9.51/4.00 YES 11.92/4.67 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 11.92/4.67 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.92/4.67 11.92/4.67 11.92/4.67 H-Termination with start terms of the given HASKELL could be proven: 11.92/4.67 11.92/4.67 (0) HASKELL 11.92/4.67 (1) LR [EQUIVALENT, 0 ms] 11.92/4.67 (2) HASKELL 11.92/4.67 (3) BR [EQUIVALENT, 0 ms] 11.92/4.67 (4) HASKELL 11.92/4.67 (5) COR [EQUIVALENT, 0 ms] 11.92/4.67 (6) HASKELL 11.92/4.67 (7) LetRed [EQUIVALENT, 6 ms] 11.92/4.67 (8) HASKELL 11.92/4.67 (9) NumRed [SOUND, 0 ms] 11.92/4.67 (10) HASKELL 11.92/4.67 (11) Narrow [SOUND, 0 ms] 11.92/4.67 (12) QDP 11.92/4.67 (13) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.92/4.67 (14) YES 11.92/4.67 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (0) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad b => Int -> b a -> b (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (1) LR (EQUIVALENT) 11.92/4.67 Lambda Reductions: 11.92/4.67 The following Lambda expression 11.92/4.67 "\_->q" 11.92/4.67 is transformed to 11.92/4.67 "gtGt0 q _ = q; 11.92/4.67 " 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (2) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad b => Int -> b a -> b (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (3) BR (EQUIVALENT) 11.92/4.67 Replaced joker patterns by fresh variables and removed binding patterns. 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (4) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad a => Int -> a b -> a (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (5) COR (EQUIVALENT) 11.92/4.67 Cond Reductions: 11.92/4.67 The following Function with conditions 11.92/4.67 "undefined |Falseundefined; 11.92/4.67 " 11.92/4.67 is transformed to 11.92/4.67 "undefined = undefined1; 11.92/4.67 " 11.92/4.67 "undefined0 True = undefined; 11.92/4.67 " 11.92/4.67 "undefined1 = undefined0 False; 11.92/4.67 " 11.92/4.67 The following Function with conditions 11.92/4.67 "take n vy|n <= 0[]; 11.92/4.67 take vz [] = []; 11.92/4.67 take n (x : xs) = x : take (n - 1) xs; 11.92/4.67 " 11.92/4.67 is transformed to 11.92/4.67 "take n vy = take3 n vy; 11.92/4.67 take vz [] = take1 vz []; 11.92/4.67 take n (x : xs) = take0 n (x : xs); 11.92/4.67 " 11.92/4.67 "take0 n (x : xs) = x : take (n - 1) xs; 11.92/4.67 " 11.92/4.67 "take1 vz [] = []; 11.92/4.67 take1 wx wy = take0 wx wy; 11.92/4.67 " 11.92/4.67 "take2 n vy True = []; 11.92/4.67 take2 n vy False = take1 n vy; 11.92/4.67 " 11.92/4.67 "take3 n vy = take2 n vy (n <= 0); 11.92/4.67 take3 wz xu = take1 wz xu; 11.92/4.67 " 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (6) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad a => Int -> a b -> a (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (7) LetRed (EQUIVALENT) 11.92/4.67 Let/Where Reductions: 11.92/4.67 The bindings of the following Let/Where expression 11.92/4.67 "xs where { 11.92/4.67 xs = x : xs; 11.92/4.67 } 11.92/4.67 " 11.92/4.67 are unpacked to the following functions on top level 11.92/4.67 "repeatXs xv = xv : repeatXs xv; 11.92/4.67 " 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (8) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad a => Int -> a b -> a (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (9) NumRed (SOUND) 11.92/4.67 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (10) 11.92/4.67 Obligation: 11.92/4.67 mainModule Main 11.92/4.67 module Maybe where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Main where { 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Monad; 11.92/4.67 import qualified Prelude; 11.92/4.67 } 11.92/4.67 module Monad where { 11.92/4.67 import qualified Main; 11.92/4.67 import qualified Maybe; 11.92/4.67 import qualified Prelude; 11.92/4.67 replicateM_ :: Monad b => Int -> b a -> b (); 11.92/4.67 replicateM_ n x = sequence_ (replicate n x); 11.92/4.67 11.92/4.67 } 11.92/4.67 11.92/4.67 ---------------------------------------- 11.92/4.67 11.92/4.67 (11) Narrow (SOUND) 11.92/4.67 Haskell To QDPs 11.92/4.67 11.92/4.67 digraph dp_graph { 11.92/4.67 node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM_",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.92/4.67 3[label="Monad.replicateM_ xw3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.92/4.67 4[label="Monad.replicateM_ xw3 xw4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 11.92/4.67 5[label="sequence_ (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 11.92/4.67 6[label="foldr (>>) (return ()) (replicate xw3 xw4)",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 11.92/4.67 7[label="foldr (>>) (return ()) (take xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 11.92/4.67 8[label="foldr (>>) (return ()) (take3 xw3 (repeat xw4))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 11.92/4.67 9[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (xw3 <= Pos Zero))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 11.92/4.67 10[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (compare xw3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 11.92/4.67 11[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (compare xw3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3]; 11.92/4.67 12[label="foldr (>>) (return ()) (take2 xw3 (repeat xw4) (not (primCmpInt xw3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];81[label="xw3/Pos xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 81[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 81 -> 13[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 82[label="xw3/Neg xw30",fontsize=10,color="white",style="solid",shape="box"];12 -> 82[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 82 -> 14[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 13[label="foldr (>>) (return ()) (take2 (Pos xw30) (repeat xw4) (not (primCmpInt (Pos xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];83[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];13 -> 83[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 83 -> 15[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 84[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 84[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 84 -> 16[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 14[label="foldr (>>) (return ()) (take2 (Neg xw30) (repeat xw4) (not (primCmpInt (Neg xw30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];85[label="xw30/Succ xw300",fontsize=10,color="white",style="solid",shape="box"];14 -> 85[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 85 -> 17[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 86[label="xw30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 86[label="",style="solid", color="burlywood", weight=9]; 11.92/4.67 86 -> 18[label="",style="solid", color="burlywood", weight=3]; 11.92/4.67 15[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpInt (Pos (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 11.92/4.67 16[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 11.92/4.67 17[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (primCmpInt (Neg (Succ xw300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 11.92/4.67 18[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 11.92/4.67 19[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (primCmpNat (Succ xw300) Zero == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 11.92/4.67 20[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 11.92/4.67 21[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 11.92/4.67 22[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 11.92/4.67 23[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 11.92/4.67 24[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 11.92/4.67 25[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 11.92/4.67 26[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) (not False))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 11.92/4.68 27[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) (not True))",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3]; 11.92/4.68 28[label="foldr (>>) (return ()) (take2 (Pos Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 11.92/4.68 29[label="foldr (>>) (return ()) (take2 (Neg (Succ xw300)) (repeat xw4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 11.92/4.68 30[label="foldr (>>) (return ()) (take2 (Neg Zero) (repeat xw4) True)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3]; 11.92/4.68 31[label="foldr (>>) (return ()) (take2 (Pos (Succ xw300)) (repeat xw4) False)",fontsize=16,color="black",shape="box"];31 -> 35[label="",style="solid", color="black", weight=3]; 11.92/4.68 32[label="foldr (>>) (return ()) []",fontsize=16,color="black",shape="triangle"];32 -> 36[label="",style="solid", color="black", weight=3]; 11.92/4.68 33 -> 32[label="",style="dashed", color="red", weight=0]; 11.92/4.68 33[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];34 -> 32[label="",style="dashed", color="red", weight=0]; 11.92/4.68 34[label="foldr (>>) (return ()) []",fontsize=16,color="magenta"];35[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeat xw4))",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 11.92/4.68 36[label="return ()",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3]; 11.92/4.68 37 -> 39[label="",style="dashed", color="red", weight=0]; 11.92/4.68 37[label="foldr (>>) (return ()) (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="magenta"];37 -> 40[label="",style="dashed", color="magenta", weight=3]; 11.92/4.68 38[label="primretIO ()",fontsize=16,color="black",shape="box"];38 -> 41[label="",style="solid", color="black", weight=3]; 11.92/4.68 40 -> 36[label="",style="dashed", color="red", weight=0]; 11.92/4.68 40[label="return ()",fontsize=16,color="magenta"];39[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (repeatXs xw4))",fontsize=16,color="black",shape="triangle"];39 -> 42[label="",style="solid", color="black", weight=3]; 11.92/4.68 41[label="AProVE_IO ()",fontsize=16,color="green",shape="box"];42[label="foldr (>>) xw5 (take1 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3]; 11.92/4.68 43[label="foldr (>>) xw5 (take0 (Pos (Succ xw300)) (xw4 : repeatXs xw4))",fontsize=16,color="black",shape="box"];43 -> 44[label="",style="solid", color="black", weight=3]; 11.92/4.68 44[label="foldr (>>) xw5 (xw4 : take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];44 -> 45[label="",style="solid", color="black", weight=3]; 11.92/4.68 45[label="(>>) xw4 foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))",fontsize=16,color="black",shape="box"];45 -> 46[label="",style="solid", color="black", weight=3]; 11.92/4.68 46[label="xw4 >>= gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4)))",fontsize=16,color="black",shape="box"];46 -> 47[label="",style="solid", color="black", weight=3]; 11.92/4.68 47[label="primbindIO xw4 (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs xw4))))",fontsize=16,color="burlywood",shape="box"];87[label="xw4/IO xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 87[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 87 -> 48[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 88[label="xw4/AProVE_IO xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 88[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 88 -> 49[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 89[label="xw4/AProVE_Exception xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 89[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 89 -> 50[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 90[label="xw4/AProVE_Error xw40",fontsize=10,color="white",style="solid",shape="box"];47 -> 90[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 90 -> 51[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 48[label="primbindIO (IO xw40) (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (IO xw40)))))",fontsize=16,color="black",shape="box"];48 -> 52[label="",style="solid", color="black", weight=3]; 11.92/4.68 49[label="primbindIO (AProVE_IO xw40) (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)))))",fontsize=16,color="black",shape="box"];49 -> 53[label="",style="solid", color="black", weight=3]; 11.92/4.68 50[label="primbindIO (AProVE_Exception xw40) (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_Exception xw40)))))",fontsize=16,color="black",shape="box"];50 -> 54[label="",style="solid", color="black", weight=3]; 11.92/4.68 51[label="primbindIO (AProVE_Error xw40) (gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_Error xw40)))))",fontsize=16,color="black",shape="box"];51 -> 55[label="",style="solid", color="black", weight=3]; 11.92/4.68 52[label="error []",fontsize=16,color="red",shape="box"];53[label="gtGt0 (foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)))) xw40",fontsize=16,color="black",shape="box"];53 -> 56[label="",style="solid", color="black", weight=3]; 11.92/4.68 54[label="AProVE_Exception xw40",fontsize=16,color="green",shape="box"];55[label="AProVE_Error xw40",fontsize=16,color="green",shape="box"];56[label="foldr (>>) xw5 (take (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)))",fontsize=16,color="black",shape="box"];56 -> 57[label="",style="solid", color="black", weight=3]; 11.92/4.68 57[label="foldr (>>) xw5 (take3 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)))",fontsize=16,color="black",shape="box"];57 -> 58[label="",style="solid", color="black", weight=3]; 11.92/4.68 58[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)) (Pos (Succ xw300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];58 -> 59[label="",style="solid", color="black", weight=3]; 11.92/4.68 59[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)) (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];59 -> 60[label="",style="solid", color="black", weight=3]; 11.92/4.68 60[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)) (not (compare (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];60 -> 61[label="",style="solid", color="black", weight=3]; 11.92/4.68 61[label="foldr (>>) xw5 (take2 (Pos (Succ xw300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (Pos (Succ xw300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];61 -> 62[label="",style="solid", color="black", weight=3]; 11.92/4.68 62[label="foldr (>>) xw5 (take2 (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (primMinusInt (Pos (Succ xw300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];62 -> 63[label="",style="solid", color="black", weight=3]; 11.92/4.68 63[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw300) (Succ Zero)) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (primMinusNat (Succ xw300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];63 -> 64[label="",style="solid", color="black", weight=3]; 11.92/4.68 64[label="foldr (>>) xw5 (take2 (primMinusNat xw300 Zero) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (primMinusNat xw300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];91[label="xw300/Succ xw3000",fontsize=10,color="white",style="solid",shape="box"];64 -> 91[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 91 -> 65[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 92[label="xw300/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 92[label="",style="solid", color="burlywood", weight=9]; 11.92/4.68 92 -> 66[label="",style="solid", color="burlywood", weight=3]; 11.92/4.68 65[label="foldr (>>) xw5 (take2 (primMinusNat (Succ xw3000) Zero) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (primMinusNat (Succ xw3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];65 -> 67[label="",style="solid", color="black", weight=3]; 11.92/4.68 66[label="foldr (>>) xw5 (take2 (primMinusNat Zero Zero) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];66 -> 68[label="",style="solid", color="black", weight=3]; 11.92/4.68 67[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (Pos (Succ xw3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];67 -> 69[label="",style="solid", color="black", weight=3]; 11.92/4.68 68[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];68 -> 70[label="",style="solid", color="black", weight=3]; 11.92/4.68 69[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (primCmpNat (Succ xw3000) Zero == GT)))",fontsize=16,color="black",shape="box"];69 -> 71[label="",style="solid", color="black", weight=3]; 11.92/4.68 70[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];70 -> 72[label="",style="solid", color="black", weight=3]; 11.92/4.68 71[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];71 -> 73[label="",style="solid", color="black", weight=3]; 11.92/4.68 72[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) (not False))",fontsize=16,color="black",shape="box"];72 -> 74[label="",style="solid", color="black", weight=3]; 11.92/4.68 73[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) (not True))",fontsize=16,color="black",shape="box"];73 -> 75[label="",style="solid", color="black", weight=3]; 11.92/4.68 74[label="foldr (>>) xw5 (take2 (Pos Zero) (repeatXs (AProVE_IO xw40)) True)",fontsize=16,color="black",shape="box"];74 -> 76[label="",style="solid", color="black", weight=3]; 11.92/4.68 75[label="foldr (>>) xw5 (take2 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)) False)",fontsize=16,color="black",shape="box"];75 -> 77[label="",style="solid", color="black", weight=3]; 11.92/4.68 76[label="foldr (>>) xw5 []",fontsize=16,color="black",shape="box"];76 -> 78[label="",style="solid", color="black", weight=3]; 11.92/4.68 77 -> 39[label="",style="dashed", color="red", weight=0]; 11.92/4.68 77[label="foldr (>>) xw5 (take1 (Pos (Succ xw3000)) (repeatXs (AProVE_IO xw40)))",fontsize=16,color="magenta"];77 -> 79[label="",style="dashed", color="magenta", weight=3]; 11.92/4.68 77 -> 80[label="",style="dashed", color="magenta", weight=3]; 11.92/4.68 78[label="xw5",fontsize=16,color="green",shape="box"];79[label="AProVE_IO xw40",fontsize=16,color="green",shape="box"];80[label="xw3000",fontsize=16,color="green",shape="box"];} 11.92/4.68 11.92/4.68 ---------------------------------------- 11.92/4.68 11.92/4.68 (12) 11.92/4.68 Obligation: 11.92/4.68 Q DP problem: 11.92/4.68 The TRS P consists of the following rules: 11.92/4.68 11.92/4.68 new_foldr(xw5, Succ(xw3000), AProVE_IO(xw40), h) -> new_foldr(xw5, xw3000, AProVE_IO(xw40), h) 11.92/4.68 11.92/4.68 R is empty. 11.92/4.68 Q is empty. 11.92/4.68 We have to consider all minimal (P,Q,R)-chains. 11.92/4.68 ---------------------------------------- 11.92/4.68 11.92/4.68 (13) QDPSizeChangeProof (EQUIVALENT) 11.92/4.68 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. 11.92/4.68 11.92/4.68 From the DPs we obtained the following set of size-change graphs: 11.92/4.68 *new_foldr(xw5, Succ(xw3000), AProVE_IO(xw40), h) -> new_foldr(xw5, xw3000, AProVE_IO(xw40), h) 11.92/4.68 The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3, 4 >= 4 11.92/4.68 11.92/4.68 11.92/4.68 ---------------------------------------- 11.92/4.68 11.92/4.68 (14) 11.92/4.68 YES 11.98/4.70 EOF