shape="box"];16186[label="vz2550000",fontsize=16,color="green",shape="box"];16203[label="primPlusNat vz2700 vz2570100",fontsize=16,color="burlywood",shape="triangle"];16331[label="vz2700/Succ vz27000",fontsize=10,color="white",style="solid",shape="box"];16203 -> 16331[label="",style="solid", color="burlywood", weight=9]; 16331 -> 16204[label="",style="solid", color="burlywood", weight=3]; 16332[label="vz2700/Zero",fontsize=10,color="white",style="solid",shape="box"];16203 -> 16332[label="",style="solid", color="burlywood", weight=9]; 16332 -> 16205[label="",style="solid", color="burlywood", weight=3]; 16204[label="primPlusNat (Succ vz27000) vz2570100",fontsize=16,color="burlywood",shape="box"];16333[label="vz2570100/Succ vz25701000",fontsize=10,color="white",style="solid",shape="box"];16204 -> 16333[label="",style="solid", color="burlywood", weight=9]; 16333 -> 16206[label="",style="solid", color="burlywood", weight=3]; 16334[label="vz2570100/Zero",fontsize=10,color="white",style="solid",shape="box"];16204 -> 16334[label="",style="solid", color="burlywood", weight=9]; 16334 -> 16207[label="",style="solid", color="burlywood", weight=3]; 16205[label="primPlusNat Zero vz2570100",fontsize=16,color="burlywood",shape="box"];16335[label="vz2570100/Succ vz25701000",fontsize=10,color="white",style="solid",shape="box"];16205 -> 16335[label="",style="solid", color="burlywood", weight=9]; 16335 -> 16208[label="",style="solid", color="burlywood", weight=3]; 16336[label="vz2570100/Zero",fontsize=10,color="white",style="solid",shape="box"];16205 -> 16336[label="",style="solid", color="burlywood", weight=9]; 16336 -> 16209[label="",style="solid", color="burlywood", weight=3]; 16206[label="primPlusNat (Succ vz27000) (Succ vz25701000)",fontsize=16,color="black",shape="box"];16206 -> 16210[label="",style="solid", color="black", weight=3]; 16207[label="primPlusNat (Succ vz27000) Zero",fontsize=16,color="black",shape="box"];16207 -> 16211[label="",style="solid", color="black", weight=3]; 16208[label="primPlusNat Zero (Succ vz25701000)",fontsize=16,color="black",shape="box"];16208 -> 16212[label="",style="solid", color="black", weight=3]; 16209[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];16209 -> 16213[label="",style="solid", color="black", weight=3]; 16210[label="Succ (Succ (primPlusNat vz27000 vz25701000))",fontsize=16,color="green",shape="box"];16210 -> 16214[label="",style="dashed", color="green", weight=3]; 16211[label="Succ vz27000",fontsize=16,color="green",shape="box"];16212[label="Succ vz25701000",fontsize=16,color="green",shape="box"];16213[label="Zero",fontsize=16,color="green",shape="box"];16214 -> 16203[label="",style="dashed", color="red", weight=0]; 16214[label="primPlusNat vz27000 vz25701000",fontsize=16,color="magenta"];16214 -> 16215[label="",style="dashed", color="magenta", weight=3]; 16214 -> 16216[label="",style="dashed", color="magenta", weight=3]; 16215[label="vz27000",fontsize=16,color="green",shape="box"];16216[label="vz25701000",fontsize=16,color="green",shape="box"];} ---------------------------------------- (8) Complex Obligation (AND) ---------------------------------------- (9) Obligation: Q DP problem: The TRS P consists of the following rules: new_primCmpNat(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat(vz2550000, vz25600000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (10) 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_primCmpNat(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat(vz2550000, vz25600000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (11) YES ---------------------------------------- (12) Obligation: Q DP problem: The TRS P consists of the following rules: new_merge_pairs(:(vz256110, :(vz2561110, vz2561111)), ba) -> new_merge_pairs(vz2561111, ba) 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_merge_pairs(:(vz256110, :(vz2561110, vz2561111)), ba) -> new_merge_pairs(vz2561111, ba) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (14) YES ---------------------------------------- (15) Obligation: Q DP problem: The TRS P consists of the following rules: new_mergesort'(vz255, :(vz2560, []), ba) -> new_mergesort'(new_merge2(vz255, vz2560, ba), [], ba) new_mergesort'(vz255, :(vz2560, :(vz25610, vz25611)), ba) -> new_mergesort'(new_merge1(vz255, vz2560, vz25610, ba), new_merge_pairs0(vz25611, ba), ba) The TRS R consists of the following rules: new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_sr2(Pos(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_compare11(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, app(app(app(ty_@3, bh), ca), cb)) -> new_compare3(vz25600, vz256100, bh, ca, cb) new_compare3(vz2550, vz25600, bh, ca, cb) -> error([]) new_compare14(vz25600, vz256100, ty_@0) -> new_compare2(vz25600, vz256100) new_primMulNat0(Zero, Zero) -> Zero new_compare(vz2550, vz25600, app(ty_Maybe, cc)) -> new_compare13(vz2550, vz25600, cc) new_compare14(vz25600, vz256100, ty_Double) -> new_compare11(vz25600, vz256100) new_compare(vz2550, vz25600, app(app(ty_@2, bd), be)) -> new_compare12(vz2550, vz25600, bd, be) new_sr(vz258010, Pos(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Succ(vz2560000)) -> new_primCmpNat1(vz255000, vz2560000) new_compare(vz2550, vz25600, ty_Int) -> new_compare4(vz2550, vz25600) new_compare12(vz2550, vz25600, bd, be) -> error([]) new_primMulNat0(Succ(vz2580000), Succ(vz2570100)) -> new_primPlusNat0(new_primMulNat0(vz2580000, Succ(vz2570100)), vz2570100) new_compare4(Pos(Zero), Neg(Succ(vz2560000))) -> GT new_sr1(Pos(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare(vz2550, vz25600, app(ty_[], cd)) -> new_compare7(vz2550, vz25600, cd) new_compare4(Pos(Zero), Neg(Zero)) -> EQ new_compare4(Neg(Zero), Pos(Zero)) -> EQ new_primCmpNat1(Zero, Zero) -> EQ new_merge_pairs0(:(vz256110, []), ba) -> :(vz256110, []) new_compare(vz2550, vz25600, ty_Char) -> new_compare1(vz2550, vz25600) new_compare4(Pos(Zero), Pos(Succ(vz2560000))) -> new_primCmpNat2(Zero, vz2560000) new_compare(vz2550, vz25600, ty_Float) -> new_compare5(vz2550, vz25600) new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr(vz258010, vz25700)) new_merge1(vz255, :(vz25600, vz25601), :(vz256100, vz256101), ba) -> new_merge2(vz255, new_merge00(vz256100, vz25600, vz25601, vz256101, new_compare14(vz25600, vz256100, ba), ba), ba) new_compare8(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, app(ty_Ratio, bc)) -> new_compare10(vz25600, vz256100, bc) new_primCmpNat1(Succ(vz2550000), Zero) -> GT new_compare4(Neg(Zero), Neg(Succ(vz2560000))) -> new_primCmpNat0(vz2560000, Zero) new_primPlusNat1(Succ(vz27000), Zero) -> Succ(vz27000) new_primPlusNat1(Zero, Succ(vz25701000)) -> Succ(vz25701000) new_merge2([], :(vz25600, vz25601), ba) -> :(vz25600, vz25601) new_compare0(vz2550, vz25600) -> error([]) new_compare4(Pos(Succ(vz255000)), Pos(vz256000)) -> new_primCmpNat0(vz255000, vz256000) new_compare(vz2550, vz25600, ty_Double) -> new_compare11(vz2550, vz25600) new_compare14(vz25600, vz256100, app(app(ty_@2, bd), be)) -> new_compare12(vz25600, vz256100, bd, be) new_compare10(vz2550, vz25600, bc) -> error([]) new_compare4(Neg(Succ(vz255000)), Pos(vz256000)) -> LT new_compare(vz2550, vz25600, ty_@0) -> new_compare2(vz2550, vz25600) new_primCmpNat2(Succ(vz2560000), vz255000) -> new_primCmpNat1(vz2560000, vz255000) new_merge_pairs0(:(vz256110, :(vz2561110, vz2561111)), ba) -> :(new_merge2(vz256110, vz2561110, ba), new_merge_pairs0(vz2561111, ba)) new_sr0(vz258010, Pos(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare14(vz25600, vz256100, ty_Float) -> new_compare5(vz25600, vz256100) new_primCmpNat0(vz255000, Zero) -> GT new_compare4(Neg(Succ(vz255000)), Neg(vz256000)) -> new_primCmpNat2(vz256000, vz255000) new_merge00(vz265, vz266, vz267, vz268, LT, bb) -> :(vz266, new_merge2(vz267, :(vz265, vz268), bb)) new_compare14(vz25600, vz256100, app(app(ty_Either, bf), bg)) -> new_compare6(vz25600, vz256100, bf, bg) new_compare(vz2550, vz25600, app(ty_Ratio, bc)) -> new_compare10(vz2550, vz25600, bc) new_compare4(Pos(Succ(vz255000)), Neg(vz256000)) -> GT new_sr(vz258010, Neg(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr(vz258010, vz25700)) new_primCmpNat2(Zero, vz255000) -> LT new_compare(vz2550, vz25600, ty_Integer) -> new_compare0(vz2550, vz25600) new_merge2(:(vz2550, vz2551), :(vz25600, vz25601), ba) -> new_merge00(vz25600, vz2550, vz2551, vz25601, new_compare(vz2550, vz25600, ba), ba) new_compare(vz2550, vz25600, ty_Bool) -> new_compare8(vz2550, vz25600) new_compare4(Neg(Zero), Pos(Succ(vz2560000))) -> LT new_primCmpNat1(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat1(vz2550000, vz25600000) new_compare14(vz25600, vz256100, ty_Int) -> new_compare4(vz25600, vz256100) new_compare4(Pos(Zero), Pos(Zero)) -> EQ new_merge2(vz255, [], ba) -> vz255 new_compare13(vz2550, vz25600, cc) -> error([]) new_merge_pairs0([], ba) -> [] new_compare14(vz25600, vz256100, app(ty_Maybe, cc)) -> new_compare13(vz25600, vz256100, cc) new_primPlusNat0(Succ(vz2700), vz2570100) -> Succ(Succ(new_primPlusNat1(vz2700, vz2570100))) new_merge1(vz255, [], :(vz256100, vz256101), ba) -> new_merge2(vz255, :(vz256100, vz256101), ba) new_compare2(vz2550, vz25600) -> error([]) new_sr1(Neg(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_sr0(vz258010, Neg(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_compare7(vz2550, vz25600, cd) -> error([]) new_primCmpNat1(Zero, Succ(vz25600000)) -> LT new_compare(vz2550, vz25600, app(app(ty_Either, bf), bg)) -> new_compare6(vz2550, vz25600, bf, bg) new_merge00(vz265, vz266, vz267, vz268, GT, bb) -> :(vz265, new_merge2(:(vz266, vz267), vz268, bb)) new_compare14(vz25600, vz256100, ty_Ordering) -> new_compare9(vz25600, vz256100) new_merge00(vz265, vz266, vz267, vz268, EQ, bb) -> :(vz266, new_merge2(vz267, :(vz265, vz268), bb)) new_primPlusNat1(Succ(vz27000), Succ(vz25701000)) -> Succ(Succ(new_primPlusNat1(vz27000, vz25701000))) new_primPlusNat1(Zero, Zero) -> Zero new_compare14(vz25600, vz256100, app(ty_[], cd)) -> new_compare7(vz25600, vz256100, cd) new_compare14(vz25600, vz256100, ty_Integer) -> new_compare0(vz25600, vz256100) new_compare14(vz25600, vz256100, ty_Bool) -> new_compare8(vz25600, vz256100) new_primMulNat0(Succ(vz2580000), Zero) -> Zero new_primMulNat0(Zero, Succ(vz2570100)) -> Zero new_primPlusNat0(Zero, vz2570100) -> Succ(vz2570100) new_compare(vz2550, vz25600, ty_Ordering) -> new_compare9(vz2550, vz25600) new_sr2(Neg(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare1(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, ty_Char) -> new_compare1(vz25600, vz256100) new_compare6(vz2550, vz25600, bf, bg) -> error([]) new_compare(vz2550, vz25600, app(app(app(ty_@3, bh), ca), cb)) -> new_compare3(vz2550, vz25600, bh, ca, cb) new_compare9(vz2550, vz25600) -> error([]) new_merge1(vz255, vz2560, [], ba) -> new_merge2(vz255, vz2560, ba) new_compare4(Neg(Zero), Neg(Zero)) -> EQ The set Q consists of the following terms: new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_compare(x0, x1, ty_Bool) new_compare8(x0, x1) new_compare6(x0, x1, x2, x3) new_compare12(x0, x1, x2, x3) new_merge1(x0, :(x1, x2), :(x3, x4), x5) new_compare10(x0, x1, x2) new_compare14(x0, x1, ty_@0) new_compare3(x0, x1, x2, x3, x4) new_compare14(x0, x1, ty_Char) new_merge2([], :(x0, x1), x2) new_primPlusNat0(Succ(x0), x1) new_merge00(x0, x1, x2, x3, GT, x4) new_compare(x0, x1, app(ty_Maybe, x2)) new_primCmpNat1(Zero, Succ(x0)) new_compare14(x0, x1, ty_Integer) new_compare9(x0, x1) new_compare(x0, x1, ty_Double) new_merge_pairs0(:(x0, :(x1, x2)), x3) new_sr2(Pos(x0), x1) new_compare(x0, x1, ty_Ordering) new_primPlusNat1(Zero, Succ(x0)) new_compare(x0, x1, ty_Char) new_compare(x0, x1, ty_@0) new_merge00(x0, x1, x2, x3, EQ, x4) new_compare5(Float(x0, Neg(x1)), Float(x2, Pos(x3))) new_compare5(Float(x0, Pos(x1)), Float(x2, Neg(x3))) new_merge00(x0, x1, x2, x3, LT, x4) new_compare7(x0, x1, x2) new_compare14(x0, x1, ty_Bool) new_sr1(Pos(x0), x1) new_compare0(x0, x1) new_compare4(Pos(Succ(x0)), Pos(x1)) new_merge1(x0, [], :(x1, x2), x3) new_primPlusNat1(Succ(x0), Succ(x1)) new_primMulNat0(Zero, Zero) new_sr0(x0, Pos(x1)) new_primPlusNat1(Zero, Zero) new_primPlusNat1(Succ(x0), Zero) new_compare(x0, x1, app(app(ty_Either, x2), x3)) new_compare14(x0, x1, ty_Float) new_primCmpNat1(Succ(x0), Succ(x1)) new_compare4(Pos(Zero), Pos(Zero)) new_primCmpNat2(Zero, x0) new_primCmpNat1(Zero, Zero) new_compare4(Pos(Zero), Pos(Succ(x0))) new_compare(x0, x1, app(ty_Ratio, x2)) new_compare11(x0, x1) new_compare5(Float(x0, Neg(x1)), Float(x2, Neg(x3))) new_compare(x0, x1, ty_Float) new_compare(x0, x1, ty_Integer) new_sr1(Neg(x0), x1) new_compare(x0, x1, ty_Int) new_primMulNat0(Succ(x0), Succ(x1)) new_merge2(:(x0, x1), :(x2, x3), x4) new_compare2(x0, x1) new_compare(x0, x1, app(ty_[], x2)) new_compare14(x0, x1, app(app(ty_@2, x2), x3)) new_compare4(Neg(Succ(x0)), Neg(x1)) new_merge2(x0, [], x1) new_primCmpNat0(x0, Succ(x1)) new_compare14(x0, x1, ty_Ordering) new_compare14(x0, x1, app(ty_Ratio, x2)) new_sr0(x0, Neg(x1)) new_compare14(x0, x1, app(ty_Maybe, x2)) new_primCmpNat0(x0, Zero) new_compare14(x0, x1, app(app(ty_Either, x2), x3)) new_compare4(Pos(Zero), Neg(Succ(x0))) new_compare4(Neg(Zero), Pos(Succ(x0))) new_compare(x0, x1, app(app(ty_@2, x2), x3)) new_sr(x0, Neg(x1)) new_sr2(Neg(x0), x1) new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_compare4(Pos(Zero), Neg(Zero)) new_compare4(Neg(Zero), Pos(Zero)) new_merge1(x0, x1, [], x2) new_compare14(x0, x1, ty_Int) new_primMulNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, x0) new_compare4(Neg(Succ(x0)), Pos(x1)) new_compare4(Pos(Succ(x0)), Neg(x1)) new_merge_pairs0(:(x0, []), x1) new_compare14(x0, x1, ty_Double) new_compare1(x0, x1) new_sr(x0, Pos(x1)) new_primCmpNat1(Succ(x0), Zero) new_primMulNat0(Succ(x0), Zero) new_compare4(Neg(Zero), Neg(Zero)) new_primCmpNat2(Succ(x0), x1) new_merge_pairs0([], x0) new_compare14(x0, x1, app(ty_[], x2)) new_compare13(x0, x1, x2) new_compare4(Neg(Zero), Neg(Succ(x0))) new_compare5(Float(x0, Pos(x1)), Float(x2, Pos(x3))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (16) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node. ---------------------------------------- (17) Obligation: Q DP problem: The TRS P consists of the following rules: new_mergesort'(vz255, :(vz2560, :(vz25610, vz25611)), ba) -> new_mergesort'(new_merge1(vz255, vz2560, vz25610, ba), new_merge_pairs0(vz25611, ba), ba) The TRS R consists of the following rules: new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_sr2(Pos(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_compare11(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, app(app(app(ty_@3, bh), ca), cb)) -> new_compare3(vz25600, vz256100, bh, ca, cb) new_compare3(vz2550, vz25600, bh, ca, cb) -> error([]) new_compare14(vz25600, vz256100, ty_@0) -> new_compare2(vz25600, vz256100) new_primMulNat0(Zero, Zero) -> Zero new_compare(vz2550, vz25600, app(ty_Maybe, cc)) -> new_compare13(vz2550, vz25600, cc) new_compare14(vz25600, vz256100, ty_Double) -> new_compare11(vz25600, vz256100) new_compare(vz2550, vz25600, app(app(ty_@2, bd), be)) -> new_compare12(vz2550, vz25600, bd, be) new_sr(vz258010, Pos(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Succ(vz2560000)) -> new_primCmpNat1(vz255000, vz2560000) new_compare(vz2550, vz25600, ty_Int) -> new_compare4(vz2550, vz25600) new_compare12(vz2550, vz25600, bd, be) -> error([]) new_primMulNat0(Succ(vz2580000), Succ(vz2570100)) -> new_primPlusNat0(new_primMulNat0(vz2580000, Succ(vz2570100)), vz2570100) new_compare4(Pos(Zero), Neg(Succ(vz2560000))) -> GT new_sr1(Pos(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare(vz2550, vz25600, app(ty_[], cd)) -> new_compare7(vz2550, vz25600, cd) new_compare4(Pos(Zero), Neg(Zero)) -> EQ new_compare4(Neg(Zero), Pos(Zero)) -> EQ new_primCmpNat1(Zero, Zero) -> EQ new_merge_pairs0(:(vz256110, []), ba) -> :(vz256110, []) new_compare(vz2550, vz25600, ty_Char) -> new_compare1(vz2550, vz25600) new_compare4(Pos(Zero), Pos(Succ(vz2560000))) -> new_primCmpNat2(Zero, vz2560000) new_compare(vz2550, vz25600, ty_Float) -> new_compare5(vz2550, vz25600) new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr(vz258010, vz25700)) new_merge1(vz255, :(vz25600, vz25601), :(vz256100, vz256101), ba) -> new_merge2(vz255, new_merge00(vz256100, vz25600, vz25601, vz256101, new_compare14(vz25600, vz256100, ba), ba), ba) new_compare8(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, app(ty_Ratio, bc)) -> new_compare10(vz25600, vz256100, bc) new_primCmpNat1(Succ(vz2550000), Zero) -> GT new_compare4(Neg(Zero), Neg(Succ(vz2560000))) -> new_primCmpNat0(vz2560000, Zero) new_primPlusNat1(Succ(vz27000), Zero) -> Succ(vz27000) new_primPlusNat1(Zero, Succ(vz25701000)) -> Succ(vz25701000) new_merge2([], :(vz25600, vz25601), ba) -> :(vz25600, vz25601) new_compare0(vz2550, vz25600) -> error([]) new_compare4(Pos(Succ(vz255000)), Pos(vz256000)) -> new_primCmpNat0(vz255000, vz256000) new_compare(vz2550, vz25600, ty_Double) -> new_compare11(vz2550, vz25600) new_compare14(vz25600, vz256100, app(app(ty_@2, bd), be)) -> new_compare12(vz25600, vz256100, bd, be) new_compare10(vz2550, vz25600, bc) -> error([]) new_compare4(Neg(Succ(vz255000)), Pos(vz256000)) -> LT new_compare(vz2550, vz25600, ty_@0) -> new_compare2(vz2550, vz25600) new_primCmpNat2(Succ(vz2560000), vz255000) -> new_primCmpNat1(vz2560000, vz255000) new_merge_pairs0(:(vz256110, :(vz2561110, vz2561111)), ba) -> :(new_merge2(vz256110, vz2561110, ba), new_merge_pairs0(vz2561111, ba)) new_sr0(vz258010, Pos(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare14(vz25600, vz256100, ty_Float) -> new_compare5(vz25600, vz256100) new_primCmpNat0(vz255000, Zero) -> GT new_compare4(Neg(Succ(vz255000)), Neg(vz256000)) -> new_primCmpNat2(vz256000, vz255000) new_merge00(vz265, vz266, vz267, vz268, LT, bb) -> :(vz266, new_merge2(vz267, :(vz265, vz268), bb)) new_compare14(vz25600, vz256100, app(app(ty_Either, bf), bg)) -> new_compare6(vz25600, vz256100, bf, bg) new_compare(vz2550, vz25600, app(ty_Ratio, bc)) -> new_compare10(vz2550, vz25600, bc) new_compare4(Pos(Succ(vz255000)), Neg(vz256000)) -> GT new_sr(vz258010, Neg(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr(vz258010, vz25700)) new_primCmpNat2(Zero, vz255000) -> LT new_compare(vz2550, vz25600, ty_Integer) -> new_compare0(vz2550, vz25600) new_merge2(:(vz2550, vz2551), :(vz25600, vz25601), ba) -> new_merge00(vz25600, vz2550, vz2551, vz25601, new_compare(vz2550, vz25600, ba), ba) new_compare(vz2550, vz25600, ty_Bool) -> new_compare8(vz2550, vz25600) new_compare4(Neg(Zero), Pos(Succ(vz2560000))) -> LT new_primCmpNat1(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat1(vz2550000, vz25600000) new_compare14(vz25600, vz256100, ty_Int) -> new_compare4(vz25600, vz256100) new_compare4(Pos(Zero), Pos(Zero)) -> EQ new_merge2(vz255, [], ba) -> vz255 new_compare13(vz2550, vz25600, cc) -> error([]) new_merge_pairs0([], ba) -> [] new_compare14(vz25600, vz256100, app(ty_Maybe, cc)) -> new_compare13(vz25600, vz256100, cc) new_primPlusNat0(Succ(vz2700), vz2570100) -> Succ(Succ(new_primPlusNat1(vz2700, vz2570100))) new_merge1(vz255, [], :(vz256100, vz256101), ba) -> new_merge2(vz255, :(vz256100, vz256101), ba) new_compare2(vz2550, vz25600) -> error([]) new_sr1(Neg(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_sr0(vz258010, Neg(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_compare7(vz2550, vz25600, cd) -> error([]) new_primCmpNat1(Zero, Succ(vz25600000)) -> LT new_compare(vz2550, vz25600, app(app(ty_Either, bf), bg)) -> new_compare6(vz2550, vz25600, bf, bg) new_merge00(vz265, vz266, vz267, vz268, GT, bb) -> :(vz265, new_merge2(:(vz266, vz267), vz268, bb)) new_compare14(vz25600, vz256100, ty_Ordering) -> new_compare9(vz25600, vz256100) new_merge00(vz265, vz266, vz267, vz268, EQ, bb) -> :(vz266, new_merge2(vz267, :(vz265, vz268), bb)) new_primPlusNat1(Succ(vz27000), Succ(vz25701000)) -> Succ(Succ(new_primPlusNat1(vz27000, vz25701000))) new_primPlusNat1(Zero, Zero) -> Zero new_compare14(vz25600, vz256100, app(ty_[], cd)) -> new_compare7(vz25600, vz256100, cd) new_compare14(vz25600, vz256100, ty_Integer) -> new_compare0(vz25600, vz256100) new_compare14(vz25600, vz256100, ty_Bool) -> new_compare8(vz25600, vz256100) new_primMulNat0(Succ(vz2580000), Zero) -> Zero new_primMulNat0(Zero, Succ(vz2570100)) -> Zero new_primPlusNat0(Zero, vz2570100) -> Succ(vz2570100) new_compare(vz2550, vz25600, ty_Ordering) -> new_compare9(vz2550, vz25600) new_sr2(Neg(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare1(vz2550, vz25600) -> error([]) new_compare14(vz25600, vz256100, ty_Char) -> new_compare1(vz25600, vz256100) new_compare6(vz2550, vz25600, bf, bg) -> error([]) new_compare(vz2550, vz25600, app(app(app(ty_@3, bh), ca), cb)) -> new_compare3(vz2550, vz25600, bh, ca, cb) new_compare9(vz2550, vz25600) -> error([]) new_merge1(vz255, vz2560, [], ba) -> new_merge2(vz255, vz2560, ba) new_compare4(Neg(Zero), Neg(Zero)) -> EQ The set Q consists of the following terms: new_compare14(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_compare(x0, x1, ty_Bool) new_compare8(x0, x1) new_compare6(x0, x1, x2, x3) new_compare12(x0, x1, x2, x3) new_merge1(x0, :(x1, x2), :(x3, x4), x5) new_compare10(x0, x1, x2) new_compare14(x0, x1, ty_@0) new_compare3(x0, x1, x2, x3, x4) new_compare14(x0, x1, ty_Char) new_merge2([], :(x0, x1), x2) new_primPlusNat0(Succ(x0), x1) new_merge00(x0, x1, x2, x3, GT, x4) new_compare(x0, x1, app(ty_Maybe, x2)) new_primCmpNat1(Zero, Succ(x0)) new_compare14(x0, x1, ty_Integer) new_compare9(x0, x1) new_compare(x0, x1, ty_Double) new_merge_pairs0(:(x0, :(x1, x2)), x3) new_sr2(Pos(x0), x1) new_compare(x0, x1, ty_Ordering) new_primPlusNat1(Zero, Succ(x0)) new_compare(x0, x1, ty_Char) new_compare(x0, x1, ty_@0) new_merge00(x0, x1, x2, x3, EQ, x4) new_compare5(Float(x0, Neg(x1)), Float(x2, Pos(x3))) new_compare5(Float(x0, Pos(x1)), Float(x2, Neg(x3))) new_merge00(x0, x1, x2, x3, LT, x4) new_compare7(x0, x1, x2) new_compare14(x0, x1, ty_Bool) new_sr1(Pos(x0), x1) new_compare0(x0, x1) new_compare4(Pos(Succ(x0)), Pos(x1)) new_merge1(x0, [], :(x1, x2), x3) new_primPlusNat1(Succ(x0), Succ(x1)) new_primMulNat0(Zero, Zero) new_sr0(x0, Pos(x1)) new_primPlusNat1(Zero, Zero) new_primPlusNat1(Succ(x0), Zero) new_compare(x0, x1, app(app(ty_Either, x2), x3)) new_compare14(x0, x1, ty_Float) new_primCmpNat1(Succ(x0), Succ(x1)) new_compare4(Pos(Zero), Pos(Zero)) new_primCmpNat2(Zero, x0) new_primCmpNat1(Zero, Zero) new_compare4(Pos(Zero), Pos(Succ(x0))) new_compare(x0, x1, app(ty_Ratio, x2)) new_compare11(x0, x1) new_compare5(Float(x0, Neg(x1)), Float(x2, Neg(x3))) new_compare(x0, x1, ty_Float) new_compare(x0, x1, ty_Integer) new_sr1(Neg(x0), x1) new_compare(x0, x1, ty_Int) new_primMulNat0(Succ(x0), Succ(x1)) new_merge2(:(x0, x1), :(x2, x3), x4) new_compare2(x0, x1) new_compare(x0, x1, app(ty_[], x2)) new_compare14(x0, x1, app(app(ty_@2, x2), x3)) new_compare4(Neg(Succ(x0)), Neg(x1)) new_merge2(x0, [], x1) new_primCmpNat0(x0, Succ(x1)) new_compare14(x0, x1, ty_Ordering) new_compare14(x0, x1, app(ty_Ratio, x2)) new_sr0(x0, Neg(x1)) new_compare14(x0, x1, app(ty_Maybe, x2)) new_primCmpNat0(x0, Zero) new_compare14(x0, x1, app(app(ty_Either, x2), x3)) new_compare4(Pos(Zero), Neg(Succ(x0))) new_compare4(Neg(Zero), Pos(Succ(x0))) new_compare(x0, x1, app(app(ty_@2, x2), x3)) new_sr(x0, Neg(x1)) new_sr2(Neg(x0), x1) new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_compare4(Pos(Zero), Neg(Zero)) new_compare4(Neg(Zero), Pos(Zero)) new_merge1(x0, x1, [], x2) new_compare14(x0, x1, ty_Int) new_primMulNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, x0) new_compare4(Neg(Succ(x0)), Pos(x1)) new_compare4(Pos(Succ(x0)), Neg(x1)) new_merge_pairs0(:(x0, []), x1) new_compare14(x0, x1, ty_Double) new_compare1(x0, x1) new_sr(x0, Pos(x1)) new_primCmpNat1(Succ(x0), Zero) new_primMulNat0(Succ(x0), Zero) new_compare4(Neg(Zero), Neg(Zero)) new_primCmpNat2(Succ(x0), x1) new_merge_pairs0([], x0) new_compare14(x0, x1, app(ty_[], x2)) new_compare13(x0, x1, x2) new_compare4(Neg(Zero), Neg(Succ(x0))) new_compare5(Float(x0, Pos(x1)), Float(x2, Pos(x3))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (18) QDPSizeChangeProof (EQUIVALENT) We used the following order together with the size-change analysis [AAECC05] to show that there are no infinite chains for this DP problem. Order:Polynomial interpretation [POLO]: POL(:(x_1, x_2)) = 1 + x_2 POL(EQ) = 0 POL(GT) = 0 POL(LT) = 0 POL(Neg(x_1)) = 1 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = x_1 POL(Zero) = 0 POL([]) = 0 POL(new_merge2(x_1, x_2, x_3)) = x_1 POL(new_merge_pairs0(x_1, x_2)) = x_1 POL(new_primCmpNat0(x_1, x_2)) = 1 + x_2 POL(new_primCmpNat1(x_1, x_2)) = 0 POL(new_primCmpNat2(x_1, x_2)) = x_1 + x_2 POL(new_primMulNat0(x_1, x_2)) = x_1 + x_2 POL(new_primPlusNat0(x_1, x_2)) = x_2 POL(new_primPlusNat1(x_1, x_2)) = 0 POL(new_sr(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr0(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr1(x_1, x_2)) = 1 POL(new_sr2(x_1, x_2)) = 1 From the DPs we obtained the following set of size-change graphs: *new_mergesort'(vz255, :(vz2560, :(vz25610, vz25611)), ba) -> new_mergesort'(new_merge1(vz255, vz2560, vz25610, ba), new_merge_pairs0(vz25611, ba), ba) (allowed arguments on rhs = {2, 3}) The graph contains the following edges 2 > 2, 3 >= 3 We oriented the following set of usable rules [AAECC05,FROCOS05]. new_merge_pairs0([], ba) -> [] new_merge_pairs0(:(vz256110, []), ba) -> :(vz256110, []) new_merge_pairs0(:(vz256110, :(vz2561110, vz2561111)), ba) -> :(new_merge2(vz256110, vz2561110, ba), new_merge_pairs0(vz2561111, ba)) ---------------------------------------- (19) YES ---------------------------------------- (20) Obligation: Q DP problem: The TRS P consists of the following rules: new_map(:(vz3110, vz3111)) -> new_map(vz3111) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (21) 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_map(:(vz3110, vz3111)) -> new_map(vz3111) The graph contains the following edges 1 > 1 ---------------------------------------- (22) YES ---------------------------------------- (23) Obligation: Q DP problem: The TRS P consists of the following rules: new_primMulNat(Succ(vz2580000), Succ(vz2570100)) -> new_primMulNat(vz2580000, Succ(vz2570100)) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (24) 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_primMulNat(Succ(vz2580000), Succ(vz2570100)) -> new_primMulNat(vz2580000, Succ(vz2570100)) The graph contains the following edges 1 > 1, 2 >= 2 ---------------------------------------- (25) YES ---------------------------------------- (26) Obligation: Q DP problem: The TRS P consists of the following rules: new_merge0(vz265, vz266, vz267, vz268, EQ, ba) -> new_merge(vz267, :(vz265, vz268), ba) new_merge0(vz265, vz266, vz267, vz268, GT, ba) -> new_merge(:(vz266, vz267), vz268, ba) new_merge0(vz265, vz266, vz267, vz268, LT, ba) -> new_merge(vz267, :(vz265, vz268), ba) new_merge(:(vz2550, vz2551), :(vz25600, vz25601), bb) -> new_merge0(vz25600, vz2550, vz2551, vz25601, new_compare(vz2550, vz25600, bb), bb) The TRS R consists of the following rules: new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_sr2(Pos(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_compare11(vz2550, vz25600) -> error([]) new_compare3(vz2550, vz25600, bc, bd, be) -> error([]) new_primMulNat0(Zero, Zero) -> Zero new_compare(vz2550, vz25600, app(ty_Maybe, cd)) -> new_compare13(vz2550, vz25600, cd) new_compare(vz2550, vz25600, app(app(ty_@2, cb), cc)) -> new_compare12(vz2550, vz25600, cb, cc) new_sr(vz258010, Pos(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Succ(vz2560000)) -> new_primCmpNat1(vz255000, vz2560000) new_compare(vz2550, vz25600, ty_Int) -> new_compare4(vz2550, vz25600) new_compare12(vz2550, vz25600, cb, cc) -> error([]) new_primMulNat0(Succ(vz2580000), Succ(vz2570100)) -> new_primPlusNat0(new_primMulNat0(vz2580000, Succ(vz2570100)), vz2570100) new_compare4(Pos(Zero), Neg(Succ(vz2560000))) -> GT new_sr1(Pos(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare(vz2550, vz25600, app(ty_[], bh)) -> new_compare7(vz2550, vz25600, bh) new_compare4(Pos(Zero), Neg(Zero)) -> EQ new_compare4(Neg(Zero), Pos(Zero)) -> EQ new_primCmpNat1(Zero, Zero) -> EQ new_compare(vz2550, vz25600, ty_Char) -> new_compare1(vz2550, vz25600) new_compare(vz2550, vz25600, ty_Float) -> new_compare5(vz2550, vz25600) new_compare4(Pos(Zero), Pos(Succ(vz2560000))) -> new_primCmpNat2(Zero, vz2560000) new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr(vz258010, vz25700)) new_compare8(vz2550, vz25600) -> error([]) new_primCmpNat1(Succ(vz2550000), Zero) -> GT new_compare4(Neg(Zero), Neg(Succ(vz2560000))) -> new_primCmpNat0(vz2560000, Zero) new_primPlusNat1(Succ(vz27000), Zero) -> Succ(vz27000) new_primPlusNat1(Zero, Succ(vz25701000)) -> Succ(vz25701000) new_compare0(vz2550, vz25600) -> error([]) new_compare4(Pos(Succ(vz255000)), Pos(vz256000)) -> new_primCmpNat0(vz255000, vz256000) new_compare(vz2550, vz25600, ty_Double) -> new_compare11(vz2550, vz25600) new_compare10(vz2550, vz25600, ca) -> error([]) new_compare4(Neg(Succ(vz255000)), Pos(vz256000)) -> LT new_compare(vz2550, vz25600, ty_@0) -> new_compare2(vz2550, vz25600) new_primCmpNat2(Succ(vz2560000), vz255000) -> new_primCmpNat1(vz2560000, vz255000) new_sr0(vz258010, Pos(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Zero) -> GT new_compare4(Neg(Succ(vz255000)), Neg(vz256000)) -> new_primCmpNat2(vz256000, vz255000) new_compare(vz2550, vz25600, app(ty_Ratio, ca)) -> new_compare10(vz2550, vz25600, ca) new_compare4(Pos(Succ(vz255000)), Neg(vz256000)) -> GT new_sr(vz258010, Neg(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr(vz258010, vz25700)) new_compare(vz2550, vz25600, ty_Integer) -> new_compare0(vz2550, vz25600) new_primCmpNat2(Zero, vz255000) -> LT new_compare(vz2550, vz25600, ty_Bool) -> new_compare8(vz2550, vz25600) new_compare4(Neg(Zero), Pos(Succ(vz2560000))) -> LT new_primCmpNat1(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat1(vz2550000, vz25600000) new_compare4(Pos(Zero), Pos(Zero)) -> EQ new_compare13(vz2550, vz25600, cd) -> error([]) new_primPlusNat0(Succ(vz2700), vz2570100) -> Succ(Succ(new_primPlusNat1(vz2700, vz2570100))) new_compare2(vz2550, vz25600) -> error([]) new_sr1(Neg(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_sr0(vz258010, Neg(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_compare7(vz2550, vz25600, bh) -> error([]) new_compare(vz2550, vz25600, app(app(ty_Either, bf), bg)) -> new_compare6(vz2550, vz25600, bf, bg) new_primCmpNat1(Zero, Succ(vz25600000)) -> LT new_primPlusNat1(Succ(vz27000), Succ(vz25701000)) -> Succ(Succ(new_primPlusNat1(vz27000, vz25701000))) new_primPlusNat1(Zero, Zero) -> Zero new_primMulNat0(Succ(vz2580000), Zero) -> Zero new_primMulNat0(Zero, Succ(vz2570100)) -> Zero new_primPlusNat0(Zero, vz2570100) -> Succ(vz2570100) new_compare(vz2550, vz25600, ty_Ordering) -> new_compare9(vz2550, vz25600) new_sr2(Neg(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare1(vz2550, vz25600) -> error([]) new_compare(vz2550, vz25600, app(app(app(ty_@3, bc), bd), be)) -> new_compare3(vz2550, vz25600, bc, bd, be) new_compare6(vz2550, vz25600, bf, bg) -> error([]) new_compare9(vz2550, vz25600) -> error([]) new_compare4(Neg(Zero), Neg(Zero)) -> EQ The set Q consists of the following terms: new_compare7(x0, x1, x2) new_compare(x0, x1, ty_Bool) new_compare8(x0, x1) new_compare6(x0, x1, x2, x3) new_primPlusNat0(Succ(x0), x1) new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_primCmpNat1(Zero, Succ(x0)) new_compare9(x0, x1) new_compare(x0, x1, ty_Double) new_sr2(Pos(x0), x1) new_compare(x0, x1, ty_Ordering) new_primPlusNat1(Zero, Succ(x0)) new_compare(x0, x1, ty_Char) new_compare(x0, x1, ty_@0) new_compare10(x0, x1, x2) new_compare5(Float(x0, Neg(x1)), Float(x2, Pos(x3))) new_compare5(Float(x0, Pos(x1)), Float(x2, Neg(x3))) new_sr1(Pos(x0), x1) new_compare0(x0, x1) new_compare4(Pos(Succ(x0)), Pos(x1)) new_primPlusNat1(Succ(x0), Succ(x1)) new_primMulNat0(Zero, Zero) new_sr0(x0, Pos(x1)) new_primPlusNat1(Zero, Zero) new_primPlusNat1(Succ(x0), Zero) new_compare(x0, x1, app(app(ty_Either, x2), x3)) new_compare(x0, x1, app(app(ty_@2, x2), x3)) new_primCmpNat1(Succ(x0), Succ(x1)) new_compare4(Pos(Zero), Pos(Zero)) new_primCmpNat2(Zero, x0) new_primCmpNat1(Zero, Zero) new_compare4(Pos(Zero), Pos(Succ(x0))) new_compare11(x0, x1) new_compare(x0, x1, app(ty_Maybe, x2)) new_compare5(Float(x0, Neg(x1)), Float(x2, Neg(x3))) new_compare(x0, x1, app(ty_[], x2)) new_compare(x0, x1, ty_Float) new_compare(x0, x1, ty_Integer) new_sr1(Neg(x0), x1) new_compare(x0, x1, ty_Int) new_primMulNat0(Succ(x0), Succ(x1)) new_compare2(x0, x1) new_compare(x0, x1, app(ty_Ratio, x2)) new_compare4(Neg(Succ(x0)), Neg(x1)) new_primCmpNat0(x0, Succ(x1)) new_sr0(x0, Neg(x1)) new_primCmpNat0(x0, Zero) new_compare4(Pos(Zero), Neg(Succ(x0))) new_compare4(Neg(Zero), Pos(Succ(x0))) new_sr(x0, Neg(x1)) new_compare13(x0, x1, x2) new_sr2(Neg(x0), x1) new_compare4(Pos(Zero), Neg(Zero)) new_compare4(Neg(Zero), Pos(Zero)) new_primMulNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, x0) new_compare4(Neg(Succ(x0)), Pos(x1)) new_compare4(Pos(Succ(x0)), Neg(x1)) new_compare1(x0, x1) new_compare3(x0, x1, x2, x3, x4) new_sr(x0, Pos(x1)) new_primCmpNat1(Succ(x0), Zero) new_primMulNat0(Succ(x0), Zero) new_compare4(Neg(Zero), Neg(Zero)) new_compare12(x0, x1, x2, x3) new_primCmpNat2(Succ(x0), x1) new_compare4(Neg(Zero), Neg(Succ(x0))) new_compare5(Float(x0, Pos(x1)), Float(x2, Pos(x3))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (27) QDPOrderProof (EQUIVALENT) We use the reduction pair processor [LPAR04,JAR06]. The following pairs can be oriented strictly and are deleted. new_merge(:(vz2550, vz2551), :(vz25600, vz25601), bb) -> new_merge0(vz25600, vz2550, vz2551, vz25601, new_compare(vz2550, vz25600, bb), bb) The remaining pairs can at least be oriented weakly. Used ordering: Polynomial interpretation [POLO]: POL(:(x_1, x_2)) = 1 + x_1 + x_2 POL(EQ) = 0 POL(Float(x_1, x_2)) = x_1 POL(GT) = 0 POL(LT) = 0 POL(Neg(x_1)) = 0 POL(Pos(x_1)) = 0 POL(Succ(x_1)) = 0 POL(Zero) = 0 POL([]) = 1 POL(app(x_1, x_2)) = 1 + x_1 + x_2 POL(error(x_1)) = 1 + x_1 POL(new_compare(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(new_compare0(x_1, x_2)) = 1 + x_1 + x_2 POL(new_compare1(x_1, x_2)) = 1 + x_1 + x_2 POL(new_compare10(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_compare11(x_1, x_2)) = 1 + x_1 + x_2 POL(new_compare12(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(new_compare13(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_compare2(x_1, x_2)) = 1 + x_1 + x_2 POL(new_compare3(x_1, x_2, x_3, x_4, x_5)) = 1 + x_1 + x_2 + x_3 + x_4 + x_5 POL(new_compare4(x_1, x_2)) = 0 POL(new_compare5(x_1, x_2)) = 1 POL(new_compare6(x_1, x_2, x_3, x_4)) = 1 + x_1 + x_2 + x_3 + x_4 POL(new_compare7(x_1, x_2, x_3)) = 1 + x_1 + x_2 + x_3 POL(new_compare8(x_1, x_2)) = 1 + x_1 + x_2 POL(new_compare9(x_1, x_2)) = 1 + x_1 + x_2 POL(new_merge(x_1, x_2, x_3)) = x_1 + x_2 + x_3 POL(new_merge0(x_1, x_2, x_3, x_4, x_5, x_6)) = 1 + x_1 + x_2 + x_3 + x_4 + x_6 POL(new_primCmpNat0(x_1, x_2)) = 1 + x_1 POL(new_primCmpNat1(x_1, x_2)) = 1 POL(new_primCmpNat2(x_1, x_2)) = 1 + x_2 POL(new_primMulNat0(x_1, x_2)) = 0 POL(new_primPlusNat0(x_1, x_2)) = x_2 POL(new_primPlusNat1(x_1, x_2)) = 0 POL(new_sr(x_1, x_2)) = x_1 + x_2 POL(new_sr0(x_1, x_2)) = x_1 + x_2 POL(new_sr1(x_1, x_2)) = 1 + x_1 + x_2 POL(new_sr2(x_1, x_2)) = 1 + x_1 + x_2 POL(ty_@0) = 1 POL(ty_@2) = 1 POL(ty_@3) = 1 POL(ty_Bool) = 1 POL(ty_Char) = 1 POL(ty_Double) = 1 POL(ty_Either) = 1 POL(ty_Float) = 1 POL(ty_Int) = 1 POL(ty_Integer) = 1 POL(ty_Maybe) = 1 POL(ty_Ordering) = 1 POL(ty_Ratio) = 1 POL(ty_[]) = 1 The following usable rules [FROCOS05] with respect to the argument filtering of the ordering [JAR06] were oriented: none ---------------------------------------- (28) Obligation: Q DP problem: The TRS P consists of the following rules: new_merge0(vz265, vz266, vz267, vz268, EQ, ba) -> new_merge(vz267, :(vz265, vz268), ba) new_merge0(vz265, vz266, vz267, vz268, GT, ba) -> new_merge(:(vz266, vz267), vz268, ba) new_merge0(vz265, vz266, vz267, vz268, LT, ba) -> new_merge(vz267, :(vz265, vz268), ba) The TRS R consists of the following rules: new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_sr2(Pos(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_compare11(vz2550, vz25600) -> error([]) new_compare3(vz2550, vz25600, bc, bd, be) -> error([]) new_primMulNat0(Zero, Zero) -> Zero new_compare(vz2550, vz25600, app(ty_Maybe, cd)) -> new_compare13(vz2550, vz25600, cd) new_compare(vz2550, vz25600, app(app(ty_@2, cb), cc)) -> new_compare12(vz2550, vz25600, cb, cc) new_sr(vz258010, Pos(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Succ(vz2560000)) -> new_primCmpNat1(vz255000, vz2560000) new_compare(vz2550, vz25600, ty_Int) -> new_compare4(vz2550, vz25600) new_compare12(vz2550, vz25600, cb, cc) -> error([]) new_primMulNat0(Succ(vz2580000), Succ(vz2570100)) -> new_primPlusNat0(new_primMulNat0(vz2580000, Succ(vz2570100)), vz2570100) new_compare4(Pos(Zero), Neg(Succ(vz2560000))) -> GT new_sr1(Pos(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare(vz2550, vz25600, app(ty_[], bh)) -> new_compare7(vz2550, vz25600, bh) new_compare4(Pos(Zero), Neg(Zero)) -> EQ new_compare4(Neg(Zero), Pos(Zero)) -> EQ new_primCmpNat1(Zero, Zero) -> EQ new_compare(vz2550, vz25600, ty_Char) -> new_compare1(vz2550, vz25600) new_compare(vz2550, vz25600, ty_Float) -> new_compare5(vz2550, vz25600) new_compare4(Pos(Zero), Pos(Succ(vz2560000))) -> new_primCmpNat2(Zero, vz2560000) new_compare5(Float(vz25800, Neg(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr1(vz25800, vz257010), new_sr(vz258010, vz25700)) new_compare8(vz2550, vz25600) -> error([]) new_primCmpNat1(Succ(vz2550000), Zero) -> GT new_compare4(Neg(Zero), Neg(Succ(vz2560000))) -> new_primCmpNat0(vz2560000, Zero) new_primPlusNat1(Succ(vz27000), Zero) -> Succ(vz27000) new_primPlusNat1(Zero, Succ(vz25701000)) -> Succ(vz25701000) new_compare0(vz2550, vz25600) -> error([]) new_compare4(Pos(Succ(vz255000)), Pos(vz256000)) -> new_primCmpNat0(vz255000, vz256000) new_compare(vz2550, vz25600, ty_Double) -> new_compare11(vz2550, vz25600) new_compare10(vz2550, vz25600, ca) -> error([]) new_compare4(Neg(Succ(vz255000)), Pos(vz256000)) -> LT new_compare(vz2550, vz25600, ty_@0) -> new_compare2(vz2550, vz25600) new_primCmpNat2(Succ(vz2560000), vz255000) -> new_primCmpNat1(vz2560000, vz255000) new_sr0(vz258010, Pos(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_primCmpNat0(vz255000, Zero) -> GT new_compare4(Neg(Succ(vz255000)), Neg(vz256000)) -> new_primCmpNat2(vz256000, vz255000) new_compare(vz2550, vz25600, app(ty_Ratio, ca)) -> new_compare10(vz2550, vz25600, ca) new_compare4(Pos(Succ(vz255000)), Neg(vz256000)) -> GT new_sr(vz258010, Neg(vz257000)) -> Pos(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Neg(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr(vz258010, vz25700)) new_compare(vz2550, vz25600, ty_Integer) -> new_compare0(vz2550, vz25600) new_primCmpNat2(Zero, vz255000) -> LT new_compare(vz2550, vz25600, ty_Bool) -> new_compare8(vz2550, vz25600) new_compare4(Neg(Zero), Pos(Succ(vz2560000))) -> LT new_primCmpNat1(Succ(vz2550000), Succ(vz25600000)) -> new_primCmpNat1(vz2550000, vz25600000) new_compare4(Pos(Zero), Pos(Zero)) -> EQ new_compare13(vz2550, vz25600, cd) -> error([]) new_primPlusNat0(Succ(vz2700), vz2570100) -> Succ(Succ(new_primPlusNat1(vz2700, vz2570100))) new_compare2(vz2550, vz25600) -> error([]) new_sr1(Neg(vz258000), vz257010) -> Pos(new_primMulNat0(vz258000, vz257010)) new_sr0(vz258010, Neg(vz257000)) -> Neg(new_primMulNat0(vz258010, vz257000)) new_compare5(Float(vz25800, Pos(vz258010)), Float(vz25700, Pos(vz257010))) -> new_compare4(new_sr2(vz25800, vz257010), new_sr0(vz258010, vz25700)) new_compare7(vz2550, vz25600, bh) -> error([]) new_compare(vz2550, vz25600, app(app(ty_Either, bf), bg)) -> new_compare6(vz2550, vz25600, bf, bg) new_primCmpNat1(Zero, Succ(vz25600000)) -> LT new_primPlusNat1(Succ(vz27000), Succ(vz25701000)) -> Succ(Succ(new_primPlusNat1(vz27000, vz25701000))) new_primPlusNat1(Zero, Zero) -> Zero new_primMulNat0(Succ(vz2580000), Zero) -> Zero new_primMulNat0(Zero, Succ(vz2570100)) -> Zero new_primPlusNat0(Zero, vz2570100) -> Succ(vz2570100) new_compare(vz2550, vz25600, ty_Ordering) -> new_compare9(vz2550, vz25600) new_sr2(Neg(vz258000), vz257010) -> Neg(new_primMulNat0(vz258000, vz257010)) new_compare1(vz2550, vz25600) -> error([]) new_compare(vz2550, vz25600, app(app(app(ty_@3, bc), bd), be)) -> new_compare3(vz2550, vz25600, bc, bd, be) new_compare6(vz2550, vz25600, bf, bg) -> error([]) new_compare9(vz2550, vz25600) -> error([]) new_compare4(Neg(Zero), Neg(Zero)) -> EQ The set Q consists of the following terms: new_compare7(x0, x1, x2) new_compare(x0, x1, ty_Bool) new_compare8(x0, x1) new_compare6(x0, x1, x2, x3) new_primPlusNat0(Succ(x0), x1) new_compare(x0, x1, app(app(app(ty_@3, x2), x3), x4)) new_primCmpNat1(Zero, Succ(x0)) new_compare9(x0, x1) new_compare(x0, x1, ty_Double) new_sr2(Pos(x0), x1) new_compare(x0, x1, ty_Ordering) new_primPlusNat1(Zero, Succ(x0)) new_compare(x0, x1, ty_Char) new_compare(x0, x1, ty_@0) new_compare10(x0, x1, x2) new_compare5(Float(x0, Neg(x1)), Float(x2, Pos(x3))) new_compare5(Float(x0, Pos(x1)), Float(x2, Neg(x3))) new_sr1(Pos(x0), x1) new_compare0(x0, x1) new_compare4(Pos(Succ(x0)), Pos(x1)) new_primPlusNat1(Succ(x0), Succ(x1)) new_primMulNat0(Zero, Zero) new_sr0(x0, Pos(x1)) new_primPlusNat1(Zero, Zero) new_primPlusNat1(Succ(x0), Zero) new_compare(x0, x1, app(app(ty_Either, x2), x3)) new_compare(x0, x1, app(app(ty_@2, x2), x3)) new_primCmpNat1(Succ(x0), Succ(x1)) new_compare4(Pos(Zero), Pos(Zero)) new_primCmpNat2(Zero, x0) new_primCmpNat1(Zero, Zero) new_compare4(Pos(Zero), Pos(Succ(x0))) new_compare11(x0, x1) new_compare(x0, x1, app(ty_Maybe, x2)) new_compare5(Float(x0, Neg(x1)), Float(x2, Neg(x3))) new_compare(x0, x1, app(ty_[], x2)) new_compare(x0, x1, ty_Float) new_compare(x0, x1, ty_Integer) new_sr1(Neg(x0), x1) new_compare(x0, x1, ty_Int) new_primMulNat0(Succ(x0), Succ(x1)) new_compare2(x0, x1) new_compare(x0, x1, app(ty_Ratio, x2)) new_compare4(Neg(Succ(x0)), Neg(x1)) new_primCmpNat0(x0, Succ(x1)) new_sr0(x0, Neg(x1)) new_primCmpNat0(x0, Zero) new_compare4(Pos(Zero), Neg(Succ(x0))) new_compare4(Neg(Zero), Pos(Succ(x0))) new_sr(x0, Neg(x1)) new_compare13(x0, x1, x2) new_sr2(Neg(x0), x1) new_compare4(Pos(Zero), Neg(Zero)) new_compare4(Neg(Zero), Pos(Zero)) new_primMulNat0(Zero, Succ(x0)) new_primPlusNat0(Zero, x0) new_compare4(Neg(Succ(x0)), Pos(x1)) new_compare4(Pos(Succ(x0)), Neg(x1)) new_compare1(x0, x1) new_compare3(x0, x1, x2, x3, x4) new_sr(x0, Pos(x1)) new_primCmpNat1(Succ(x0), Zero) new_primMulNat0(Succ(x0), Zero) new_compare4(Neg(Zero), Neg(Zero)) new_compare12(x0, x1, x2, x3) new_primCmpNat2(Succ(x0), x1) new_compare4(Neg(Zero), Neg(Succ(x0))) new_compare5(Float(x0, Pos(x1)), Float(x2, Pos(x3))) We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (29) DependencyGraphProof (EQUIVALENT) The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes. ---------------------------------------- (30) TRUE ---------------------------------------- (31) Obligation: Q DP problem: The TRS P consists of the following rules: new_primPlusNat(Succ(vz27000), Succ(vz25701000)) -> new_primPlusNat(vz27000, vz25701000) R is empty. Q is empty. We have to consider all minimal (P,Q,R)-chains. ---------------------------------------- (32) 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_primPlusNat(Succ(vz27000), Succ(vz25701000)) -> new_primPlusNat(vz27000, vz25701000) The graph contains the following edges 1 > 1, 2 > 2 ---------------------------------------- (33) YES