7.90/3.79	YES
10.12/4.36	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.12/4.36	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.12/4.36	
10.12/4.36	
10.12/4.36	H-Termination with start terms of the given HASKELL could be proven:
10.12/4.36	
10.12/4.36	(0) HASKELL
10.12/4.36	(1) BR [EQUIVALENT, 0 ms]
10.12/4.36	(2) HASKELL
10.12/4.36	(3) COR [EQUIVALENT, 0 ms]
10.12/4.36	(4) HASKELL
10.12/4.36	(5) Narrow [SOUND, 0 ms]
10.12/4.36	(6) AND
10.12/4.36	    (7) QDP
10.12/4.36	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.12/4.36	        (9) YES
10.12/4.36	    (10) QDP
10.12/4.36	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.12/4.36	        (12) YES
10.12/4.36	
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(0)
10.12/4.36	Obligation:
10.12/4.36	mainModule Main 
10.12/4.36	module Main where {
10.12/4.36	import qualified Prelude;
10.12/4.36	data Main.Char = Char MyInt ;
10.12/4.36	
10.12/4.36	data List a = Cons a (List a)  | Nil ;
10.12/4.36	
10.12/4.36	data MyBool = MyTrue  | MyFalse ;
10.12/4.36	
10.12/4.36	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.12/4.36	
10.12/4.36	data Main.Nat = Succ Main.Nat  | Zero ;
10.12/4.36	
10.12/4.36	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.12/4.36	any p = pt or (map p);
10.12/4.36	
10.12/4.36	elemChar :: Main.Char  ->  List Main.Char  ->  MyBool;
10.12/4.36	elemChar = pt any esEsChar;
10.12/4.36	
10.12/4.36	esEsChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	esEsChar = primEqChar;
10.12/4.36	
10.12/4.36	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
10.12/4.36	foldr f z Nil = z;
10.12/4.36	foldr f z (Cons x xs) = f x (foldr f z xs);
10.12/4.36	
10.12/4.36	map :: (a  ->  b)  ->  List a  ->  List b;
10.12/4.36	map f Nil = Nil;
10.12/4.36	map f (Cons x xs) = Cons (f x) (map f xs);
10.12/4.36	
10.12/4.36	or :: List MyBool  ->  MyBool;
10.12/4.36	or = foldr pePe MyFalse;
10.12/4.36	
10.12/4.36	pePe :: MyBool  ->  MyBool  ->  MyBool;
10.12/4.36	pePe MyFalse x = x;
10.12/4.36	pePe MyTrue x = MyTrue;
10.12/4.36	
10.12/4.36	primEqChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	primEqChar (Main.Char x) (Main.Char y) = primEqInt x y;
10.12/4.36	
10.12/4.36	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.12/4.36	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt vv vw = MyFalse;
10.12/4.36	
10.12/4.36	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.12/4.36	primEqNat Main.Zero Main.Zero = MyTrue;
10.12/4.36	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.12/4.36	
10.12/4.36	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
10.12/4.36	pt f g x = f (g x);
10.12/4.36	
10.12/4.36	}
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(1) BR (EQUIVALENT)
10.12/4.36	Replaced joker patterns by fresh variables and removed binding patterns.
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(2)
10.12/4.36	Obligation:
10.12/4.36	mainModule Main 
10.12/4.36	module Main where {
10.12/4.36	import qualified Prelude;
10.12/4.36	data Main.Char = Char MyInt ;
10.12/4.36	
10.12/4.36	data List a = Cons a (List a)  | Nil ;
10.12/4.36	
10.12/4.36	data MyBool = MyTrue  | MyFalse ;
10.12/4.36	
10.12/4.36	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.12/4.36	
10.12/4.36	data Main.Nat = Succ Main.Nat  | Zero ;
10.12/4.36	
10.12/4.36	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.12/4.36	any p = pt or (map p);
10.12/4.36	
10.12/4.36	elemChar :: Main.Char  ->  List Main.Char  ->  MyBool;
10.12/4.36	elemChar = pt any esEsChar;
10.12/4.36	
10.12/4.36	esEsChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	esEsChar = primEqChar;
10.12/4.36	
10.12/4.36	foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
10.12/4.36	foldr f z Nil = z;
10.12/4.36	foldr f z (Cons x xs) = f x (foldr f z xs);
10.12/4.36	
10.12/4.36	map :: (b  ->  a)  ->  List b  ->  List a;
10.12/4.36	map f Nil = Nil;
10.12/4.36	map f (Cons x xs) = Cons (f x) (map f xs);
10.12/4.36	
10.12/4.36	or :: List MyBool  ->  MyBool;
10.12/4.36	or = foldr pePe MyFalse;
10.12/4.36	
10.12/4.36	pePe :: MyBool  ->  MyBool  ->  MyBool;
10.12/4.36	pePe MyFalse x = x;
10.12/4.36	pePe MyTrue x = MyTrue;
10.12/4.36	
10.12/4.36	primEqChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	primEqChar (Main.Char x) (Main.Char y) = primEqInt x y;
10.12/4.36	
10.12/4.36	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.12/4.36	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt vv vw = MyFalse;
10.12/4.36	
10.12/4.36	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.12/4.36	primEqNat Main.Zero Main.Zero = MyTrue;
10.12/4.36	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.12/4.36	
10.12/4.36	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
10.12/4.36	pt f g x = f (g x);
10.12/4.36	
10.12/4.36	}
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(3) COR (EQUIVALENT)
10.12/4.36	Cond Reductions:
10.12/4.36	The following Function with conditions
10.12/4.36	"undefined |Falseundefined;
10.12/4.36	"
10.12/4.36	is transformed to
10.12/4.36	"undefined  = undefined1;
10.12/4.36	"
10.12/4.36	"undefined0 True = undefined;
10.12/4.36	"
10.12/4.36	"undefined1  = undefined0 False;
10.12/4.36	"
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(4)
10.12/4.36	Obligation:
10.12/4.36	mainModule Main 
10.12/4.36	module Main where {
10.12/4.36	import qualified Prelude;
10.12/4.36	data Main.Char = Char MyInt ;
10.12/4.36	
10.12/4.36	data List a = Cons a (List a)  | Nil ;
10.12/4.36	
10.12/4.36	data MyBool = MyTrue  | MyFalse ;
10.12/4.36	
10.12/4.36	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.12/4.36	
10.12/4.36	data Main.Nat = Succ Main.Nat  | Zero ;
10.12/4.36	
10.12/4.36	any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
10.12/4.36	any p = pt or (map p);
10.12/4.36	
10.12/4.36	elemChar :: Main.Char  ->  List Main.Char  ->  MyBool;
10.12/4.36	elemChar = pt any esEsChar;
10.12/4.36	
10.12/4.36	esEsChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	esEsChar = primEqChar;
10.12/4.36	
10.12/4.36	foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
10.12/4.36	foldr f z Nil = z;
10.12/4.36	foldr f z (Cons x xs) = f x (foldr f z xs);
10.12/4.36	
10.12/4.36	map :: (a  ->  b)  ->  List a  ->  List b;
10.12/4.36	map f Nil = Nil;
10.12/4.36	map f (Cons x xs) = Cons (f x) (map f xs);
10.12/4.36	
10.12/4.36	or :: List MyBool  ->  MyBool;
10.12/4.36	or = foldr pePe MyFalse;
10.12/4.36	
10.12/4.36	pePe :: MyBool  ->  MyBool  ->  MyBool;
10.12/4.36	pePe MyFalse x = x;
10.12/4.36	pePe MyTrue x = MyTrue;
10.12/4.36	
10.12/4.36	primEqChar :: Main.Char  ->  Main.Char  ->  MyBool;
10.12/4.36	primEqChar (Main.Char x) (Main.Char y) = primEqInt x y;
10.12/4.36	
10.12/4.36	primEqInt :: MyInt  ->  MyInt  ->  MyBool;
10.12/4.36	primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
10.12/4.36	primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
10.12/4.36	primEqInt vv vw = MyFalse;
10.12/4.36	
10.12/4.36	primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
10.12/4.36	primEqNat Main.Zero Main.Zero = MyTrue;
10.12/4.36	primEqNat Main.Zero (Main.Succ y) = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) Main.Zero = MyFalse;
10.12/4.36	primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;
10.12/4.36	
10.12/4.36	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
10.12/4.36	pt f g x = f (g x);
10.12/4.36	
10.12/4.36	}
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(5) Narrow (SOUND)
10.12/4.36	Haskell To QDPs
10.12/4.36	
10.12/4.36	digraph dp_graph {
10.12/4.36	node [outthreshold=100, inthreshold=100];1[label="elemChar",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.12/4.36	3[label="elemChar vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.12/4.36	4[label="elemChar vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.12/4.36	5[label="pt any esEsChar vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.12/4.36	6[label="any (esEsChar vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.12/4.36	7[label="pt or (map (esEsChar vz3)) vz4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.12/4.36	8[label="or (map (esEsChar vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.12/4.36	9[label="foldr pePe MyFalse (map (esEsChar 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];
10.12/4.36	79 -> 10[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	80[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 80[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	80 -> 11[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	10[label="foldr pePe MyFalse (map (esEsChar vz3) (Cons vz40 vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
10.12/4.36	11[label="foldr pePe MyFalse (map (esEsChar vz3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
10.12/4.36	12[label="foldr pePe MyFalse (Cons (esEsChar vz3 vz40) (map (esEsChar vz3) vz41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
10.12/4.36	13[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
10.12/4.36	14 -> 16[label="",style="dashed", color="red", weight=0];
10.12/4.36	14[label="pePe (esEsChar vz3 vz40) (foldr pePe MyFalse (map (esEsChar vz3) vz41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	15[label="MyFalse",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
10.12/4.36	17[label="foldr pePe MyFalse (map (esEsChar vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	16[label="pePe (esEsChar vz3 vz40) vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
10.12/4.36	18[label="vz41",fontsize=16,color="green",shape="box"];19[label="pePe (primEqChar vz3 vz40) vz5",fontsize=16,color="burlywood",shape="box"];81[label="vz3/Char vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 81[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	81 -> 20[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	20[label="pePe (primEqChar (Char vz30) vz40) vz5",fontsize=16,color="burlywood",shape="box"];82[label="vz40/Char vz400",fontsize=10,color="white",style="solid",shape="box"];20 -> 82[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	82 -> 21[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	21[label="pePe (primEqChar (Char vz30) (Char vz400)) vz5",fontsize=16,color="black",shape="box"];21 -> 22[label="",style="solid", color="black", weight=3];
10.12/4.36	22[label="pePe (primEqInt vz30 vz400) vz5",fontsize=16,color="burlywood",shape="box"];83[label="vz30/Pos vz300",fontsize=10,color="white",style="solid",shape="box"];22 -> 83[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	83 -> 23[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	84[label="vz30/Neg vz300",fontsize=10,color="white",style="solid",shape="box"];22 -> 84[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	84 -> 24[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	23[label="pePe (primEqInt (Pos vz300) vz400) vz5",fontsize=16,color="burlywood",shape="box"];85[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];23 -> 85[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	85 -> 25[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	86[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 86[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	86 -> 26[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	24[label="pePe (primEqInt (Neg vz300) vz400) vz5",fontsize=16,color="burlywood",shape="box"];87[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];24 -> 87[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	87 -> 27[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	88[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	88 -> 28[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	25[label="pePe (primEqInt (Pos (Succ vz3000)) vz400) vz5",fontsize=16,color="burlywood",shape="box"];89[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 89[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	89 -> 29[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	90[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];25 -> 90[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	90 -> 30[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	26[label="pePe (primEqInt (Pos Zero) vz400) vz5",fontsize=16,color="burlywood",shape="box"];91[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 91[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	91 -> 31[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	92[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 92[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	92 -> 32[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	27[label="pePe (primEqInt (Neg (Succ vz3000)) vz400) vz5",fontsize=16,color="burlywood",shape="box"];93[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 93[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	93 -> 33[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	94[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];27 -> 94[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	94 -> 34[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	28[label="pePe (primEqInt (Neg Zero) vz400) vz5",fontsize=16,color="burlywood",shape="box"];95[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 95[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	95 -> 35[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	96[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 96[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	96 -> 36[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	29[label="pePe (primEqInt (Pos (Succ vz3000)) (Pos vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];97[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	97 -> 37[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	98[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 98[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	98 -> 38[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	30[label="pePe (primEqInt (Pos (Succ vz3000)) (Neg vz4000)) vz5",fontsize=16,color="black",shape="box"];30 -> 39[label="",style="solid", color="black", weight=3];
10.12/4.36	31[label="pePe (primEqInt (Pos Zero) (Pos vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];99[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];31 -> 99[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	99 -> 40[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	100[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 100[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	100 -> 41[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	32[label="pePe (primEqInt (Pos Zero) (Neg vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];101[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];32 -> 101[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	101 -> 42[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	102[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 102[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	102 -> 43[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	33[label="pePe (primEqInt (Neg (Succ vz3000)) (Pos vz4000)) vz5",fontsize=16,color="black",shape="box"];33 -> 44[label="",style="solid", color="black", weight=3];
10.12/4.36	34[label="pePe (primEqInt (Neg (Succ vz3000)) (Neg vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];103[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];34 -> 103[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	103 -> 45[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	104[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];34 -> 104[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	104 -> 46[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	35[label="pePe (primEqInt (Neg Zero) (Pos vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];105[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];35 -> 105[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	105 -> 47[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	106[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];35 -> 106[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	106 -> 48[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	36[label="pePe (primEqInt (Neg Zero) (Neg vz4000)) vz5",fontsize=16,color="burlywood",shape="box"];107[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];36 -> 107[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	107 -> 49[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	108[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];36 -> 108[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	108 -> 50[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	37[label="pePe (primEqInt (Pos (Succ vz3000)) (Pos (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
10.12/4.36	38[label="pePe (primEqInt (Pos (Succ vz3000)) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
10.12/4.36	39[label="pePe MyFalse vz5",fontsize=16,color="black",shape="triangle"];39 -> 53[label="",style="solid", color="black", weight=3];
10.12/4.36	40[label="pePe (primEqInt (Pos Zero) (Pos (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
10.12/4.36	41[label="pePe (primEqInt (Pos Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];41 -> 55[label="",style="solid", color="black", weight=3];
10.12/4.36	42[label="pePe (primEqInt (Pos Zero) (Neg (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];42 -> 56[label="",style="solid", color="black", weight=3];
10.12/4.36	43[label="pePe (primEqInt (Pos Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];43 -> 57[label="",style="solid", color="black", weight=3];
10.12/4.36	44 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	44[label="pePe MyFalse vz5",fontsize=16,color="magenta"];45[label="pePe (primEqInt (Neg (Succ vz3000)) (Neg (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3];
10.12/4.36	46[label="pePe (primEqInt (Neg (Succ vz3000)) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
10.12/4.36	47[label="pePe (primEqInt (Neg Zero) (Pos (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
10.12/4.36	48[label="pePe (primEqInt (Neg Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];48 -> 61[label="",style="solid", color="black", weight=3];
10.12/4.36	49[label="pePe (primEqInt (Neg Zero) (Neg (Succ vz40000))) vz5",fontsize=16,color="black",shape="box"];49 -> 62[label="",style="solid", color="black", weight=3];
10.12/4.36	50[label="pePe (primEqInt (Neg Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];50 -> 63[label="",style="solid", color="black", weight=3];
10.12/4.36	51[label="pePe (primEqNat vz3000 vz40000) vz5",fontsize=16,color="burlywood",shape="triangle"];109[label="vz3000/Succ vz30000",fontsize=10,color="white",style="solid",shape="box"];51 -> 109[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	109 -> 64[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	110[label="vz3000/Zero",fontsize=10,color="white",style="solid",shape="box"];51 -> 110[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	110 -> 65[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	52 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	52[label="pePe MyFalse vz5",fontsize=16,color="magenta"];53[label="vz5",fontsize=16,color="green",shape="box"];54 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	54[label="pePe MyFalse vz5",fontsize=16,color="magenta"];55[label="pePe MyTrue vz5",fontsize=16,color="black",shape="triangle"];55 -> 66[label="",style="solid", color="black", weight=3];
10.12/4.36	56 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	56[label="pePe MyFalse vz5",fontsize=16,color="magenta"];57 -> 55[label="",style="dashed", color="red", weight=0];
10.12/4.36	57[label="pePe MyTrue vz5",fontsize=16,color="magenta"];58 -> 51[label="",style="dashed", color="red", weight=0];
10.12/4.36	58[label="pePe (primEqNat vz3000 vz40000) vz5",fontsize=16,color="magenta"];58 -> 67[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	58 -> 68[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	59 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	59[label="pePe MyFalse vz5",fontsize=16,color="magenta"];60 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	60[label="pePe MyFalse vz5",fontsize=16,color="magenta"];61 -> 55[label="",style="dashed", color="red", weight=0];
10.12/4.36	61[label="pePe MyTrue vz5",fontsize=16,color="magenta"];62 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	62[label="pePe MyFalse vz5",fontsize=16,color="magenta"];63 -> 55[label="",style="dashed", color="red", weight=0];
10.12/4.36	63[label="pePe MyTrue vz5",fontsize=16,color="magenta"];64[label="pePe (primEqNat (Succ vz30000) vz40000) vz5",fontsize=16,color="burlywood",shape="box"];111[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];64 -> 111[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	111 -> 69[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	112[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 112[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	112 -> 70[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	65[label="pePe (primEqNat Zero vz40000) vz5",fontsize=16,color="burlywood",shape="box"];113[label="vz40000/Succ vz400000",fontsize=10,color="white",style="solid",shape="box"];65 -> 113[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	113 -> 71[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	114[label="vz40000/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 114[label="",style="solid", color="burlywood", weight=9];
10.12/4.36	114 -> 72[label="",style="solid", color="burlywood", weight=3];
10.12/4.36	66[label="MyTrue",fontsize=16,color="green",shape="box"];67[label="vz3000",fontsize=16,color="green",shape="box"];68[label="vz40000",fontsize=16,color="green",shape="box"];69[label="pePe (primEqNat (Succ vz30000) (Succ vz400000)) vz5",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
10.12/4.36	70[label="pePe (primEqNat (Succ vz30000) Zero) vz5",fontsize=16,color="black",shape="box"];70 -> 74[label="",style="solid", color="black", weight=3];
10.12/4.36	71[label="pePe (primEqNat Zero (Succ vz400000)) vz5",fontsize=16,color="black",shape="box"];71 -> 75[label="",style="solid", color="black", weight=3];
10.12/4.36	72[label="pePe (primEqNat Zero Zero) vz5",fontsize=16,color="black",shape="box"];72 -> 76[label="",style="solid", color="black", weight=3];
10.12/4.36	73 -> 51[label="",style="dashed", color="red", weight=0];
10.12/4.36	73[label="pePe (primEqNat vz30000 vz400000) vz5",fontsize=16,color="magenta"];73 -> 77[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	73 -> 78[label="",style="dashed", color="magenta", weight=3];
10.12/4.36	74 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	74[label="pePe MyFalse vz5",fontsize=16,color="magenta"];75 -> 39[label="",style="dashed", color="red", weight=0];
10.12/4.36	75[label="pePe MyFalse vz5",fontsize=16,color="magenta"];76 -> 55[label="",style="dashed", color="red", weight=0];
10.12/4.36	76[label="pePe MyTrue vz5",fontsize=16,color="magenta"];77[label="vz30000",fontsize=16,color="green",shape="box"];78[label="vz400000",fontsize=16,color="green",shape="box"];}
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(6)
10.12/4.36	Complex Obligation (AND)
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(7)
10.12/4.36	Obligation:
10.12/4.36	Q DP problem:
10.12/4.36	The TRS P consists of the following rules:
10.12/4.36	
10.12/4.36	   new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)
10.12/4.36	
10.12/4.36	R is empty.
10.12/4.36	Q is empty.
10.12/4.36	We have to consider all minimal (P,Q,R)-chains.
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(8) QDPSizeChangeProof (EQUIVALENT)
10.12/4.36	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. 
10.12/4.36	
10.12/4.36	From the DPs we obtained the following set of size-change graphs:
10.12/4.36	*new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)
10.12/4.36	The graph contains the following edges 1 >= 1, 2 > 2
10.12/4.36	
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(9)
10.12/4.36	YES
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(10)
10.12/4.36	Obligation:
10.12/4.36	Q DP problem:
10.12/4.36	The TRS P consists of the following rules:
10.12/4.36	
10.12/4.36	   new_pePe(Main.Succ(vz30000), Main.Succ(vz400000), vz5) -> new_pePe(vz30000, vz400000, vz5)
10.12/4.36	
10.12/4.36	R is empty.
10.12/4.36	Q is empty.
10.12/4.36	We have to consider all minimal (P,Q,R)-chains.
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(11) QDPSizeChangeProof (EQUIVALENT)
10.12/4.36	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. 
10.12/4.36	
10.12/4.36	From the DPs we obtained the following set of size-change graphs:
10.12/4.36	*new_pePe(Main.Succ(vz30000), Main.Succ(vz400000), vz5) -> new_pePe(vz30000, vz400000, vz5)
10.12/4.36	The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3
10.12/4.36	
10.12/4.36	
10.12/4.36	----------------------------------------
10.12/4.36	
10.12/4.36	(12)
10.12/4.36	YES
10.12/4.40	EOF