9.12/3.94	YES
11.39/4.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.39/4.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.39/4.55	
11.39/4.55	
11.39/4.55	H-Termination with start terms of the given HASKELL could be proven:
11.39/4.55	
11.39/4.55	(0) HASKELL
11.39/4.55	(1) LR [EQUIVALENT, 0 ms]
11.39/4.55	(2) HASKELL
11.39/4.55	(3) BR [EQUIVALENT, 0 ms]
11.39/4.55	(4) HASKELL
11.39/4.55	(5) COR [EQUIVALENT, 0 ms]
11.39/4.55	(6) HASKELL
11.39/4.55	(7) LetRed [EQUIVALENT, 3 ms]
11.39/4.55	(8) HASKELL
11.39/4.55	(9) NumRed [SOUND, 0 ms]
11.39/4.55	(10) HASKELL
11.39/4.55	(11) Narrow [SOUND, 0 ms]
11.39/4.55	(12) QDP
11.39/4.55	(13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.39/4.55	(14) YES
11.39/4.55	
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(0)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad b => Int  ->  b a  ->  b [a];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(1) LR (EQUIVALENT)
11.39/4.55	Lambda Reductions:
11.39/4.55	The following Lambda expression
11.39/4.55	"\xs->return (x : xs)"
11.39/4.55	is transformed to
11.39/4.55	"sequence0 x xs = return (x : xs);
11.39/4.55	"
11.39/4.55	The following Lambda expression
11.39/4.55	"\x->sequence cs >>= sequence0 x"
11.39/4.55	is transformed to
11.39/4.55	"sequence1 cs x = sequence cs >>= sequence0 x;
11.39/4.55	"
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(2)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad a => Int  ->  a b  ->  a [b];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(3) BR (EQUIVALENT)
11.39/4.55	Replaced joker patterns by fresh variables and removed binding patterns.
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(4)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad b => Int  ->  b a  ->  b [a];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(5) COR (EQUIVALENT)
11.39/4.55	Cond Reductions:
11.39/4.55	The following Function with conditions
11.39/4.55	"undefined |Falseundefined;
11.39/4.55	"
11.39/4.55	is transformed to
11.39/4.55	"undefined  = undefined1;
11.39/4.55	"
11.39/4.55	"undefined0 True = undefined;
11.39/4.55	"
11.39/4.55	"undefined1  = undefined0 False;
11.39/4.55	"
11.39/4.55	The following Function with conditions
11.39/4.55	"take n vy|n <= 0[];
11.39/4.55	take vz [] = [];
11.39/4.55	take n (x : xs) = x : take (n - 1) xs;
11.39/4.55	"
11.39/4.55	is transformed to
11.39/4.55	"take n vy = take3 n vy;
11.39/4.55	take vz [] = take1 vz [];
11.39/4.55	take n (x : xs) = take0 n (x : xs);
11.39/4.55	"
11.39/4.55	"take0 n (x : xs) = x : take (n - 1) xs;
11.39/4.55	"
11.39/4.55	"take1 vz [] = [];
11.39/4.55	take1 ww wx = take0 ww wx;
11.39/4.55	"
11.39/4.55	"take2 n vy True = [];
11.39/4.55	take2 n vy False = take1 n vy;
11.39/4.55	"
11.39/4.55	"take3 n vy = take2 n vy (n <= 0);
11.39/4.55	take3 wy wz = take1 wy wz;
11.39/4.55	"
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(6)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad b => Int  ->  b a  ->  b [a];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(7) LetRed (EQUIVALENT)
11.39/4.55	Let/Where Reductions:
11.39/4.55	The bindings of the following Let/Where expression
11.39/4.55	"xs where {
11.39/4.55	xs  = x : xs;
11.39/4.55	} 
11.39/4.55	"
11.39/4.55	are unpacked to the following functions on top level
11.39/4.55	"repeatXs xu = xu : repeatXs xu;
11.39/4.55	"
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(8)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad a => Int  ->  a b  ->  a [b];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(9) NumRed (SOUND)
11.39/4.55	Num Reduction:All numbers are transformed to their corresponding representation with Succ, Pred and Zero.
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(10)
11.39/4.55	Obligation:
11.39/4.55	mainModule Main 
11.39/4.55	module Maybe where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Main where {
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Monad;
11.39/4.55	import qualified Prelude;
11.39/4.55	}
11.39/4.55	module Monad where {
11.39/4.55	import qualified Main;
11.39/4.55	import qualified Maybe;
11.39/4.55	import qualified Prelude;
11.39/4.55	replicateM :: Monad a => Int  ->  a b  ->  a [b];
11.39/4.55	replicateM n x = sequence (replicate n x);
11.39/4.55	
11.39/4.55	}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(11) Narrow (SOUND)
11.39/4.55	Haskell To QDPs
11.39/4.55	
11.39/4.55	digraph dp_graph {
11.39/4.55	node [outthreshold=100, inthreshold=100];1[label="Monad.replicateM",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.39/4.55	3[label="Monad.replicateM xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.39/4.55	4[label="Monad.replicateM xv3 xv4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.39/4.55	5[label="sequence (replicate xv3 xv4)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.39/4.55	6[label="sequence (take xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
11.39/4.55	7[label="sequence (take3 xv3 (repeat xv4))",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	11[label="sequence (take2 xv3 (repeat xv4) (not (primCmpInt xv3 (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];130[label="xv3/Pos xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 130[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	130 -> 12[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	131[label="xv3/Neg xv30",fontsize=10,color="white",style="solid",shape="box"];11 -> 131[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	131 -> 13[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	12[label="sequence (take2 (Pos xv30) (repeat xv4) (not (primCmpInt (Pos xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];132[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];12 -> 132[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	132 -> 14[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	133[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 133[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	133 -> 15[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	13[label="sequence (take2 (Neg xv30) (repeat xv4) (not (primCmpInt (Neg xv30) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];134[label="xv30/Succ xv300",fontsize=10,color="white",style="solid",shape="box"];13 -> 134[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	134 -> 16[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	135[label="xv30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 135[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	135 -> 17[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	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];
11.39/4.55	27[label="sequence (take2 (Pos Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
11.39/4.55	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];
11.39/4.55	29[label="sequence (take2 (Neg Zero) (repeat xv4) True)",fontsize=16,color="black",shape="box"];29 -> 33[label="",style="solid", color="black", weight=3];
11.39/4.55	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];
11.39/4.55	31[label="sequence []",fontsize=16,color="black",shape="triangle"];31 -> 35[label="",style="solid", color="black", weight=3];
11.39/4.55	32 -> 31[label="",style="dashed", color="red", weight=0];
11.39/4.55	32[label="sequence []",fontsize=16,color="magenta"];33 -> 31[label="",style="dashed", color="red", weight=0];
11.39/4.55	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];
11.39/4.55	35[label="return []",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
11.39/4.55	36[label="sequence (take1 (Pos (Succ xv300)) (repeatXs xv4))",fontsize=16,color="black",shape="triangle"];36 -> 38[label="",style="solid", color="black", weight=3];
11.39/4.55	37[label="primretIO []",fontsize=16,color="black",shape="box"];37 -> 39[label="",style="solid", color="black", weight=3];
11.39/4.55	38[label="sequence (take1 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
11.39/4.55	39[label="AProVE_IO []",fontsize=16,color="green",shape="box"];40[label="sequence (take0 (Pos (Succ xv300)) (xv4 : repeatXs xv4))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
11.39/4.55	41[label="sequence (xv4 : take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
11.39/4.55	42[label="xv4 >>= sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4))",fontsize=16,color="black",shape="box"];42 -> 43[label="",style="solid", color="black", weight=3];
11.39/4.55	43[label="primbindIO xv4 (sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs xv4)))",fontsize=16,color="burlywood",shape="box"];136[label="xv4/IO xv40",fontsize=10,color="white",style="solid",shape="box"];43 -> 136[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	136 -> 44[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	137[label="xv4/AProVE_IO xv40",fontsize=10,color="white",style="solid",shape="box"];43 -> 137[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	137 -> 45[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	138[label="xv4/AProVE_Exception xv40",fontsize=10,color="white",style="solid",shape="box"];43 -> 138[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	138 -> 46[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	139[label="xv4/AProVE_Error xv40",fontsize=10,color="white",style="solid",shape="box"];43 -> 139[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	139 -> 47[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	44[label="primbindIO (IO xv40) (sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (IO xv40))))",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
11.39/4.55	45[label="primbindIO (AProVE_IO xv40) (sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40))))",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
11.39/4.55	46[label="primbindIO (AProVE_Exception xv40) (sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_Exception xv40))))",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
11.39/4.55	47[label="primbindIO (AProVE_Error xv40) (sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_Error xv40))))",fontsize=16,color="black",shape="box"];47 -> 51[label="",style="solid", color="black", weight=3];
11.39/4.55	48[label="error []",fontsize=16,color="red",shape="box"];49[label="sequence1 (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40))) xv40",fontsize=16,color="black",shape="box"];49 -> 52[label="",style="solid", color="black", weight=3];
11.39/4.55	50[label="AProVE_Exception xv40",fontsize=16,color="green",shape="box"];51[label="AProVE_Error xv40",fontsize=16,color="green",shape="box"];52[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40))) >>= sequence0 xv40",fontsize=16,color="black",shape="box"];52 -> 53[label="",style="solid", color="black", weight=3];
11.39/4.55	53 -> 75[label="",style="dashed", color="red", weight=0];
11.39/4.55	53[label="primbindIO (sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)))) (sequence0 xv40)",fontsize=16,color="magenta"];53 -> 76[label="",style="dashed", color="magenta", weight=3];
11.39/4.55	76[label="sequence (take (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)))",fontsize=16,color="black",shape="box"];76 -> 96[label="",style="solid", color="black", weight=3];
11.39/4.55	75[label="primbindIO xv5 (sequence0 xv40)",fontsize=16,color="burlywood",shape="triangle"];140[label="xv5/IO xv50",fontsize=10,color="white",style="solid",shape="box"];75 -> 140[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	140 -> 97[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	141[label="xv5/AProVE_IO xv50",fontsize=10,color="white",style="solid",shape="box"];75 -> 141[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	141 -> 98[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	142[label="xv5/AProVE_Exception xv50",fontsize=10,color="white",style="solid",shape="box"];75 -> 142[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	142 -> 99[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	143[label="xv5/AProVE_Error xv50",fontsize=10,color="white",style="solid",shape="box"];75 -> 143[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	143 -> 100[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	96[label="sequence (take3 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)))",fontsize=16,color="black",shape="box"];96 -> 101[label="",style="solid", color="black", weight=3];
11.39/4.55	97[label="primbindIO (IO xv50) (sequence0 xv40)",fontsize=16,color="black",shape="box"];97 -> 102[label="",style="solid", color="black", weight=3];
11.39/4.55	98[label="primbindIO (AProVE_IO xv50) (sequence0 xv40)",fontsize=16,color="black",shape="box"];98 -> 103[label="",style="solid", color="black", weight=3];
11.39/4.55	99[label="primbindIO (AProVE_Exception xv50) (sequence0 xv40)",fontsize=16,color="black",shape="box"];99 -> 104[label="",style="solid", color="black", weight=3];
11.39/4.55	100[label="primbindIO (AProVE_Error xv50) (sequence0 xv40)",fontsize=16,color="black",shape="box"];100 -> 105[label="",style="solid", color="black", weight=3];
11.39/4.55	101[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)) (Pos (Succ xv300) - Pos (Succ Zero) <= Pos Zero))",fontsize=16,color="black",shape="box"];101 -> 106[label="",style="solid", color="black", weight=3];
11.39/4.55	102[label="error []",fontsize=16,color="red",shape="box"];103[label="sequence0 xv40 xv50",fontsize=16,color="black",shape="box"];103 -> 107[label="",style="solid", color="black", weight=3];
11.39/4.55	104[label="AProVE_Exception xv50",fontsize=16,color="green",shape="box"];105[label="AProVE_Error xv50",fontsize=16,color="green",shape="box"];106[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)) (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) /= GT))",fontsize=16,color="black",shape="box"];106 -> 108[label="",style="solid", color="black", weight=3];
11.39/4.55	107[label="return (xv40 : xv50)",fontsize=16,color="black",shape="box"];107 -> 109[label="",style="solid", color="black", weight=3];
11.39/4.55	108[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)) (not (compare (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];108 -> 110[label="",style="solid", color="black", weight=3];
11.39/4.55	109[label="primretIO (xv40 : xv50)",fontsize=16,color="black",shape="box"];109 -> 111[label="",style="solid", color="black", weight=3];
11.39/4.55	110[label="sequence (take2 (Pos (Succ xv300) - Pos (Succ Zero)) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (Pos (Succ xv300) - Pos (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];110 -> 112[label="",style="solid", color="black", weight=3];
11.39/4.55	111[label="AProVE_IO (xv40 : xv50)",fontsize=16,color="green",shape="box"];112[label="sequence (take2 (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (primMinusInt (Pos (Succ xv300)) (Pos (Succ Zero))) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];112 -> 113[label="",style="solid", color="black", weight=3];
11.39/4.55	113[label="sequence (take2 (primMinusNat (Succ xv300) (Succ Zero)) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (primMinusNat (Succ xv300) (Succ Zero)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];113 -> 114[label="",style="solid", color="black", weight=3];
11.39/4.55	114[label="sequence (take2 (primMinusNat xv300 Zero) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (primMinusNat xv300 Zero) (Pos Zero) == GT)))",fontsize=16,color="burlywood",shape="box"];144[label="xv300/Succ xv3000",fontsize=10,color="white",style="solid",shape="box"];114 -> 144[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	144 -> 115[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	145[label="xv300/Zero",fontsize=10,color="white",style="solid",shape="box"];114 -> 145[label="",style="solid", color="burlywood", weight=9];
11.39/4.55	145 -> 116[label="",style="solid", color="burlywood", weight=3];
11.39/4.55	115[label="sequence (take2 (primMinusNat (Succ xv3000) Zero) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (primMinusNat (Succ xv3000) Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];115 -> 117[label="",style="solid", color="black", weight=3];
11.39/4.55	116[label="sequence (take2 (primMinusNat Zero Zero) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (primMinusNat Zero Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];116 -> 118[label="",style="solid", color="black", weight=3];
11.39/4.55	117[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (Pos (Succ xv3000)) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];117 -> 119[label="",style="solid", color="black", weight=3];
11.39/4.55	118[label="sequence (take2 (Pos Zero) (repeatXs (AProVE_IO xv40)) (not (primCmpInt (Pos Zero) (Pos Zero) == GT)))",fontsize=16,color="black",shape="box"];118 -> 120[label="",style="solid", color="black", weight=3];
11.39/4.55	119[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)) (not (primCmpNat (Succ xv3000) Zero == GT)))",fontsize=16,color="black",shape="box"];119 -> 121[label="",style="solid", color="black", weight=3];
11.39/4.55	120[label="sequence (take2 (Pos Zero) (repeatXs (AProVE_IO xv40)) (not (EQ == GT)))",fontsize=16,color="black",shape="box"];120 -> 122[label="",style="solid", color="black", weight=3];
11.39/4.55	121[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)) (not (GT == GT)))",fontsize=16,color="black",shape="box"];121 -> 123[label="",style="solid", color="black", weight=3];
11.39/4.55	122[label="sequence (take2 (Pos Zero) (repeatXs (AProVE_IO xv40)) (not False))",fontsize=16,color="black",shape="box"];122 -> 124[label="",style="solid", color="black", weight=3];
11.39/4.55	123[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)) (not True))",fontsize=16,color="black",shape="box"];123 -> 125[label="",style="solid", color="black", weight=3];
11.39/4.55	124[label="sequence (take2 (Pos Zero) (repeatXs (AProVE_IO xv40)) True)",fontsize=16,color="black",shape="box"];124 -> 126[label="",style="solid", color="black", weight=3];
11.39/4.55	125[label="sequence (take2 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)) False)",fontsize=16,color="black",shape="box"];125 -> 127[label="",style="solid", color="black", weight=3];
11.39/4.55	126 -> 31[label="",style="dashed", color="red", weight=0];
11.39/4.55	126[label="sequence []",fontsize=16,color="magenta"];127 -> 36[label="",style="dashed", color="red", weight=0];
11.39/4.55	127[label="sequence (take1 (Pos (Succ xv3000)) (repeatXs (AProVE_IO xv40)))",fontsize=16,color="magenta"];127 -> 128[label="",style="dashed", color="magenta", weight=3];
11.39/4.55	127 -> 129[label="",style="dashed", color="magenta", weight=3];
11.39/4.55	128[label="AProVE_IO xv40",fontsize=16,color="green",shape="box"];129[label="xv3000",fontsize=16,color="green",shape="box"];}
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(12)
11.39/4.55	Obligation:
11.39/4.55	Q DP problem:
11.39/4.55	The TRS P consists of the following rules:
11.39/4.55	
11.39/4.55	   new_sequence(Succ(xv3000), AProVE_IO(xv40), h) -> new_sequence(xv3000, AProVE_IO(xv40), h)
11.39/4.55	
11.39/4.55	R is empty.
11.39/4.55	Q is empty.
11.39/4.55	We have to consider all minimal (P,Q,R)-chains.
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(13) QDPSizeChangeProof (EQUIVALENT)
11.39/4.55	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. 
11.39/4.55	
11.39/4.55	From the DPs we obtained the following set of size-change graphs:
11.39/4.55	*new_sequence(Succ(xv3000), AProVE_IO(xv40), h) -> new_sequence(xv3000, AProVE_IO(xv40), h)
11.39/4.55	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.39/4.55	
11.39/4.55	
11.39/4.55	----------------------------------------
11.39/4.55	
11.39/4.55	(14)
11.39/4.55	YES
11.62/4.60	EOF