9.32/4.14	YES
11.27/4.73	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
11.27/4.73	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
11.27/4.73	
11.27/4.73	
11.27/4.73	H-Termination with start terms of the given HASKELL could be proven:
11.27/4.73	
11.27/4.73	(0) HASKELL
11.27/4.73	(1) LR [EQUIVALENT, 0 ms]
11.27/4.73	(2) HASKELL
11.27/4.73	(3) BR [EQUIVALENT, 0 ms]
11.27/4.73	(4) HASKELL
11.27/4.73	(5) COR [EQUIVALENT, 0 ms]
11.27/4.73	(6) HASKELL
11.27/4.73	(7) Narrow [SOUND, 0 ms]
11.27/4.73	(8) AND
11.27/4.73	    (9) QDP
11.27/4.73	        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.27/4.73	        (11) YES
11.27/4.73	    (12) QDP
11.27/4.73	        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.27/4.73	        (14) YES
11.27/4.73	    (15) QDP
11.27/4.73	        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
11.27/4.73	        (17) YES
11.27/4.73	
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(0)
11.27/4.73	Obligation:
11.27/4.73	mainModule Main 
11.27/4.73	module Maybe where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Main where {
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Monad where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Prelude;
11.27/4.73	ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
11.27/4.73	ap = liftM2 id;
11.27/4.73	
11.27/4.73	liftM2 :: Monad d => (c  ->  b  ->  a)  ->  d c  ->  d b  ->  d a;
11.27/4.73	liftM2 f m1 m2 = m1 >>= (\x1 ->m2 >>= (\x2 ->return (f x1 x2)));
11.27/4.73	
11.27/4.73	}
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(1) LR (EQUIVALENT)
11.27/4.73	Lambda Reductions:
11.27/4.73	The following Lambda expression
11.27/4.73	"\x2->return (f x1 x2)"
11.27/4.73	is transformed to
11.27/4.73	"liftM20 f x1 x2 = return (f x1 x2);
11.27/4.73	"
11.27/4.73	The following Lambda expression
11.27/4.73	"\x1->m2 >>= liftM20 f x1"
11.27/4.73	is transformed to
11.27/4.73	"liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.27/4.73	"
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(2)
11.27/4.73	Obligation:
11.27/4.73	mainModule Main 
11.27/4.73	module Maybe where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Main where {
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Monad where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Prelude;
11.27/4.73	ap :: Monad b => b (a  ->  c)  ->  b a  ->  b c;
11.27/4.73	ap = liftM2 id;
11.27/4.73	
11.27/4.73	liftM2 :: Monad a => (b  ->  d  ->  c)  ->  a b  ->  a d  ->  a c;
11.27/4.73	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.27/4.73	
11.27/4.73	liftM20 f x1 x2 = return (f x1 x2);
11.27/4.73	
11.27/4.73	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.27/4.73	
11.27/4.73	}
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(3) BR (EQUIVALENT)
11.27/4.73	Replaced joker patterns by fresh variables and removed binding patterns.
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(4)
11.27/4.73	Obligation:
11.27/4.73	mainModule Main 
11.27/4.73	module Maybe where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Main where {
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Monad where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Prelude;
11.27/4.73	ap :: Monad c => c (a  ->  b)  ->  c a  ->  c b;
11.27/4.73	ap = liftM2 id;
11.27/4.73	
11.27/4.73	liftM2 :: Monad d => (c  ->  a  ->  b)  ->  d c  ->  d a  ->  d b;
11.27/4.73	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.27/4.73	
11.27/4.73	liftM20 f x1 x2 = return (f x1 x2);
11.27/4.73	
11.27/4.73	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.27/4.73	
11.27/4.73	}
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(5) COR (EQUIVALENT)
11.27/4.73	Cond Reductions:
11.27/4.73	The following Function with conditions
11.27/4.73	"undefined |Falseundefined;
11.27/4.73	"
11.27/4.73	is transformed to
11.27/4.73	"undefined  = undefined1;
11.27/4.73	"
11.27/4.73	"undefined0 True = undefined;
11.27/4.73	"
11.27/4.73	"undefined1  = undefined0 False;
11.27/4.73	"
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(6)
11.27/4.73	Obligation:
11.27/4.73	mainModule Main 
11.27/4.73	module Maybe where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Main where {
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Monad;
11.27/4.73	import qualified Prelude;
11.27/4.73	}
11.27/4.73	module Monad where {
11.27/4.73	import qualified Main;
11.27/4.73	import qualified Maybe;
11.27/4.73	import qualified Prelude;
11.27/4.73	ap :: Monad c => c (a  ->  b)  ->  c a  ->  c b;
11.27/4.73	ap = liftM2 id;
11.27/4.73	
11.27/4.73	liftM2 :: Monad a => (b  ->  c  ->  d)  ->  a b  ->  a c  ->  a d;
11.27/4.73	liftM2 f m1 m2 = m1 >>= liftM21 m2 f;
11.27/4.73	
11.27/4.73	liftM20 f x1 x2 = return (f x1 x2);
11.27/4.73	
11.27/4.73	liftM21 m2 f x1 = m2 >>= liftM20 f x1;
11.27/4.73	
11.27/4.73	}
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(7) Narrow (SOUND)
11.27/4.73	Haskell To QDPs
11.27/4.73	
11.27/4.73	digraph dp_graph {
11.27/4.73	node [outthreshold=100, inthreshold=100];1[label="Monad.ap",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
11.27/4.73	3[label="Monad.ap vy3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
11.27/4.73	4[label="Monad.ap vy3 vy4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
11.27/4.73	5[label="Monad.liftM2 id vy3 vy4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
11.27/4.73	6[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="blue",shape="box"];94[label=">>= :: (IO (a -> b)) -> ((a -> b) -> IO b) -> IO b",fontsize=10,color="white",style="solid",shape="box"];6 -> 94[label="",style="solid", color="blue", weight=9];
11.27/4.73	94 -> 7[label="",style="solid", color="blue", weight=3];
11.27/4.73	95[label=">>= :: (Maybe (a -> b)) -> ((a -> b) -> Maybe b) -> Maybe b",fontsize=10,color="white",style="solid",shape="box"];6 -> 95[label="",style="solid", color="blue", weight=9];
11.27/4.73	95 -> 8[label="",style="solid", color="blue", weight=3];
11.27/4.73	96[label=">>= :: ([] (a -> b)) -> ((a -> b) -> [] b) -> [] b",fontsize=10,color="white",style="solid",shape="box"];6 -> 96[label="",style="solid", color="blue", weight=9];
11.27/4.73	96 -> 9[label="",style="solid", color="blue", weight=3];
11.27/4.73	7[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
11.27/4.73	8[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="box"];97[label="vy3/Nothing",fontsize=10,color="white",style="solid",shape="box"];8 -> 97[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	97 -> 11[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	98[label="vy3/Just vy30",fontsize=10,color="white",style="solid",shape="box"];8 -> 98[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	98 -> 12[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	9[label="vy3 >>= Monad.liftM21 vy4 id",fontsize=16,color="burlywood",shape="triangle"];99[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];9 -> 99[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	99 -> 13[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	100[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 100[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	100 -> 14[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	10[label="primbindIO vy3 (Monad.liftM21 vy4 id)",fontsize=16,color="burlywood",shape="box"];101[label="vy3/IO vy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 101[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	101 -> 15[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	102[label="vy3/AProVE_IO vy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 102[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	102 -> 16[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	103[label="vy3/AProVE_Exception vy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 103[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	103 -> 17[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	104[label="vy3/AProVE_Error vy30",fontsize=10,color="white",style="solid",shape="box"];10 -> 104[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	104 -> 18[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	11[label="Nothing >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];11 -> 19[label="",style="solid", color="black", weight=3];
11.27/4.73	12[label="Just vy30 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];12 -> 20[label="",style="solid", color="black", weight=3];
11.27/4.73	13[label="vy30 : vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];13 -> 21[label="",style="solid", color="black", weight=3];
11.27/4.73	14[label="[] >>= Monad.liftM21 vy4 id",fontsize=16,color="black",shape="box"];14 -> 22[label="",style="solid", color="black", weight=3];
11.27/4.73	15[label="primbindIO (IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];15 -> 23[label="",style="solid", color="black", weight=3];
11.27/4.73	16[label="primbindIO (AProVE_IO vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];16 -> 24[label="",style="solid", color="black", weight=3];
11.27/4.73	17[label="primbindIO (AProVE_Exception vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];17 -> 25[label="",style="solid", color="black", weight=3];
11.27/4.73	18[label="primbindIO (AProVE_Error vy30) (Monad.liftM21 vy4 id)",fontsize=16,color="black",shape="box"];18 -> 26[label="",style="solid", color="black", weight=3];
11.27/4.73	19[label="Nothing",fontsize=16,color="green",shape="box"];20[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];20 -> 27[label="",style="solid", color="black", weight=3];
11.27/4.73	21 -> 28[label="",style="dashed", color="red", weight=0];
11.27/4.73	21[label="Monad.liftM21 vy4 id vy30 ++ (vy31 >>= Monad.liftM21 vy4 id)",fontsize=16,color="magenta"];21 -> 29[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	22[label="[]",fontsize=16,color="green",shape="box"];23[label="error []",fontsize=16,color="red",shape="box"];24[label="Monad.liftM21 vy4 id vy30",fontsize=16,color="black",shape="box"];24 -> 30[label="",style="solid", color="black", weight=3];
11.27/4.73	25[label="AProVE_Exception vy30",fontsize=16,color="green",shape="box"];26[label="AProVE_Error vy30",fontsize=16,color="green",shape="box"];27[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="box"];105[label="vy4/Nothing",fontsize=10,color="white",style="solid",shape="box"];27 -> 105[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	105 -> 31[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	106[label="vy4/Just vy40",fontsize=10,color="white",style="solid",shape="box"];27 -> 106[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	106 -> 32[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	29 -> 9[label="",style="dashed", color="red", weight=0];
11.27/4.73	29[label="vy31 >>= Monad.liftM21 vy4 id",fontsize=16,color="magenta"];29 -> 33[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	28[label="Monad.liftM21 vy4 id vy30 ++ vy5",fontsize=16,color="black",shape="triangle"];28 -> 34[label="",style="solid", color="black", weight=3];
11.27/4.73	30[label="vy4 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];30 -> 35[label="",style="solid", color="black", weight=3];
11.27/4.73	31[label="Nothing >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];31 -> 36[label="",style="solid", color="black", weight=3];
11.27/4.73	32[label="Just vy40 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
11.27/4.73	33[label="vy31",fontsize=16,color="green",shape="box"];34[label="(vy4 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="burlywood",shape="box"];107[label="vy4/vy40 : vy41",fontsize=10,color="white",style="solid",shape="box"];34 -> 107[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	107 -> 38[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	108[label="vy4/[]",fontsize=10,color="white",style="solid",shape="box"];34 -> 108[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	108 -> 39[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	35[label="primbindIO vy4 (Monad.liftM20 id vy30)",fontsize=16,color="burlywood",shape="box"];109[label="vy4/IO vy40",fontsize=10,color="white",style="solid",shape="box"];35 -> 109[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	109 -> 40[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	110[label="vy4/AProVE_IO vy40",fontsize=10,color="white",style="solid",shape="box"];35 -> 110[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	110 -> 41[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	111[label="vy4/AProVE_Exception vy40",fontsize=10,color="white",style="solid",shape="box"];35 -> 111[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	111 -> 42[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	112[label="vy4/AProVE_Error vy40",fontsize=10,color="white",style="solid",shape="box"];35 -> 112[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	112 -> 43[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	36[label="Nothing",fontsize=16,color="green",shape="box"];37[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];37 -> 44[label="",style="solid", color="black", weight=3];
11.27/4.73	38[label="(vy40 : vy41 >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];38 -> 45[label="",style="solid", color="black", weight=3];
11.27/4.73	39[label="([] >>= Monad.liftM20 id vy30) ++ vy5",fontsize=16,color="black",shape="box"];39 -> 46[label="",style="solid", color="black", weight=3];
11.27/4.73	40[label="primbindIO (IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];40 -> 47[label="",style="solid", color="black", weight=3];
11.27/4.73	41[label="primbindIO (AProVE_IO vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];41 -> 48[label="",style="solid", color="black", weight=3];
11.27/4.73	42[label="primbindIO (AProVE_Exception vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];42 -> 49[label="",style="solid", color="black", weight=3];
11.27/4.73	43[label="primbindIO (AProVE_Error vy40) (Monad.liftM20 id vy30)",fontsize=16,color="black",shape="box"];43 -> 50[label="",style="solid", color="black", weight=3];
11.27/4.73	44[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];44 -> 51[label="",style="solid", color="black", weight=3];
11.27/4.73	45[label="(Monad.liftM20 id vy30 vy40 ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];45 -> 52[label="",style="solid", color="black", weight=3];
11.27/4.73	46[label="[] ++ vy5",fontsize=16,color="black",shape="triangle"];46 -> 53[label="",style="solid", color="black", weight=3];
11.27/4.73	47[label="error []",fontsize=16,color="red",shape="box"];48[label="Monad.liftM20 id vy30 vy40",fontsize=16,color="black",shape="box"];48 -> 54[label="",style="solid", color="black", weight=3];
11.27/4.73	49[label="AProVE_Exception vy40",fontsize=16,color="green",shape="box"];50[label="AProVE_Error vy40",fontsize=16,color="green",shape="box"];51[label="Just (id vy30 vy40)",fontsize=16,color="green",shape="box"];51 -> 55[label="",style="dashed", color="green", weight=3];
11.27/4.73	52[label="(return (id vy30 vy40) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="box"];52 -> 56[label="",style="solid", color="black", weight=3];
11.27/4.73	53[label="vy5",fontsize=16,color="green",shape="box"];54[label="return (id vy30 vy40)",fontsize=16,color="black",shape="box"];54 -> 57[label="",style="solid", color="black", weight=3];
11.27/4.73	55[label="id vy30 vy40",fontsize=16,color="black",shape="triangle"];55 -> 58[label="",style="solid", color="black", weight=3];
11.27/4.73	56 -> 59[label="",style="dashed", color="red", weight=0];
11.27/4.73	56[label="((id vy30 vy40 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];56 -> 60[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	57 -> 61[label="",style="dashed", color="red", weight=0];
11.27/4.73	57[label="primretIO (id vy30 vy40)",fontsize=16,color="magenta"];57 -> 62[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	58[label="vy30 vy40",fontsize=16,color="green",shape="box"];58 -> 63[label="",style="dashed", color="green", weight=3];
11.27/4.73	60 -> 55[label="",style="dashed", color="red", weight=0];
11.27/4.73	60[label="id vy30 vy40",fontsize=16,color="magenta"];60 -> 64[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	60 -> 65[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	59[label="((vy6 : []) ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="black",shape="triangle"];59 -> 66[label="",style="solid", color="black", weight=3];
11.27/4.73	62 -> 55[label="",style="dashed", color="red", weight=0];
11.27/4.73	62[label="id vy30 vy40",fontsize=16,color="magenta"];62 -> 67[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	62 -> 68[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	61[label="primretIO vy7",fontsize=16,color="black",shape="triangle"];61 -> 69[label="",style="solid", color="black", weight=3];
11.27/4.73	63[label="vy40",fontsize=16,color="green",shape="box"];64[label="vy40",fontsize=16,color="green",shape="box"];65[label="vy30",fontsize=16,color="green",shape="box"];66 -> 70[label="",style="dashed", color="red", weight=0];
11.27/4.73	66[label="(vy6 : [] ++ (vy41 >>= Monad.liftM20 id vy30)) ++ vy5",fontsize=16,color="magenta"];66 -> 71[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	67[label="vy40",fontsize=16,color="green",shape="box"];68[label="vy30",fontsize=16,color="green",shape="box"];69[label="AProVE_IO vy7",fontsize=16,color="green",shape="box"];71 -> 46[label="",style="dashed", color="red", weight=0];
11.27/4.73	71[label="[] ++ (vy41 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];71 -> 72[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	70[label="(vy6 : vy8) ++ vy5",fontsize=16,color="black",shape="triangle"];70 -> 73[label="",style="solid", color="black", weight=3];
11.27/4.73	72[label="vy41 >>= Monad.liftM20 id vy30",fontsize=16,color="burlywood",shape="triangle"];113[label="vy41/vy410 : vy411",fontsize=10,color="white",style="solid",shape="box"];72 -> 113[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	113 -> 74[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	114[label="vy41/[]",fontsize=10,color="white",style="solid",shape="box"];72 -> 114[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	114 -> 75[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	73[label="vy6 : vy8 ++ vy5",fontsize=16,color="green",shape="box"];73 -> 76[label="",style="dashed", color="green", weight=3];
11.27/4.73	74[label="vy410 : vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];74 -> 77[label="",style="solid", color="black", weight=3];
11.27/4.73	75[label="[] >>= Monad.liftM20 id vy30",fontsize=16,color="black",shape="box"];75 -> 78[label="",style="solid", color="black", weight=3];
11.27/4.73	76[label="vy8 ++ vy5",fontsize=16,color="burlywood",shape="triangle"];115[label="vy8/vy80 : vy81",fontsize=10,color="white",style="solid",shape="box"];76 -> 115[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	115 -> 79[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	116[label="vy8/[]",fontsize=10,color="white",style="solid",shape="box"];76 -> 116[label="",style="solid", color="burlywood", weight=9];
11.27/4.73	116 -> 80[label="",style="solid", color="burlywood", weight=3];
11.27/4.73	77 -> 76[label="",style="dashed", color="red", weight=0];
11.27/4.73	77[label="Monad.liftM20 id vy30 vy410 ++ (vy411 >>= Monad.liftM20 id vy30)",fontsize=16,color="magenta"];77 -> 81[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	77 -> 82[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	78[label="[]",fontsize=16,color="green",shape="box"];79[label="(vy80 : vy81) ++ vy5",fontsize=16,color="black",shape="box"];79 -> 83[label="",style="solid", color="black", weight=3];
11.27/4.73	80[label="[] ++ vy5",fontsize=16,color="black",shape="box"];80 -> 84[label="",style="solid", color="black", weight=3];
11.27/4.73	81 -> 72[label="",style="dashed", color="red", weight=0];
11.27/4.73	81[label="vy411 >>= Monad.liftM20 id vy30",fontsize=16,color="magenta"];81 -> 85[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	82[label="Monad.liftM20 id vy30 vy410",fontsize=16,color="black",shape="box"];82 -> 86[label="",style="solid", color="black", weight=3];
11.27/4.73	83[label="vy80 : vy81 ++ vy5",fontsize=16,color="green",shape="box"];83 -> 87[label="",style="dashed", color="green", weight=3];
11.27/4.73	84[label="vy5",fontsize=16,color="green",shape="box"];85[label="vy411",fontsize=16,color="green",shape="box"];86 -> 88[label="",style="dashed", color="red", weight=0];
11.27/4.73	86[label="return (id vy30 vy410)",fontsize=16,color="magenta"];86 -> 89[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	87 -> 76[label="",style="dashed", color="red", weight=0];
11.27/4.73	87[label="vy81 ++ vy5",fontsize=16,color="magenta"];87 -> 90[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	89 -> 55[label="",style="dashed", color="red", weight=0];
11.27/4.73	89[label="id vy30 vy410",fontsize=16,color="magenta"];89 -> 91[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	89 -> 92[label="",style="dashed", color="magenta", weight=3];
11.27/4.73	88[label="return vy9",fontsize=16,color="black",shape="triangle"];88 -> 93[label="",style="solid", color="black", weight=3];
11.27/4.73	90[label="vy81",fontsize=16,color="green",shape="box"];91[label="vy410",fontsize=16,color="green",shape="box"];92[label="vy30",fontsize=16,color="green",shape="box"];93[label="vy9 : []",fontsize=16,color="green",shape="box"];}
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(8)
11.27/4.73	Complex Obligation (AND)
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(9)
11.27/4.73	Obligation:
11.27/4.73	Q DP problem:
11.27/4.73	The TRS P consists of the following rules:
11.27/4.73	
11.27/4.73	   new_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
11.27/4.73	
11.27/4.73	R is empty.
11.27/4.73	Q is empty.
11.27/4.73	We have to consider all minimal (P,Q,R)-chains.
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(10) QDPSizeChangeProof (EQUIVALENT)
11.27/4.73	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. 
11.27/4.73	
11.27/4.73	From the DPs we obtained the following set of size-change graphs:
11.27/4.73	*new_gtGtEs0(:(vy30, vy31), vy4, h, ba) -> new_gtGtEs0(vy31, vy4, h, ba)
11.27/4.73	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
11.27/4.73	
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(11)
11.27/4.73	YES
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(12)
11.27/4.73	Obligation:
11.27/4.73	Q DP problem:
11.27/4.73	The TRS P consists of the following rules:
11.27/4.73	
11.27/4.73	   new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
11.27/4.73	
11.27/4.73	R is empty.
11.27/4.73	Q is empty.
11.27/4.73	We have to consider all minimal (P,Q,R)-chains.
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(13) QDPSizeChangeProof (EQUIVALENT)
11.27/4.73	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. 
11.27/4.73	
11.27/4.73	From the DPs we obtained the following set of size-change graphs:
11.27/4.73	*new_gtGtEs(:(vy410, vy411), vy30, h, ba) -> new_gtGtEs(vy411, vy30, h, ba)
11.27/4.73	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3, 4 >= 4
11.27/4.73	
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(14)
11.27/4.73	YES
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(15)
11.27/4.73	Obligation:
11.27/4.73	Q DP problem:
11.27/4.73	The TRS P consists of the following rules:
11.27/4.73	
11.27/4.73	   new_psPs(:(vy80, vy81), vy5, h) -> new_psPs(vy81, vy5, h)
11.27/4.73	
11.27/4.73	R is empty.
11.27/4.73	Q is empty.
11.27/4.73	We have to consider all minimal (P,Q,R)-chains.
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(16) QDPSizeChangeProof (EQUIVALENT)
11.27/4.73	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. 
11.27/4.73	
11.27/4.73	From the DPs we obtained the following set of size-change graphs:
11.27/4.73	*new_psPs(:(vy80, vy81), vy5, h) -> new_psPs(vy81, vy5, h)
11.27/4.73	The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3
11.27/4.73	
11.27/4.73	
11.27/4.73	----------------------------------------
11.27/4.73	
11.27/4.73	(17)
11.27/4.73	YES
11.62/6.51	EOF