8.17/3.66	YES
10.13/4.18	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
10.13/4.18	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
10.13/4.18	
10.13/4.18	
10.13/4.18	H-Termination with start terms of the given HASKELL could be proven:
10.13/4.18	
10.13/4.18	(0) HASKELL
10.13/4.18	(1) BR [EQUIVALENT, 0 ms]
10.13/4.18	(2) HASKELL
10.13/4.18	(3) COR [EQUIVALENT, 0 ms]
10.13/4.18	(4) HASKELL
10.13/4.18	(5) Narrow [SOUND, 0 ms]
10.13/4.18	(6) QDP
10.13/4.18	(7) TransformationProof [EQUIVALENT, 0 ms]
10.13/4.18	(8) QDP
10.13/4.18	(9) DependencyGraphProof [EQUIVALENT, 0 ms]
10.13/4.18	(10) QDP
10.13/4.18	(11) UsableRulesProof [EQUIVALENT, 0 ms]
10.13/4.18	(12) QDP
10.13/4.18	(13) QReductionProof [EQUIVALENT, 0 ms]
10.13/4.18	(14) QDP
10.13/4.18	(15) QDPSizeChangeProof [EQUIVALENT, 0 ms]
10.13/4.18	(16) YES
10.13/4.18	
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(0)
10.13/4.18	Obligation:
10.13/4.18	mainModule Main 
10.13/4.18	module Main where {
10.13/4.18	import qualified Prelude;
10.13/4.18	data List a = Cons a (List a)  | Nil ;
10.13/4.18	
10.13/4.18	data MyBool = MyTrue  | MyFalse ;
10.13/4.18	
10.13/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.13/4.18	
10.13/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.13/4.18	
10.13/4.18	data Ordering = LT  | EQ  | GT ;
10.13/4.18	
10.13/4.18	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	compareMyInt = primCmpInt;
10.13/4.18	
10.13/4.18	drop :: MyInt  ->  List a  ->  List a;
10.13/4.18	drop n xs = drop3 n xs;
10.13/4.18	drop vv Nil = drop1 vv Nil;
10.13/4.18	drop n (Cons vw xs) = drop0 n (Cons vw xs);
10.13/4.18	
10.13/4.18	drop0 n (Cons vw xs) = drop (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs;
10.13/4.18	
10.13/4.18	drop1 vv Nil = Nil;
10.13/4.18	drop1 vz wu = drop0 vz wu;
10.13/4.18	
10.13/4.18	drop2 n xs MyTrue = xs;
10.13/4.18	drop2 n xs MyFalse = drop1 n xs;
10.13/4.18	
10.13/4.18	drop3 n xs = drop2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
10.13/4.18	drop3 wv ww = drop1 wv ww;
10.13/4.18	
10.13/4.18	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	esEsOrdering LT LT = MyTrue;
10.13/4.18	esEsOrdering LT EQ = MyFalse;
10.13/4.18	esEsOrdering LT GT = MyFalse;
10.13/4.18	esEsOrdering EQ LT = MyFalse;
10.13/4.18	esEsOrdering EQ EQ = MyTrue;
10.13/4.18	esEsOrdering EQ GT = MyFalse;
10.13/4.18	esEsOrdering GT LT = MyFalse;
10.13/4.18	esEsOrdering GT EQ = MyFalse;
10.13/4.18	esEsOrdering GT GT = MyTrue;
10.13/4.18	
10.13/4.18	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	fsEsOrdering x y = not (esEsOrdering x y);
10.13/4.18	
10.13/4.18	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.13/4.18	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.13/4.18	
10.13/4.18	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	msMyInt = primMinusInt;
10.13/4.18	
10.13/4.18	not :: MyBool  ->  MyBool;
10.13/4.18	not MyTrue = MyFalse;
10.13/4.18	not MyFalse = MyTrue;
10.13/4.18	
10.13/4.18	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.13/4.18	
10.13/4.18	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.13/4.18	primCmpNat Main.Zero Main.Zero = EQ;
10.13/4.18	primCmpNat Main.Zero (Main.Succ y) = LT;
10.13/4.18	primCmpNat (Main.Succ x) Main.Zero = GT;
10.13/4.18	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.13/4.18	
10.13/4.18	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.13/4.18	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.13/4.18	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.13/4.18	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.13/4.18	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.13/4.18	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.13/4.18	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.13/4.18	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.13/4.18	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.13/4.18	
10.13/4.18	}
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(1) BR (EQUIVALENT)
10.13/4.18	Replaced joker patterns by fresh variables and removed binding patterns.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(2)
10.13/4.18	Obligation:
10.13/4.18	mainModule Main 
10.13/4.18	module Main where {
10.13/4.18	import qualified Prelude;
10.13/4.18	data List a = Cons a (List a)  | Nil ;
10.13/4.18	
10.13/4.18	data MyBool = MyTrue  | MyFalse ;
10.13/4.18	
10.13/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.13/4.18	
10.13/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.13/4.18	
10.13/4.18	data Ordering = LT  | EQ  | GT ;
10.13/4.18	
10.13/4.18	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	compareMyInt = primCmpInt;
10.13/4.18	
10.13/4.18	drop :: MyInt  ->  List a  ->  List a;
10.13/4.18	drop n xs = drop3 n xs;
10.13/4.18	drop vv Nil = drop1 vv Nil;
10.13/4.18	drop n (Cons vw xs) = drop0 n (Cons vw xs);
10.13/4.18	
10.13/4.18	drop0 n (Cons vw xs) = drop (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs;
10.13/4.18	
10.13/4.18	drop1 vv Nil = Nil;
10.13/4.18	drop1 vz wu = drop0 vz wu;
10.13/4.18	
10.13/4.18	drop2 n xs MyTrue = xs;
10.13/4.18	drop2 n xs MyFalse = drop1 n xs;
10.13/4.18	
10.13/4.18	drop3 n xs = drop2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
10.13/4.18	drop3 wv ww = drop1 wv ww;
10.13/4.18	
10.13/4.18	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	esEsOrdering LT LT = MyTrue;
10.13/4.18	esEsOrdering LT EQ = MyFalse;
10.13/4.18	esEsOrdering LT GT = MyFalse;
10.13/4.18	esEsOrdering EQ LT = MyFalse;
10.13/4.18	esEsOrdering EQ EQ = MyTrue;
10.13/4.18	esEsOrdering EQ GT = MyFalse;
10.13/4.18	esEsOrdering GT LT = MyFalse;
10.13/4.18	esEsOrdering GT EQ = MyFalse;
10.13/4.18	esEsOrdering GT GT = MyTrue;
10.13/4.18	
10.13/4.18	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	fsEsOrdering x y = not (esEsOrdering x y);
10.13/4.18	
10.13/4.18	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.13/4.18	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.13/4.18	
10.13/4.18	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	msMyInt = primMinusInt;
10.13/4.18	
10.13/4.18	not :: MyBool  ->  MyBool;
10.13/4.18	not MyTrue = MyFalse;
10.13/4.18	not MyFalse = MyTrue;
10.13/4.18	
10.13/4.18	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.13/4.18	
10.13/4.18	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.13/4.18	primCmpNat Main.Zero Main.Zero = EQ;
10.13/4.18	primCmpNat Main.Zero (Main.Succ y) = LT;
10.13/4.18	primCmpNat (Main.Succ x) Main.Zero = GT;
10.13/4.18	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.13/4.18	
10.13/4.18	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.13/4.18	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.13/4.18	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.13/4.18	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.13/4.18	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.13/4.18	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.13/4.18	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.13/4.18	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.13/4.18	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.13/4.18	
10.13/4.18	}
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(3) COR (EQUIVALENT)
10.13/4.18	Cond Reductions:
10.13/4.18	The following Function with conditions
10.13/4.18	"undefined |Falseundefined;
10.13/4.18	"
10.13/4.18	is transformed to
10.13/4.18	"undefined  = undefined1;
10.13/4.18	"
10.13/4.18	"undefined0 True = undefined;
10.13/4.18	"
10.13/4.18	"undefined1  = undefined0 False;
10.13/4.18	"
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(4)
10.13/4.18	Obligation:
10.13/4.18	mainModule Main 
10.13/4.18	module Main where {
10.13/4.18	import qualified Prelude;
10.13/4.18	data List a = Cons a (List a)  | Nil ;
10.13/4.18	
10.13/4.18	data MyBool = MyTrue  | MyFalse ;
10.13/4.18	
10.13/4.18	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
10.13/4.18	
10.13/4.18	data Main.Nat = Succ Main.Nat  | Zero ;
10.13/4.18	
10.13/4.18	data Ordering = LT  | EQ  | GT ;
10.13/4.18	
10.13/4.18	compareMyInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	compareMyInt = primCmpInt;
10.13/4.18	
10.13/4.18	drop :: MyInt  ->  List a  ->  List a;
10.13/4.18	drop n xs = drop3 n xs;
10.13/4.18	drop vv Nil = drop1 vv Nil;
10.13/4.18	drop n (Cons vw xs) = drop0 n (Cons vw xs);
10.13/4.18	
10.13/4.18	drop0 n (Cons vw xs) = drop (msMyInt n (Main.Pos (Main.Succ Main.Zero))) xs;
10.13/4.18	
10.13/4.18	drop1 vv Nil = Nil;
10.13/4.18	drop1 vz wu = drop0 vz wu;
10.13/4.18	
10.13/4.18	drop2 n xs MyTrue = xs;
10.13/4.18	drop2 n xs MyFalse = drop1 n xs;
10.13/4.18	
10.13/4.18	drop3 n xs = drop2 n xs (ltEsMyInt n (Main.Pos Main.Zero));
10.13/4.18	drop3 wv ww = drop1 wv ww;
10.13/4.18	
10.13/4.18	esEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	esEsOrdering LT LT = MyTrue;
10.13/4.18	esEsOrdering LT EQ = MyFalse;
10.13/4.18	esEsOrdering LT GT = MyFalse;
10.13/4.18	esEsOrdering EQ LT = MyFalse;
10.13/4.18	esEsOrdering EQ EQ = MyTrue;
10.13/4.18	esEsOrdering EQ GT = MyFalse;
10.13/4.18	esEsOrdering GT LT = MyFalse;
10.13/4.18	esEsOrdering GT EQ = MyFalse;
10.13/4.18	esEsOrdering GT GT = MyTrue;
10.13/4.18	
10.13/4.18	fsEsOrdering :: Ordering  ->  Ordering  ->  MyBool;
10.13/4.18	fsEsOrdering x y = not (esEsOrdering x y);
10.13/4.18	
10.13/4.18	ltEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
10.13/4.18	ltEsMyInt x y = fsEsOrdering (compareMyInt x y) GT;
10.13/4.18	
10.13/4.18	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	msMyInt = primMinusInt;
10.13/4.18	
10.13/4.18	not :: MyBool  ->  MyBool;
10.13/4.18	not MyTrue = MyFalse;
10.13/4.18	not MyFalse = MyTrue;
10.13/4.18	
10.13/4.18	primCmpInt :: MyInt  ->  MyInt  ->  Ordering;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = EQ;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Pos y) = primCmpNat x y;
10.13/4.18	primCmpInt (Main.Pos x) (Main.Neg y) = GT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Pos y) = LT;
10.13/4.18	primCmpInt (Main.Neg x) (Main.Neg y) = primCmpNat y x;
10.13/4.18	
10.13/4.18	primCmpNat :: Main.Nat  ->  Main.Nat  ->  Ordering;
10.13/4.18	primCmpNat Main.Zero Main.Zero = EQ;
10.13/4.18	primCmpNat Main.Zero (Main.Succ y) = LT;
10.13/4.18	primCmpNat (Main.Succ x) Main.Zero = GT;
10.13/4.18	primCmpNat (Main.Succ x) (Main.Succ y) = primCmpNat x y;
10.13/4.18	
10.13/4.18	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
10.13/4.18	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
10.13/4.18	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
10.13/4.18	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
10.13/4.18	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
10.13/4.18	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
10.13/4.18	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
10.13/4.18	
10.13/4.18	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
10.13/4.18	primPlusNat Main.Zero Main.Zero = Main.Zero;
10.13/4.18	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
10.13/4.18	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
10.13/4.18	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
10.13/4.18	
10.13/4.18	}
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(5) Narrow (SOUND)
10.13/4.18	Haskell To QDPs
10.13/4.18	
10.13/4.18	digraph dp_graph {
10.13/4.18	node [outthreshold=100, inthreshold=100];1[label="drop",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
10.13/4.18	3[label="drop wx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
10.13/4.18	4[label="drop wx3 wx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
10.13/4.18	5[label="drop3 wx3 wx4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
10.13/4.18	6[label="drop2 wx3 wx4 (ltEsMyInt wx3 (Pos Zero))",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
10.13/4.18	7[label="drop2 wx3 wx4 (fsEsOrdering (compareMyInt wx3 (Pos Zero)) GT)",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
10.13/4.18	8[label="drop2 wx3 wx4 (not (esEsOrdering (compareMyInt wx3 (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
10.13/4.18	9[label="drop2 wx3 wx4 (not (esEsOrdering (primCmpInt wx3 (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];47[label="wx3/Pos wx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 47[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	47 -> 10[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	48[label="wx3/Neg wx30",fontsize=10,color="white",style="solid",shape="box"];9 -> 48[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	48 -> 11[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	10[label="drop2 (Pos wx30) wx4 (not (esEsOrdering (primCmpInt (Pos wx30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];49[label="wx30/Succ wx300",fontsize=10,color="white",style="solid",shape="box"];10 -> 49[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	49 -> 12[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	50[label="wx30/Zero",fontsize=10,color="white",style="solid",shape="box"];10 -> 50[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	50 -> 13[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	11[label="drop2 (Neg wx30) wx4 (not (esEsOrdering (primCmpInt (Neg wx30) (Pos Zero)) GT))",fontsize=16,color="burlywood",shape="box"];51[label="wx30/Succ wx300",fontsize=10,color="white",style="solid",shape="box"];11 -> 51[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	51 -> 14[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	52[label="wx30/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 52[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	52 -> 15[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	12[label="drop2 (Pos (Succ wx300)) wx4 (not (esEsOrdering (primCmpInt (Pos (Succ wx300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
10.13/4.18	13[label="drop2 (Pos Zero) wx4 (not (esEsOrdering (primCmpInt (Pos Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];13 -> 17[label="",style="solid", color="black", weight=3];
10.13/4.18	14[label="drop2 (Neg (Succ wx300)) wx4 (not (esEsOrdering (primCmpInt (Neg (Succ wx300)) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
10.13/4.18	15[label="drop2 (Neg Zero) wx4 (not (esEsOrdering (primCmpInt (Neg Zero) (Pos Zero)) GT))",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
10.13/4.18	16[label="drop2 (Pos (Succ wx300)) wx4 (not (esEsOrdering (primCmpNat (Succ wx300) Zero) GT))",fontsize=16,color="black",shape="box"];16 -> 20[label="",style="solid", color="black", weight=3];
10.13/4.18	17[label="drop2 (Pos Zero) wx4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];17 -> 21[label="",style="solid", color="black", weight=3];
10.13/4.18	18[label="drop2 (Neg (Succ wx300)) wx4 (not (esEsOrdering LT GT))",fontsize=16,color="black",shape="box"];18 -> 22[label="",style="solid", color="black", weight=3];
10.13/4.18	19[label="drop2 (Neg Zero) wx4 (not (esEsOrdering EQ GT))",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
10.13/4.18	20[label="drop2 (Pos (Succ wx300)) wx4 (not (esEsOrdering GT GT))",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
10.13/4.18	21[label="drop2 (Pos Zero) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
10.13/4.18	22[label="drop2 (Neg (Succ wx300)) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
10.13/4.18	23[label="drop2 (Neg Zero) wx4 (not MyFalse)",fontsize=16,color="black",shape="box"];23 -> 27[label="",style="solid", color="black", weight=3];
10.13/4.18	24[label="drop2 (Pos (Succ wx300)) wx4 (not MyTrue)",fontsize=16,color="black",shape="box"];24 -> 28[label="",style="solid", color="black", weight=3];
10.13/4.18	25[label="drop2 (Pos Zero) wx4 MyTrue",fontsize=16,color="black",shape="box"];25 -> 29[label="",style="solid", color="black", weight=3];
10.13/4.18	26[label="drop2 (Neg (Succ wx300)) wx4 MyTrue",fontsize=16,color="black",shape="box"];26 -> 30[label="",style="solid", color="black", weight=3];
10.13/4.18	27[label="drop2 (Neg Zero) wx4 MyTrue",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
10.13/4.18	28[label="drop2 (Pos (Succ wx300)) wx4 MyFalse",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
10.13/4.18	29[label="wx4",fontsize=16,color="green",shape="box"];30[label="wx4",fontsize=16,color="green",shape="box"];31[label="wx4",fontsize=16,color="green",shape="box"];32[label="drop1 (Pos (Succ wx300)) wx4",fontsize=16,color="burlywood",shape="box"];53[label="wx4/Cons wx40 wx41",fontsize=10,color="white",style="solid",shape="box"];32 -> 53[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	53 -> 33[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	54[label="wx4/Nil",fontsize=10,color="white",style="solid",shape="box"];32 -> 54[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	54 -> 34[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	33[label="drop1 (Pos (Succ wx300)) (Cons wx40 wx41)",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
10.13/4.18	34[label="drop1 (Pos (Succ wx300)) Nil",fontsize=16,color="black",shape="box"];34 -> 36[label="",style="solid", color="black", weight=3];
10.13/4.18	35[label="drop0 (Pos (Succ wx300)) (Cons wx40 wx41)",fontsize=16,color="black",shape="box"];35 -> 37[label="",style="solid", color="black", weight=3];
10.13/4.18	36[label="Nil",fontsize=16,color="green",shape="box"];37 -> 4[label="",style="dashed", color="red", weight=0];
10.13/4.18	37[label="drop (msMyInt (Pos (Succ wx300)) (Pos (Succ Zero))) wx41",fontsize=16,color="magenta"];37 -> 38[label="",style="dashed", color="magenta", weight=3];
10.13/4.18	37 -> 39[label="",style="dashed", color="magenta", weight=3];
10.13/4.18	38[label="wx41",fontsize=16,color="green",shape="box"];39[label="msMyInt (Pos (Succ wx300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];39 -> 40[label="",style="solid", color="black", weight=3];
10.13/4.18	40[label="primMinusInt (Pos (Succ wx300)) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];40 -> 41[label="",style="solid", color="black", weight=3];
10.13/4.18	41[label="primMinusNat (Succ wx300) (Succ Zero)",fontsize=16,color="black",shape="box"];41 -> 42[label="",style="solid", color="black", weight=3];
10.13/4.18	42[label="primMinusNat wx300 Zero",fontsize=16,color="burlywood",shape="box"];55[label="wx300/Succ wx3000",fontsize=10,color="white",style="solid",shape="box"];42 -> 55[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	55 -> 43[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	56[label="wx300/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 56[label="",style="solid", color="burlywood", weight=9];
10.13/4.18	56 -> 44[label="",style="solid", color="burlywood", weight=3];
10.13/4.18	43[label="primMinusNat (Succ wx3000) Zero",fontsize=16,color="black",shape="box"];43 -> 45[label="",style="solid", color="black", weight=3];
10.13/4.18	44[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];44 -> 46[label="",style="solid", color="black", weight=3];
10.13/4.18	45[label="Pos (Succ wx3000)",fontsize=16,color="green",shape="box"];46[label="Pos Zero",fontsize=16,color="green",shape="box"];}
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(6)
10.13/4.18	Obligation:
10.13/4.18	Q DP problem:
10.13/4.18	The TRS P consists of the following rules:
10.13/4.18	
10.13/4.18	   new_drop(Main.Pos(Main.Succ(wx300)), Cons(wx40, wx41), h) -> new_drop(new_primMinusNat(wx300), wx41, h)
10.13/4.18	
10.13/4.18	The TRS R consists of the following rules:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
10.13/4.18	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
10.13/4.18	
10.13/4.18	The set Q consists of the following terms:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Zero)
10.13/4.18	   new_primMinusNat(Main.Succ(x0))
10.13/4.18	
10.13/4.18	We have to consider all minimal (P,Q,R)-chains.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(7) TransformationProof (EQUIVALENT)
10.13/4.18	By narrowing [LPAR04] the rule new_drop(Main.Pos(Main.Succ(wx300)), Cons(wx40, wx41), h) -> new_drop(new_primMinusNat(wx300), wx41, h) at position [0] we obtained the following new rules [LPAR04]:
10.13/4.18	
10.13/4.18	   (new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3),new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3))
10.13/4.18	   (new_drop(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Zero), y2, y3),new_drop(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Zero), y2, y3))
10.13/4.18	
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(8)
10.13/4.18	Obligation:
10.13/4.18	Q DP problem:
10.13/4.18	The TRS P consists of the following rules:
10.13/4.18	
10.13/4.18	   new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3)
10.13/4.18	   new_drop(Main.Pos(Main.Succ(Main.Zero)), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Zero), y2, y3)
10.13/4.18	
10.13/4.18	The TRS R consists of the following rules:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
10.13/4.18	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
10.13/4.18	
10.13/4.18	The set Q consists of the following terms:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Zero)
10.13/4.18	   new_primMinusNat(Main.Succ(x0))
10.13/4.18	
10.13/4.18	We have to consider all minimal (P,Q,R)-chains.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(9) DependencyGraphProof (EQUIVALENT)
10.13/4.18	The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 1 less node.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(10)
10.13/4.18	Obligation:
10.13/4.18	Q DP problem:
10.13/4.18	The TRS P consists of the following rules:
10.13/4.18	
10.13/4.18	   new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3)
10.13/4.18	
10.13/4.18	The TRS R consists of the following rules:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Succ(wx3000)) -> Main.Pos(Main.Succ(wx3000))
10.13/4.18	   new_primMinusNat(Main.Zero) -> Main.Pos(Main.Zero)
10.13/4.18	
10.13/4.18	The set Q consists of the following terms:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Zero)
10.13/4.18	   new_primMinusNat(Main.Succ(x0))
10.13/4.18	
10.13/4.18	We have to consider all minimal (P,Q,R)-chains.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(11) UsableRulesProof (EQUIVALENT)
10.13/4.18	As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(12)
10.13/4.18	Obligation:
10.13/4.18	Q DP problem:
10.13/4.18	The TRS P consists of the following rules:
10.13/4.18	
10.13/4.18	   new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3)
10.13/4.18	
10.13/4.18	R is empty.
10.13/4.18	The set Q consists of the following terms:
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Zero)
10.13/4.18	   new_primMinusNat(Main.Succ(x0))
10.13/4.18	
10.13/4.18	We have to consider all minimal (P,Q,R)-chains.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(13) QReductionProof (EQUIVALENT)
10.13/4.18	We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
10.13/4.18	
10.13/4.18	   new_primMinusNat(Main.Zero)
10.13/4.18	   new_primMinusNat(Main.Succ(x0))
10.13/4.18	
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(14)
10.13/4.18	Obligation:
10.13/4.18	Q DP problem:
10.13/4.18	The TRS P consists of the following rules:
10.13/4.18	
10.13/4.18	   new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3)
10.13/4.18	
10.13/4.18	R is empty.
10.13/4.18	Q is empty.
10.13/4.18	We have to consider all minimal (P,Q,R)-chains.
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(15) QDPSizeChangeProof (EQUIVALENT)
10.13/4.18	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. 
10.13/4.18	
10.13/4.18	From the DPs we obtained the following set of size-change graphs:
10.13/4.18	*new_drop(Main.Pos(Main.Succ(Main.Succ(x0))), Cons(y1, y2), y3) -> new_drop(Main.Pos(Main.Succ(x0)), y2, y3)
10.13/4.18	The graph contains the following edges 2 > 2, 3 >= 3
10.13/4.18	
10.13/4.18	
10.13/4.18	----------------------------------------
10.13/4.18	
10.13/4.18	(16)
10.13/4.18	YES
10.17/4.22	EOF