8.08/3.65	YES
10.07/4.18	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.07/4.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.07/4.18	
10.07/4.18	
10.07/4.18	H-Termination with start terms of the given HASKELL could be proven:
10.07/4.18	
10.07/4.18	(0) HASKELL
10.07/4.18	(1) BR [EQUIVALENT, 0 ms]
10.07/4.18	(2) HASKELL
10.07/4.18	(3) COR [EQUIVALENT, 0 ms]
10.07/4.18	(4) HASKELL
10.07/4.18	(5) Narrow [SOUND, 0 ms]
10.07/4.18	(6) AND
10.07/4.18	    (7) QDP
10.07/4.18	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.07/4.18	        (9) YES
10.07/4.18	    (10) QDP
10.07/4.18	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.07/4.18	        (12) YES
10.07/4.18	
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(0)
10.07/4.18	Obligation:
10.07/4.18	mainModule Main 
10.07/4.18	module Main where {
10.07/4.18	import qualified Prelude;
10.07/4.18	data List a = Cons a (List a)  | Nil ;
10.07/4.18	
10.07/4.18	data MyBool = MyTrue  | MyFalse ;
10.07/4.18	
10.07/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.07/4.18	
10.07/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.07/4.18	
10.07/4.18	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.07/4.18	all p = pt and (map p);
10.07/4.18	
10.07/4.18	and :: List MyBool  ->  MyBool;
10.07/4.18	and = foldr asAs MyTrue;
10.07/4.18	
10.07/4.18	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.07/4.18	asAs MyFalse x = MyFalse;
10.07/4.18	asAs MyTrue x = x;
10.07/4.18	
10.07/4.18	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	esEsMyInt = primEqInt;
10.07/4.18	
10.07/4.18	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.07/4.18	foldr f z Nil = z;
10.07/4.18	foldr f z (Cons x xs) = f x (foldr f z xs);
10.07/4.18	
10.07/4.18	fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	fsEsMyInt x y = not (esEsMyInt x y);
10.07/4.18	
10.07/4.18	map :: (b  ->  a)  ->  List b  ->  List a;
10.07/4.18	map f Nil = Nil;
10.07/4.18	map f (Cons x xs) = Cons (f x) (map f xs);
10.07/4.18	
10.07/4.18	not :: MyBool  ->  MyBool;
10.07/4.18	not MyTrue = MyFalse;
10.07/4.18	not MyFalse = MyTrue;
10.07/4.18	
10.07/4.18	notElemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
10.07/4.18	notElemMyInt = pt all fsEsMyInt;
10.07/4.18	
10.07/4.18	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt vv vw = MyFalse;
10.07/4.18	
10.07/4.18	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.07/4.18	primEqNat Main.Zero Main.Zero = MyTrue;
10.07/4.18	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.07/4.18	
10.07/4.18	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
10.07/4.18	pt f g x = f (g x);
10.07/4.18	
10.07/4.18	}
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(1) BR (EQUIVALENT)
10.07/4.18	Replaced joker patterns by fresh variables and removed binding patterns.
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(2)
10.07/4.18	Obligation:
10.07/4.18	mainModule Main 
10.07/4.18	module Main where {
10.07/4.18	import qualified Prelude;
10.07/4.18	data List a = Cons a (List a)  | Nil ;
10.07/4.18	
10.07/4.18	data MyBool = MyTrue  | MyFalse ;
10.07/4.18	
10.07/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.07/4.18	
10.07/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.07/4.18	
10.07/4.18	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.07/4.18	all p = pt and (map p);
10.07/4.18	
10.07/4.18	and :: List MyBool  ->  MyBool;
10.07/4.18	and = foldr asAs MyTrue;
10.07/4.18	
10.07/4.18	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.07/4.18	asAs MyFalse x = MyFalse;
10.07/4.18	asAs MyTrue x = x;
10.07/4.18	
10.07/4.18	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	esEsMyInt = primEqInt;
10.07/4.18	
10.07/4.18	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.07/4.18	foldr f z Nil = z;
10.07/4.18	foldr f z (Cons x xs) = f x (foldr f z xs);
10.07/4.18	
10.07/4.18	fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	fsEsMyInt x y = not (esEsMyInt x y);
10.07/4.18	
10.07/4.18	map :: (a  ->  b)  ->  List a  ->  List b;
10.07/4.18	map f Nil = Nil;
10.07/4.18	map f (Cons x xs) = Cons (f x) (map f xs);
10.07/4.18	
10.07/4.18	not :: MyBool  ->  MyBool;
10.07/4.18	not MyTrue = MyFalse;
10.07/4.18	not MyFalse = MyTrue;
10.07/4.18	
10.07/4.18	notElemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
10.07/4.18	notElemMyInt = pt all fsEsMyInt;
10.07/4.18	
10.07/4.18	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt vv vw = MyFalse;
10.07/4.18	
10.07/4.18	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.07/4.18	primEqNat Main.Zero Main.Zero = MyTrue;
10.07/4.18	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.07/4.18	
10.07/4.18	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
10.07/4.18	pt f g x = f (g x);
10.07/4.18	
10.07/4.18	}
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(3) COR (EQUIVALENT)
10.07/4.18	Cond Reductions:
10.07/4.18	The following Function with conditions
10.07/4.18	"undefined |Falseundefined;
10.07/4.18	"
10.07/4.18	is transformed to
10.07/4.18	"undefined  = undefined1;
10.07/4.18	"
10.07/4.18	"undefined0 True = undefined;
10.07/4.18	"
10.07/4.18	"undefined1  = undefined0 False;
10.07/4.18	"
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(4)
10.07/4.18	Obligation:
10.07/4.18	mainModule Main 
10.07/4.18	module Main where {
10.07/4.18	import qualified Prelude;
10.07/4.18	data List a = Cons a (List a)  | Nil ;
10.07/4.18	
10.07/4.18	data MyBool = MyTrue  | MyFalse ;
10.07/4.18	
10.07/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.07/4.18	
10.07/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.07/4.18	
10.07/4.18	all :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.07/4.18	all p = pt and (map p);
10.07/4.18	
10.07/4.18	and :: List MyBool  ->  MyBool;
10.07/4.18	and = foldr asAs MyTrue;
10.07/4.18	
10.07/4.18	asAs :: MyBool  ->  MyBool  ->  MyBool;
10.07/4.18	asAs MyFalse x = MyFalse;
10.07/4.18	asAs MyTrue x = x;
10.07/4.18	
10.07/4.18	esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	esEsMyInt = primEqInt;
10.07/4.18	
10.07/4.18	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.07/4.18	foldr f z Nil = z;
10.07/4.18	foldr f z (Cons x xs) = f x (foldr f z xs);
10.07/4.18	
10.07/4.18	fsEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	fsEsMyInt x y = not (esEsMyInt x y);
10.07/4.18	
10.07/4.18	map :: (b  ->  a)  ->  List b  ->  List a;
10.07/4.18	map f Nil = Nil;
10.07/4.18	map f (Cons x xs) = Cons (f x) (map f xs);
10.07/4.18	
10.07/4.18	not :: MyBool  ->  MyBool;
10.07/4.18	not MyTrue = MyFalse;
10.07/4.18	not MyFalse = MyTrue;
10.07/4.18	
10.07/4.18	notElemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
10.07/4.18	notElemMyInt = pt all fsEsMyInt;
10.07/4.18	
10.07/4.18	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.07/4.18	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.07/4.18	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.07/4.18	primEqInt vv vw = MyFalse;
10.07/4.18	
10.07/4.18	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.07/4.18	primEqNat Main.Zero Main.Zero = MyTrue;
10.07/4.18	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.07/4.18	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.07/4.18	
10.07/4.18	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
10.07/4.18	pt f g x = f (g x);
10.07/4.18	
10.07/4.18	}
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(5) Narrow (SOUND)
10.07/4.18	Haskell To QDPs
10.07/4.18	
10.07/4.18	digraph dp_graph {
10.07/4.18	node [outthreshold=100, inthreshold=100];1[label="notElemMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.07/4.18	3[label="notElemMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.07/4.18	4[label="notElemMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.07/4.18	5[label="pt all fsEsMyInt vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.07/4.18	6[label="all (fsEsMyInt vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.07/4.18	7[label="pt and (map (fsEsMyInt vz3)) vz4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.07/4.18	8[label="and (map (fsEsMyInt vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.07/4.18	9[label="foldr asAs MyTrue (map (fsEsMyInt vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];79[label="vz4/Cons vz40 vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 79[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	79 -> 10[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	80[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 80[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	80 -> 11[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	10[label="foldr asAs MyTrue (map (fsEsMyInt vz3) (Cons vz40 vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.07/4.18	11[label="foldr asAs MyTrue (map (fsEsMyInt vz3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.07/4.18	12[label="foldr asAs MyTrue (Cons (fsEsMyInt vz3 vz40) (map (fsEsMyInt vz3) vz41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.07/4.18	13[label="foldr asAs MyTrue Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.07/4.18	14 -> 16[label="",style="dashed", color="red", weight=0];
10.07/4.18	14[label="asAs (fsEsMyInt vz3 vz40) (foldr asAs MyTrue (map (fsEsMyInt vz3) vz41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	15[label="MyTrue",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
10.07/4.18	17[label="foldr asAs MyTrue (map (fsEsMyInt vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	16[label="asAs (fsEsMyInt vz3 vz40) vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
10.07/4.18	18[label="vz41",fontsize=16,color="green",shape="box"];19[label="asAs (not (esEsMyInt vz3 vz40)) vz5",fontsize=16,color="black",shape="box"];19 -> 20[label="",style="solid", color="black", weight=3];
10.07/4.18	20[label="asAs (not (primEqInt vz3 vz40)) vz5",fontsize=16,color="burlywood",shape="box"];81[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 81[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	81 -> 21[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	82[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];20 -> 82[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	82 -> 22[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	21[label="asAs (not (primEqInt (Pos vz30) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];83[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];21 -> 83[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	83 -> 23[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	84[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 84[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	84 -> 24[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	22[label="asAs (not (primEqInt (Neg vz30) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];85[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];22 -> 85[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	85 -> 25[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	86[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 86[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	86 -> 26[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	23[label="asAs (not (primEqInt (Pos (Succ vz300)) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];87[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 87[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	87 -> 27[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	88[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 88[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	88 -> 28[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	24[label="asAs (not (primEqInt (Pos Zero) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];89[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	89 -> 29[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	90[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 90[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	90 -> 30[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	25[label="asAs (not (primEqInt (Neg (Succ vz300)) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];91[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 91[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	91 -> 31[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	92[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 92[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	92 -> 32[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	26[label="asAs (not (primEqInt (Neg Zero) vz40)) vz5",fontsize=16,color="burlywood",shape="box"];93[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];26 -> 93[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	93 -> 33[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	94[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];26 -> 94[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	94 -> 34[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	27[label="asAs (not (primEqInt (Pos (Succ vz300)) (Pos vz400))) vz5",fontsize=16,color="burlywood",shape="box"];95[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 95[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	95 -> 35[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	96[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 96[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	96 -> 36[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	28[label="asAs (not (primEqInt (Pos (Succ vz300)) (Neg vz400))) vz5",fontsize=16,color="black",shape="box"];28 -> 37[label="",style="solid", color="black", weight=3];
10.07/4.18	29[label="asAs (not (primEqInt (Pos Zero) (Pos vz400))) vz5",fontsize=16,color="burlywood",shape="box"];97[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	97 -> 38[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	98[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	98 -> 39[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	30[label="asAs (not (primEqInt (Pos Zero) (Neg vz400))) vz5",fontsize=16,color="burlywood",shape="box"];99[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];30 -> 99[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	99 -> 40[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	100[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];30 -> 100[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	100 -> 41[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	31[label="asAs (not (primEqInt (Neg (Succ vz300)) (Pos vz400))) vz5",fontsize=16,color="black",shape="box"];31 -> 42[label="",style="solid", color="black", weight=3];
10.07/4.18	32[label="asAs (not (primEqInt (Neg (Succ vz300)) (Neg vz400))) vz5",fontsize=16,color="burlywood",shape="box"];101[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 101[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	101 -> 43[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	102[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 102[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	102 -> 44[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	33[label="asAs (not (primEqInt (Neg Zero) (Pos vz400))) vz5",fontsize=16,color="burlywood",shape="box"];103[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	103 -> 45[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	104[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 104[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	104 -> 46[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	34[label="asAs (not (primEqInt (Neg Zero) (Neg vz400))) vz5",fontsize=16,color="burlywood",shape="box"];105[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];34 -> 105[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	105 -> 47[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	106[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 106[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	106 -> 48[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	35[label="asAs (not (primEqInt (Pos (Succ vz300)) (Pos (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
10.07/4.18	36[label="asAs (not (primEqInt (Pos (Succ vz300)) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
10.07/4.18	37[label="asAs (not MyFalse) vz5",fontsize=16,color="black",shape="triangle"];37 -> 51[label="",style="solid", color="black", weight=3];
10.07/4.18	38[label="asAs (not (primEqInt (Pos Zero) (Pos (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
10.07/4.18	39[label="asAs (not (primEqInt (Pos Zero) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
10.07/4.18	40[label="asAs (not (primEqInt (Pos Zero) (Neg (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
10.07/4.18	41[label="asAs (not (primEqInt (Pos Zero) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
10.07/4.18	42 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	42[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];43[label="asAs (not (primEqInt (Neg (Succ vz300)) (Neg (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
10.07/4.18	44[label="asAs (not (primEqInt (Neg (Succ vz300)) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3];
10.07/4.18	45[label="asAs (not (primEqInt (Neg Zero) (Pos (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3];
10.07/4.18	46[label="asAs (not (primEqInt (Neg Zero) (Pos Zero))) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
10.07/4.18	47[label="asAs (not (primEqInt (Neg Zero) (Neg (Succ vz4000)))) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
10.07/4.18	48[label="asAs (not (primEqInt (Neg Zero) (Neg Zero))) vz5",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3];
10.07/4.18	49[label="asAs (not (primEqNat vz300 vz4000)) vz5",fontsize=16,color="burlywood",shape="triangle"];107[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];49 -> 107[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	107 -> 62[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	108[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];49 -> 108[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	108 -> 63[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	50 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	50[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];51[label="asAs MyTrue vz5",fontsize=16,color="black",shape="box"];51 -> 64[label="",style="solid", color="black", weight=3];
10.07/4.18	52 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	52[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];53[label="asAs (not MyTrue) vz5",fontsize=16,color="black",shape="triangle"];53 -> 65[label="",style="solid", color="black", weight=3];
10.07/4.18	54 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	54[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];55 -> 53[label="",style="dashed", color="red", weight=0];
10.07/4.18	55[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];56 -> 49[label="",style="dashed", color="red", weight=0];
10.07/4.18	56[label="asAs (not (primEqNat vz300 vz4000)) vz5",fontsize=16,color="magenta"];56 -> 66[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	56 -> 67[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	57 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	57[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];58 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	58[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];59 -> 53[label="",style="dashed", color="red", weight=0];
10.07/4.18	59[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];60 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	60[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];61 -> 53[label="",style="dashed", color="red", weight=0];
10.07/4.18	61[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];62[label="asAs (not (primEqNat (Succ vz3000) vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];109[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 109[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	109 -> 68[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	110[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 110[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	110 -> 69[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	63[label="asAs (not (primEqNat Zero vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];111[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];63 -> 111[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	111 -> 70[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	112[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];63 -> 112[label="",style="solid", color="burlywood", weight=9];
10.07/4.18	112 -> 71[label="",style="solid", color="burlywood", weight=3];
10.07/4.18	64[label="vz5",fontsize=16,color="green",shape="box"];65[label="asAs MyFalse vz5",fontsize=16,color="black",shape="box"];65 -> 72[label="",style="solid", color="black", weight=3];
10.07/4.18	66[label="vz300",fontsize=16,color="green",shape="box"];67[label="vz4000",fontsize=16,color="green",shape="box"];68[label="asAs (not (primEqNat (Succ vz3000) (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];68 -> 73[label="",style="solid", color="black", weight=3];
10.07/4.18	69[label="asAs (not (primEqNat (Succ vz3000) Zero)) vz5",fontsize=16,color="black",shape="box"];69 -> 74[label="",style="solid", color="black", weight=3];
10.07/4.18	70[label="asAs (not (primEqNat Zero (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];70 -> 75[label="",style="solid", color="black", weight=3];
10.07/4.18	71[label="asAs (not (primEqNat Zero Zero)) vz5",fontsize=16,color="black",shape="box"];71 -> 76[label="",style="solid", color="black", weight=3];
10.07/4.18	72[label="MyFalse",fontsize=16,color="green",shape="box"];73 -> 49[label="",style="dashed", color="red", weight=0];
10.07/4.18	73[label="asAs (not (primEqNat vz3000 vz40000)) vz5",fontsize=16,color="magenta"];73 -> 77[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	73 -> 78[label="",style="dashed", color="magenta", weight=3];
10.07/4.18	74 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	74[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];75 -> 37[label="",style="dashed", color="red", weight=0];
10.07/4.18	75[label="asAs (not MyFalse) vz5",fontsize=16,color="magenta"];76 -> 53[label="",style="dashed", color="red", weight=0];
10.07/4.18	76[label="asAs (not MyTrue) vz5",fontsize=16,color="magenta"];77[label="vz3000",fontsize=16,color="green",shape="box"];78[label="vz40000",fontsize=16,color="green",shape="box"];}
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(6)
10.07/4.18	Complex Obligation (AND)
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(7)
10.07/4.18	Obligation:
10.07/4.18	Q DP problem:
10.07/4.18	The TRS P consists of the following rules:
10.07/4.18	
10.07/4.18	   new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)
10.07/4.18	
10.07/4.18	R is empty.
10.07/4.18	Q is empty.
10.07/4.18	We have to consider all minimal (P,Q,R)-chains.
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(8) QDPSizeChangeProof (EQUIVALENT)
10.07/4.18	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. 
10.07/4.18	
10.07/4.18	From the DPs we obtained the following set of size-change graphs:
10.07/4.18	*new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)
10.07/4.18	The graph contains the following edges 1 >= 1, 2 > 2
10.07/4.18	
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(9)
10.07/4.18	YES
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(10)
10.07/4.18	Obligation:
10.07/4.18	Q DP problem:
10.07/4.18	The TRS P consists of the following rules:
10.07/4.18	
10.07/4.18	   new_asAs(Main.Succ(vz3000), Main.Succ(vz40000), vz5) -> new_asAs(vz3000, vz40000, vz5)
10.07/4.18	
10.07/4.18	R is empty.
10.07/4.18	Q is empty.
10.07/4.18	We have to consider all minimal (P,Q,R)-chains.
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(11) QDPSizeChangeProof (EQUIVALENT)
10.07/4.18	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. 
10.07/4.18	
10.07/4.18	From the DPs we obtained the following set of size-change graphs:
10.07/4.18	*new_asAs(Main.Succ(vz3000), Main.Succ(vz40000), vz5) -> new_asAs(vz3000, vz40000, vz5)
10.07/4.18	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.07/4.18	
10.07/4.18	
10.07/4.18	----------------------------------------
10.07/4.18	
10.07/4.18	(12)
10.07/4.18	YES
10.07/4.22	EOF