7.46/3.51	YES
9.26/3.99	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.26/3.99	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.26/3.99	
9.26/3.99	
9.26/3.99	H-Termination with start terms of the given HASKELL could be proven:
9.26/3.99	
9.26/3.99	(0) HASKELL
9.26/3.99	(1) BR [EQUIVALENT, 0 ms]
9.26/3.99	(2) HASKELL
9.26/3.99	(3) COR [EQUIVALENT, 0 ms]
9.26/3.99	(4) HASKELL
9.26/3.99	(5) Narrow [SOUND, 0 ms]
9.26/3.99	(6) QDP
9.26/3.99	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.26/3.99	(8) YES
9.26/3.99	
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(0)
9.26/3.99	Obligation:
9.26/3.99	mainModule Main 
9.26/3.99	module Main where {
9.26/3.99	import qualified Prelude;
9.26/3.99	data MyBool = MyTrue  | MyFalse ;
9.26/3.99	
9.26/3.99	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.26/3.99	
9.26/3.99	data Main.Nat = Succ Main.Nat  | Zero ;
9.26/3.99	
9.26/3.99	evenMyInt :: MyInt  ->  MyBool;
9.26/3.99	evenMyInt = primEvenInt;
9.26/3.99	
9.26/3.99	not :: MyBool  ->  MyBool;
9.26/3.99	not MyTrue = MyFalse;
9.26/3.99	not MyFalse = MyTrue;
9.26/3.99	
9.26/3.99	oddMyInt :: MyInt  ->  MyBool;
9.26/3.99	oddMyInt = pt not evenMyInt;
9.26/3.99	
9.26/3.99	primEvenInt :: MyInt  ->  MyBool;
9.26/3.99	primEvenInt (Main.Pos x) = primEvenNat x;
9.26/3.99	primEvenInt (Main.Neg x) = primEvenNat x;
9.26/3.99	
9.26/3.99	primEvenNat :: Main.Nat  ->  MyBool;
9.26/3.99	primEvenNat Main.Zero = MyTrue;
9.26/3.99	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.26/3.99	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.26/3.99	
9.26/3.99	pt :: (c  ->  b)  ->  (a  ->  c)  ->  a  ->  b;
9.26/3.99	pt f g x = f (g x);
9.26/3.99	
9.26/3.99	}
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(1) BR (EQUIVALENT)
9.26/3.99	Replaced joker patterns by fresh variables and removed binding patterns.
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(2)
9.26/3.99	Obligation:
9.26/3.99	mainModule Main 
9.26/3.99	module Main where {
9.26/3.99	import qualified Prelude;
9.26/3.99	data MyBool = MyTrue  | MyFalse ;
9.26/3.99	
9.26/3.99	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.26/3.99	
9.26/3.99	data Main.Nat = Succ Main.Nat  | Zero ;
9.26/3.99	
9.26/3.99	evenMyInt :: MyInt  ->  MyBool;
9.26/3.99	evenMyInt = primEvenInt;
9.26/3.99	
9.26/3.99	not :: MyBool  ->  MyBool;
9.26/3.99	not MyTrue = MyFalse;
9.26/3.99	not MyFalse = MyTrue;
9.26/3.99	
9.26/3.99	oddMyInt :: MyInt  ->  MyBool;
9.26/3.99	oddMyInt = pt not evenMyInt;
9.26/3.99	
9.26/3.99	primEvenInt :: MyInt  ->  MyBool;
9.26/3.99	primEvenInt (Main.Pos x) = primEvenNat x;
9.26/3.99	primEvenInt (Main.Neg x) = primEvenNat x;
9.26/3.99	
9.26/3.99	primEvenNat :: Main.Nat  ->  MyBool;
9.26/3.99	primEvenNat Main.Zero = MyTrue;
9.26/3.99	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.26/3.99	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.26/3.99	
9.26/3.99	pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
9.26/3.99	pt f g x = f (g x);
9.26/3.99	
9.26/3.99	}
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(3) COR (EQUIVALENT)
9.26/3.99	Cond Reductions:
9.26/3.99	The following Function with conditions
9.26/3.99	"undefined |Falseundefined;
9.26/3.99	"
9.26/3.99	is transformed to
9.26/3.99	"undefined  = undefined1;
9.26/3.99	"
9.26/3.99	"undefined0 True = undefined;
9.26/3.99	"
9.26/3.99	"undefined1  = undefined0 False;
9.26/3.99	"
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(4)
9.26/3.99	Obligation:
9.26/3.99	mainModule Main 
9.26/3.99	module Main where {
9.26/3.99	import qualified Prelude;
9.26/3.99	data MyBool = MyTrue  | MyFalse ;
9.26/3.99	
9.26/3.99	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.26/3.99	
9.26/3.99	data Main.Nat = Succ Main.Nat  | Zero ;
9.26/3.99	
9.26/3.99	evenMyInt :: MyInt  ->  MyBool;
9.26/3.99	evenMyInt = primEvenInt;
9.26/3.99	
9.26/3.99	not :: MyBool  ->  MyBool;
9.26/3.99	not MyTrue = MyFalse;
9.26/3.99	not MyFalse = MyTrue;
9.26/3.99	
9.26/3.99	oddMyInt :: MyInt  ->  MyBool;
9.26/3.99	oddMyInt = pt not evenMyInt;
9.26/3.99	
9.26/3.99	primEvenInt :: MyInt  ->  MyBool;
9.26/3.99	primEvenInt (Main.Pos x) = primEvenNat x;
9.26/3.99	primEvenInt (Main.Neg x) = primEvenNat x;
9.26/3.99	
9.26/3.99	primEvenNat :: Main.Nat  ->  MyBool;
9.26/3.99	primEvenNat Main.Zero = MyTrue;
9.26/3.99	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.26/3.99	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.26/3.99	
9.26/3.99	pt :: (b  ->  c)  ->  (a  ->  b)  ->  a  ->  c;
9.26/3.99	pt f g x = f (g x);
9.26/3.99	
9.26/3.99	}
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(5) Narrow (SOUND)
9.26/3.99	Haskell To QDPs
9.26/3.99	
9.26/3.99	digraph dp_graph {
9.26/3.99	node [outthreshold=100, inthreshold=100];1[label="oddMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.26/3.99	3[label="oddMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.26/3.99	4[label="pt not evenMyInt vx3",fontsize=16,color="black",shape="box"];4 -> 5[label="",style="solid", color="black", weight=3];
9.26/3.99	5[label="not (evenMyInt vx3)",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
9.26/3.99	6[label="not (primEvenInt vx3)",fontsize=16,color="burlywood",shape="box"];22[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 22[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	22 -> 7[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	23[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];6 -> 23[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	23 -> 8[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	7[label="not (primEvenInt (Pos vx30))",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
9.26/3.99	8[label="not (primEvenInt (Neg vx30))",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9.26/3.99	9[label="not (primEvenNat vx30)",fontsize=16,color="burlywood",shape="triangle"];24[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	24 -> 11[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	25[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 25[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	25 -> 12[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	10 -> 9[label="",style="dashed", color="red", weight=0];
9.26/3.99	10[label="not (primEvenNat vx30)",fontsize=16,color="magenta"];10 -> 13[label="",style="dashed", color="magenta", weight=3];
9.26/3.99	11[label="not (primEvenNat (Succ vx300))",fontsize=16,color="burlywood",shape="box"];26[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];11 -> 26[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	26 -> 14[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	27[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];11 -> 27[label="",style="solid", color="burlywood", weight=9];
9.26/3.99	27 -> 15[label="",style="solid", color="burlywood", weight=3];
9.26/3.99	12[label="not (primEvenNat Zero)",fontsize=16,color="black",shape="box"];12 -> 16[label="",style="solid", color="black", weight=3];
9.26/3.99	13[label="vx30",fontsize=16,color="green",shape="box"];14[label="not (primEvenNat (Succ (Succ vx3000)))",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
9.26/3.99	15[label="not (primEvenNat (Succ Zero))",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
9.26/3.99	16[label="not MyTrue",fontsize=16,color="black",shape="box"];16 -> 19[label="",style="solid", color="black", weight=3];
9.26/3.99	17 -> 9[label="",style="dashed", color="red", weight=0];
9.26/3.99	17[label="not (primEvenNat vx3000)",fontsize=16,color="magenta"];17 -> 20[label="",style="dashed", color="magenta", weight=3];
9.26/3.99	18[label="not MyFalse",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
9.26/3.99	19[label="MyFalse",fontsize=16,color="green",shape="box"];20[label="vx3000",fontsize=16,color="green",shape="box"];21[label="MyTrue",fontsize=16,color="green",shape="box"];}
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(6)
9.26/3.99	Obligation:
9.26/3.99	Q DP problem:
9.26/3.99	The TRS P consists of the following rules:
9.26/3.99	
9.26/3.99	   new_not(Main.Succ(Main.Succ(vx3000))) -> new_not(vx3000)
9.26/3.99	
9.26/3.99	R is empty.
9.26/3.99	Q is empty.
9.26/3.99	We have to consider all minimal (P,Q,R)-chains.
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(7) QDPSizeChangeProof (EQUIVALENT)
9.26/3.99	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. 
9.26/3.99	
9.26/3.99	From the DPs we obtained the following set of size-change graphs:
9.26/3.99	*new_not(Main.Succ(Main.Succ(vx3000))) -> new_not(vx3000)
9.26/3.99	The graph contains the following edges 1 > 1
9.26/3.99	
9.26/3.99	
9.26/3.99	----------------------------------------
9.26/3.99	
9.26/3.99	(8)
9.26/3.99	YES
9.26/4.02	EOF