9.60/4.09 YES 11.87/4.74 proof of /export/starexec/sandbox/benchmark/theBenchmark.hs 11.87/4.74 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 11.87/4.74 11.87/4.74 11.87/4.74 H-Termination with start terms of the given HASKELL could be proven: 11.87/4.74 11.87/4.74 (0) HASKELL 11.87/4.74 (1) LR [EQUIVALENT, 0 ms] 11.87/4.74 (2) HASKELL 11.87/4.74 (3) BR [EQUIVALENT, 0 ms] 11.87/4.74 (4) HASKELL 11.87/4.74 (5) COR [EQUIVALENT, 0 ms] 11.87/4.74 (6) HASKELL 11.87/4.74 (7) LetRed [EQUIVALENT, 0 ms] 11.87/4.74 (8) HASKELL 11.87/4.74 (9) NumRed [SOUND, 0 ms] 11.87/4.74 (10) HASKELL 11.87/4.74 (11) Narrow [SOUND, 0 ms] 11.87/4.74 (12) AND 11.87/4.74 (13) QDP 11.87/4.74 (14) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.87/4.74 (15) YES 11.87/4.74 (16) QDP 11.87/4.74 (17) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.87/4.74 (18) YES 11.87/4.74 (19) QDP 11.87/4.74 (20) QDPSizeChangeProof [EQUIVALENT, 0 ms] 11.87/4.74 (21) YES 11.87/4.74 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (0) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad a => Int -> a b -> a [b]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (1) LR (EQUIVALENT) 11.87/4.74 Lambda Reductions: 11.87/4.74 The following Lambda expression 11.87/4.74 "\xs->return (x : xs)" 11.87/4.74 is transformed to 11.87/4.74 "sequence0 x xs = return (x : xs); 11.87/4.74 " 11.87/4.74 The following Lambda expression 11.87/4.74 "\x->sequence cs >>= sequence0 x" 11.87/4.74 is transformed to 11.87/4.74 "sequence1 cs x = sequence cs >>= sequence0 x; 11.87/4.74 " 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (2) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad a => Int -> a b -> a [b]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (3) BR (EQUIVALENT) 11.87/4.74 Replaced joker patterns by fresh variables and removed binding patterns. 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (4) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad b => Int -> b a -> b [a]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (5) COR (EQUIVALENT) 11.87/4.74 Cond Reductions: 11.87/4.74 The following Function with conditions 11.87/4.74 "undefined |Falseundefined; 11.87/4.74 " 11.87/4.74 is transformed to 11.87/4.74 "undefined = undefined1; 11.87/4.74 " 11.87/4.74 "undefined0 True = undefined; 11.87/4.74 " 11.87/4.74 "undefined1 = undefined0 False; 11.87/4.74 " 11.87/4.74 The following Function with conditions 11.87/4.74 "take n vy|n <= 0[]; 11.87/4.74 take vz [] = []; 11.87/4.74 take n (x : xs) = x : take (n - 1) xs; 11.87/4.74 " 11.87/4.74 is transformed to 11.87/4.74 "take n vy = take3 n vy; 11.87/4.74 take vz [] = take1 vz []; 11.87/4.74 take n (x : xs) = take0 n (x : xs); 11.87/4.74 " 11.87/4.74 "take0 n (x : xs) = x : take (n - 1) xs; 11.87/4.74 " 11.87/4.74 "take1 vz [] = []; 11.87/4.74 take1 ww wx = take0 ww wx; 11.87/4.74 " 11.87/4.74 "take2 n vy True = []; 11.87/4.74 take2 n vy False = take1 n vy; 11.87/4.74 " 11.87/4.74 "take3 n vy = take2 n vy (n <= 0); 11.87/4.74 take3 wy wz = take1 wy wz; 11.87/4.74 " 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (6) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad a => Int -> a b -> a [b]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (7) LetRed (EQUIVALENT) 11.87/4.74 Let/Where Reductions: 11.87/4.74 The bindings of the following Let/Where expression 11.87/4.74 "xs where { 11.87/4.74 xs = x : xs; 11.87/4.74 } 11.87/4.74 " 11.87/4.74 are unpacked to the following functions on top level 11.87/4.74 "repeatXs xu = xu : repeatXs xu; 11.87/4.74 " 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (8) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad b => Int -> b a -> b [a]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (9) NumRed (SOUND) 11.87/4.74 Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero. 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (10) 11.87/4.74 Obligation: 11.87/4.74 mainModule Main 11.87/4.74 module Maybe where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Main where { 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Monad; 11.87/4.74 import qualified Prelude; 11.87/4.74 } 11.87/4.74 module Monad where { 11.87/4.74 import qualified Main; 11.87/4.74 import qualified Maybe; 11.87/4.74 import qualified Prelude; 11.87/4.74 replicateM :: Monad a => Int -> a b -> a [b]; 11.87/4.74 replicateM n x = sequence (replicate n x); 11.87/4.74 11.87/4.74 } 11.87/4.74 11.87/4.74 ---------------------------------------- 11.87/4.74 11.87/4.74 (11) Narrow (SOUND) 11.87/4.74 Haskell To QDPs 11.87/4.74 11.87/4.74 digraph dp_graph { 11.87/4.74 node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3]; 11.87/4.74 3[label="Monad.replicateM xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3]; 11.87/4.74 4[label="Monad.replicateM xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3]; 11.87/4.74 5[label="sequence (replicate xv3 xv4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3]; 11.87/4.74 6[label="sequence (take xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3]; 11.87/4.74 7[label="sequence (take3 xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3]; 11.87/4.74 8[label="sequence (take2 xv3 (repeat xv4) (xv3 <= Pos Zero))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3]; 11.87/4.74 9[label="sequence (take2 xv3 (repeat xv4) (compare xv3 (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];9 -> 10[label="",style="solid", color="black", weight=3]; 11.87/4.74 10[label="sequence (take2 xv3 (repeat xv4) (not (compare xv3 (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];10 -> 11[label="",style="solid", color="black", weight=3]; 11.87/4.74 11[label="sequence (take2 xv3 (repeat xv4) (not (primCmpInt xv3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];315[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 315[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 315 -> 12[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 316[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 316[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 316 -> 13[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 12[label="sequence (take2 (Pos xv30) (repeat xv4) (not (primCmpInt (Pos xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];317[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];12 -> 317[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 317 -> 14[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 318[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 318[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 318 -> 15[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 13[label="sequence (take2 (Neg xv30) (repeat xv4) (not (primCmpInt (Neg xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];319[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];13 -> 319[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 319 -> 16[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 320[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 320[label="",style="solid", color="burlywood", weight=9]; 11.87/4.74 320 -> 17[label="",style="solid", color="burlywood", weight=3]; 11.87/4.74 14[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpInt (Pos (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3]; 11.87/4.74 15[label="sequence (take2 (Pos Zero) (repeat xv4) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3]; 11.87/4.74 16[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (primCmpInt (Neg (Succ xv300)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3]; 11.87/4.74 17[label="sequence (take2 (Neg Zero) (repeat xv4) (not (primCmpInt (Neg Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3]; 11.87/4.74 18[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (primCmpNat (Succ xv300) Zero == GT)))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3]; 11.87/4.74 19[label="sequence (take2 (Pos Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3]; 11.87/4.74 20[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not (LT == GT)))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3]; 11.87/4.74 21[label="sequence (take2 (Neg Zero) (repeat xv4) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3]; 11.87/4.74 22[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not (GT == GT)))",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3]; 11.87/4.74 23[label="sequence (take2 (Pos Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3]; 11.87/4.74 24[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3]; 11.87/4.74 25[label="sequence (take2 (Neg Zero) (repeat xv4) (not False))",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3]; 12.10/4.74 26[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) (not True))",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3]; 12.10/4.74 27[label="sequence (take2 (Pos Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3]; 12.10/4.74 28[label="sequence (take2 (Neg (Succ xv300)) (repeat xv4) True)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3]; 12.10/4.74 29[label="sequence (take2 (Neg Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3]; 12.10/4.74 30[label="sequence (take2 (Pos (Succ xv300)) (repeat xv4) False)",fontsize=16,color="black",shape="box"];30 -> 34[label="",style="solid", color="black", weight=3]; 12.10/4.74 31[label="sequence []",fontsize=16,color="black",shape="triangle"];31 -> 35[label="",style="solid", color="black", weight=3]; 12.10/4.74 32 -> 31[label="",style="dashed", color="red", weight=0]; 12.10/4.74 32[label="sequence []",fontsize=16,color="magenta"];33 -> 31[label="",style="dashed", color="red", weight=0]; 12.10/4.74 33[label="sequence []",fontsize=16,color="magenta"];34[label="sequence (take1 (Pos (Succ xv300)) (repeat xv4))",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3]; 12.10/4.74 35[label="return []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3]; 12.10/4.74 36[label="sequence (take1 (Pos (Succ xv300)) (repeatXs xv4))",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3]; 12.10/4.74 37[label="[] : []",fontsize=16,color="green",shape="box"];38[label="sequence (take1 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];38 -> 39[label="",style="solid", color="black", weight=3]; 12.10/4.74 39[label="sequence (take0 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3]; 12.10/4.74 40[label="sequence (xv4 : take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3]; 12.10/4.74 41[label="xv4 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="burlywood",shape="box"];321[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];41 -> 321[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 321 -> 42[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 322[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];41 -> 322[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 322 -> 43[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 42[label="xv40 : xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="box"];42 -> 44[label="",style="solid", color="black", weight=3]; 12.10/4.74 43[label="[] >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs []))",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3]; 12.10/4.74 44[label="sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))) xv40 ++ (xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))))",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3]; 12.10/4.74 45[label="[]",fontsize=16,color="green",shape="box"];46 -> 111[label="",style="dashed", color="red", weight=0]; 12.10/4.74 46[label="(sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))) >>= sequence0 xv40) ++ (xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41))))",fontsize=16,color="magenta"];46 -> 112[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 46 -> 113[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 46 -> 114[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 112 -> 277[label="",style="dashed", color="red", weight=0]; 12.10/4.74 112[label="xv41 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="magenta"];112 -> 278[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 112 -> 279[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 112 -> 280[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 112 -> 281[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 113[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="triangle"];113 -> 185[label="",style="solid", color="black", weight=3]; 12.10/4.74 114[label="xv40",fontsize=16,color="green",shape="box"];111[label="(xv10 >>= sequence0 xv410) ++ xv8",fontsize=16,color="burlywood",shape="triangle"];323[label="xv10/xv100 : xv101",fontsize=10,color="white",style="solid",shape="box"];111 -> 323[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 323 -> 186[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 324[label="xv10/[]",fontsize=10,color="white",style="solid",shape="box"];111 -> 324[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 324 -> 187[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 278[label="xv300",fontsize=16,color="green",shape="box"];279[label="xv40",fontsize=16,color="green",shape="box"];280[label="xv41",fontsize=16,color="green",shape="box"];281[label="xv41",fontsize=16,color="green",shape="box"];277[label="xv13 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="burlywood",shape="triangle"];325[label="xv13/xv130 : xv131",fontsize=10,color="white",style="solid",shape="box"];277 -> 325[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 325 -> 302[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 326[label="xv13/[]",fontsize=10,color="white",style="solid",shape="box"];277 -> 326[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 326 -> 303[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 185[label="sequence (take3 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)))",fontsize=16,color="black",shape="box"];185 -> 190[label="",style="solid", color="black", weight=3]; 12.10/4.74 186[label="(xv100 : xv101 >>= sequence0 xv410) ++ xv8",fontsize=16,color="black",shape="box"];186 -> 191[label="",style="solid", color="black", weight=3]; 12.10/4.74 187[label="([] >>= sequence0 xv410) ++ xv8",fontsize=16,color="black",shape="box"];187 -> 192[label="",style="solid", color="black", weight=3]; 12.10/4.74 302[label="xv130 : xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="black",shape="box"];302 -> 304[label="",style="solid", color="black", weight=3]; 12.10/4.74 303[label="[] >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="black",shape="box"];303 -> 305[label="",style="solid", color="black", weight=3]; 12.10/4.74 190[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (Pos (Succ xv300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];190 -> 194[label="",style="solid", color="black", weight=3]; 12.10/4.74 191[label="(sequence0 xv410 xv100 ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];191 -> 195[label="",style="solid", color="black", weight=3]; 12.10/4.74 192[label="[] ++ xv8",fontsize=16,color="black",shape="triangle"];192 -> 196[label="",style="solid", color="black", weight=3]; 12.10/4.74 304 -> 221[label="",style="dashed", color="red", weight=0]; 12.10/4.74 304[label="sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) xv130 ++ (xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))))",fontsize=16,color="magenta"];304 -> 306[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 304 -> 307[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 305[label="[]",fontsize=16,color="green",shape="box"];194[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];194 -> 199[label="",style="solid", color="black", weight=3]; 12.10/4.74 195[label="(return (xv410 : xv100) ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];195 -> 200[label="",style="solid", color="black", weight=3]; 12.10/4.74 196[label="xv8",fontsize=16,color="green",shape="box"];306 -> 277[label="",style="dashed", color="red", weight=0]; 12.10/4.74 306[label="xv131 >>= sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="magenta"];306 -> 308[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 307[label="sequence1 (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) xv130",fontsize=16,color="black",shape="box"];307 -> 309[label="",style="solid", color="black", weight=3]; 12.10/4.74 221[label="xv11 ++ xv8",fontsize=16,color="burlywood",shape="triangle"];327[label="xv11/xv110 : xv111",fontsize=10,color="white",style="solid",shape="box"];221 -> 327[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 327 -> 229[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 328[label="xv11/[]",fontsize=10,color="white",style="solid",shape="box"];221 -> 328[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 328 -> 230[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 199[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (not (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];199 -> 204[label="",style="solid", color="black", weight=3]; 12.10/4.74 200[label="(((xv410 : xv100) : []) ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="black",shape="box"];200 -> 205[label="",style="solid", color="black", weight=3]; 12.10/4.74 308[label="xv131",fontsize=16,color="green",shape="box"];309 -> 213[label="",style="dashed", color="red", weight=0]; 12.10/4.74 309[label="sequence (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16))) >>= sequence0 xv130",fontsize=16,color="magenta"];309 -> 310[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 309 -> 311[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 229[label="(xv110 : xv111) ++ xv8",fontsize=16,color="black",shape="box"];229 -> 237[label="",style="solid", color="black", weight=3]; 12.10/4.74 230[label="[] ++ xv8",fontsize=16,color="black",shape="box"];230 -> 238[label="",style="solid", color="black", weight=3]; 12.10/4.74 204[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];204 -> 208[label="",style="solid", color="black", weight=3]; 12.10/4.74 205 -> 209[label="",style="dashed", color="red", weight=0]; 12.10/4.74 205[label="((xv410 : xv100) : [] ++ (xv101 >>= sequence0 xv410)) ++ xv8",fontsize=16,color="magenta"];205 -> 210[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 310 -> 113[label="",style="dashed", color="red", weight=0]; 12.10/4.74 310[label="sequence (take (Pos (Succ xv14) - Pos (Succ Zero)) (repeatXs (xv15 : xv16)))",fontsize=16,color="magenta"];310 -> 312[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 310 -> 313[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 310 -> 314[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 311[label="xv130",fontsize=16,color="green",shape="box"];213[label="xv101 >>= sequence0 xv410",fontsize=16,color="burlywood",shape="triangle"];329[label="xv101/xv1010 : xv1011",fontsize=10,color="white",style="solid",shape="box"];213 -> 329[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 329 -> 219[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 330[label="xv101/[]",fontsize=10,color="white",style="solid",shape="box"];213 -> 330[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 330 -> 220[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 237[label="xv110 : xv111 ++ xv8",fontsize=16,color="green",shape="box"];237 -> 245[label="",style="dashed", color="green", weight=3]; 12.10/4.74 238[label="xv8",fontsize=16,color="green",shape="box"];208[label="sequence (take2 (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];208 -> 212[label="",style="solid", color="black", weight=3]; 12.10/4.74 210 -> 192[label="",style="dashed", color="red", weight=0]; 12.10/4.74 210[label="[] ++ (xv101 >>= sequence0 xv410)",fontsize=16,color="magenta"];210 -> 213[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 209[label="((xv410 : xv100) : xv11) ++ xv8",fontsize=16,color="black",shape="triangle"];209 -> 214[label="",style="solid", color="black", weight=3]; 12.10/4.74 312[label="xv14",fontsize=16,color="green",shape="box"];313[label="xv16",fontsize=16,color="green",shape="box"];314[label="xv15",fontsize=16,color="green",shape="box"];219[label="xv1010 : xv1011 >>= sequence0 xv410",fontsize=16,color="black",shape="box"];219 -> 227[label="",style="solid", color="black", weight=3]; 12.10/4.74 220[label="[] >>= sequence0 xv410",fontsize=16,color="black",shape="box"];220 -> 228[label="",style="solid", color="black", weight=3]; 12.10/4.74 245 -> 221[label="",style="dashed", color="red", weight=0]; 12.10/4.74 245[label="xv111 ++ xv8",fontsize=16,color="magenta"];245 -> 252[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 212[label="sequence (take2 (primMinusNat (Succ xv300) (Succ Zero)) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];212 -> 218[label="",style="solid", color="black", weight=3]; 12.10/4.74 214[label="(xv410 : xv100) : xv11 ++ xv8",fontsize=16,color="green",shape="box"];214 -> 221[label="",style="dashed", color="green", weight=3]; 12.10/4.74 227 -> 221[label="",style="dashed", color="red", weight=0]; 12.10/4.74 227[label="sequence0 xv410 xv1010 ++ (xv1011 >>= sequence0 xv410)",fontsize=16,color="magenta"];227 -> 235[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 227 -> 236[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 228[label="[]",fontsize=16,color="green",shape="box"];252[label="xv111",fontsize=16,color="green",shape="box"];218[label="sequence (take2 (primMinusNat xv300 Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat xv300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];331[label="xv300/Succ xv3000",fontsize=10,color="white",style="solid",shape="box"];218 -> 331[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 331 -> 225[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 332[label="xv300/Zero",fontsize=10,color="white",style="solid",shape="box"];218 -> 332[label="",style="solid", color="burlywood", weight=9]; 12.10/4.74 332 -> 226[label="",style="solid", color="burlywood", weight=3]; 12.10/4.74 235 -> 213[label="",style="dashed", color="red", weight=0]; 12.10/4.74 235[label="xv1011 >>= sequence0 xv410",fontsize=16,color="magenta"];235 -> 243[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 236[label="sequence0 xv410 xv1010",fontsize=16,color="black",shape="box"];236 -> 244[label="",style="solid", color="black", weight=3]; 12.10/4.74 225[label="sequence (take2 (primMinusNat (Succ xv3000) Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat (Succ xv3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];225 -> 233[label="",style="solid", color="black", weight=3]; 12.10/4.74 226[label="sequence (take2 (primMinusNat Zero Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];226 -> 234[label="",style="solid", color="black", weight=3]; 12.10/4.74 243[label="xv1011",fontsize=16,color="green",shape="box"];244[label="return (xv410 : xv1010)",fontsize=16,color="black",shape="box"];244 -> 251[label="",style="solid", color="black", weight=3]; 12.10/4.74 233[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos (Succ xv3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];233 -> 241[label="",style="solid", color="black", weight=3]; 12.10/4.74 234[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];234 -> 242[label="",style="solid", color="black", weight=3]; 12.10/4.74 251[label="(xv410 : xv1010) : []",fontsize=16,color="green",shape="box"];241[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (primCmpNat (Succ xv3000) Zero == GT)))",fontsize=16,color="black",shape="box"];241 -> 249[label="",style="solid", color="black", weight=3]; 12.10/4.74 242[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];242 -> 250[label="",style="solid", color="black", weight=3]; 12.10/4.74 249[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];249 -> 257[label="",style="solid", color="black", weight=3]; 12.10/4.74 250[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) (not False))",fontsize=16,color="black",shape="box"];250 -> 258[label="",style="solid", color="black", weight=3]; 12.10/4.74 257[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) (not True))",fontsize=16,color="black",shape="box"];257 -> 262[label="",style="solid", color="black", weight=3]; 12.10/4.74 258[label="sequence (take2 (Pos Zero) (repeatXs (xv40 : xv41)) True)",fontsize=16,color="black",shape="box"];258 -> 263[label="",style="solid", color="black", weight=3]; 12.10/4.74 262[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)) False)",fontsize=16,color="black",shape="box"];262 -> 267[label="",style="solid", color="black", weight=3]; 12.10/4.74 263 -> 31[label="",style="dashed", color="red", weight=0]; 12.10/4.74 263[label="sequence []",fontsize=16,color="magenta"];267 -> 36[label="",style="dashed", color="red", weight=0]; 12.10/4.74 267[label="sequence (take1 (Pos (Succ xv3000)) (repeatXs (xv40 : xv41)))",fontsize=16,color="magenta"];267 -> 272[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 267 -> 273[label="",style="dashed", color="magenta", weight=3]; 12.10/4.74 272[label="xv3000",fontsize=16,color="green",shape="box"];273[label="xv40 : xv41",fontsize=16,color="green",shape="box"];} 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (12) 12.10/4.74 Complex Obligation (AND) 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (13) 12.10/4.74 Obligation: 12.10/4.74 Q DP problem: 12.10/4.74 The TRS P consists of the following rules: 12.10/4.74 12.10/4.74 new_gtGtEs(:(xv1010, xv1011), xv410, h) -> new_gtGtEs(xv1011, xv410, h) 12.10/4.74 12.10/4.74 R is empty. 12.10/4.74 Q is empty. 12.10/4.74 We have to consider all minimal (P,Q,R)-chains. 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (14) QDPSizeChangeProof (EQUIVALENT) 12.10/4.74 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. 12.10/4.74 12.10/4.74 From the DPs we obtained the following set of size-change graphs: 12.10/4.74 *new_gtGtEs(:(xv1010, xv1011), xv410, h) -> new_gtGtEs(xv1011, xv410, h) 12.10/4.74 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 12.10/4.74 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (15) 12.10/4.74 YES 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (16) 12.10/4.74 Obligation: 12.10/4.74 Q DP problem: 12.10/4.74 The TRS P consists of the following rules: 12.10/4.74 12.10/4.74 new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_sequence(xv14, xv15, xv16, h) 12.10/4.74 new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_gtGtEs0(xv131, xv14, xv15, xv16, h) 12.10/4.74 new_sequence(Succ(xv3000), xv40, xv41, ba) -> new_sequence0(xv3000, :(xv40, xv41), ba) 12.10/4.74 new_sequence0(xv300, :(xv40, xv41), ba) -> new_gtGtEs0(xv41, xv300, xv40, xv41, ba) 12.10/4.74 new_sequence0(Succ(xv3000), :(xv40, xv41), ba) -> new_sequence0(xv3000, :(xv40, xv41), ba) 12.10/4.74 12.10/4.74 R is empty. 12.10/4.74 Q is empty. 12.10/4.74 We have to consider all minimal (P,Q,R)-chains. 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (17) QDPSizeChangeProof (EQUIVALENT) 12.10/4.74 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. 12.10/4.74 12.10/4.74 From the DPs we obtained the following set of size-change graphs: 12.10/4.74 *new_sequence(Succ(xv3000), xv40, xv41, ba) -> new_sequence0(xv3000, :(xv40, xv41), ba) 12.10/4.74 The graph contains the following edges 1 > 1, 4 >= 3 12.10/4.74 12.10/4.74 12.10/4.74 *new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_gtGtEs0(xv131, xv14, xv15, xv16, h) 12.10/4.74 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5 12.10/4.74 12.10/4.74 12.10/4.74 *new_sequence0(xv300, :(xv40, xv41), ba) -> new_gtGtEs0(xv41, xv300, xv40, xv41, ba) 12.10/4.74 The graph contains the following edges 2 > 1, 1 >= 2, 2 > 3, 2 > 4, 3 >= 5 12.10/4.74 12.10/4.74 12.10/4.74 *new_gtGtEs0(:(xv130, xv131), xv14, xv15, xv16, h) -> new_sequence(xv14, xv15, xv16, h) 12.10/4.74 The graph contains the following edges 2 >= 1, 3 >= 2, 4 >= 3, 5 >= 4 12.10/4.74 12.10/4.74 12.10/4.74 *new_sequence0(Succ(xv3000), :(xv40, xv41), ba) -> new_sequence0(xv3000, :(xv40, xv41), ba) 12.10/4.74 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 12.10/4.74 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (18) 12.10/4.74 YES 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (19) 12.10/4.74 Obligation: 12.10/4.74 Q DP problem: 12.10/4.74 The TRS P consists of the following rules: 12.10/4.74 12.10/4.74 new_psPs(:(xv110, xv111), xv8, h) -> new_psPs(xv111, xv8, h) 12.10/4.74 12.10/4.74 R is empty. 12.10/4.74 Q is empty. 12.10/4.74 We have to consider all minimal (P,Q,R)-chains. 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (20) QDPSizeChangeProof (EQUIVALENT) 12.10/4.74 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. 12.10/4.74 12.10/4.74 From the DPs we obtained the following set of size-change graphs: 12.10/4.74 *new_psPs(:(xv110, xv111), xv8, h) -> new_psPs(xv111, xv8, h) 12.10/4.74 The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3 12.10/4.74 12.10/4.74 12.10/4.74 ---------------------------------------- 12.10/4.74 12.10/4.74 (21) 12.10/4.74 YES 12.17/4.83 EOF