7.62/3.60	YES
9.05/4.05	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.05/4.05	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.05/4.05	
9.05/4.05	
9.05/4.05	H-Termination with start terms of the given HASKELL could be proven:
9.05/4.05	
9.05/4.05	(0) HASKELL
9.05/4.05	(1) BR [EQUIVALENT, 0 ms]
9.05/4.05	(2) HASKELL
9.05/4.05	(3) COR [EQUIVALENT, 0 ms]
9.05/4.05	(4) HASKELL
9.05/4.05	(5) Narrow [SOUND, 0 ms]
9.05/4.05	(6) QDP
9.05/4.05	(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.05/4.05	(8) YES
9.05/4.05	
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(0)
9.05/4.05	Obligation:
9.05/4.05	mainModule Main 
9.05/4.05	module Main where {
9.05/4.05	import qualified Prelude;
9.05/4.05	data MyBool = MyTrue  | MyFalse ;
9.05/4.05	
9.05/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.05/4.05	
9.05/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.05/4.05	
9.05/4.05	evenMyInt :: MyInt  ->  MyBool;
9.05/4.05	evenMyInt = primEvenInt;
9.05/4.05	
9.05/4.05	primEvenInt :: MyInt  ->  MyBool;
9.05/4.05	primEvenInt (Main.Pos x) = primEvenNat x;
9.05/4.05	primEvenInt (Main.Neg x) = primEvenNat x;
9.05/4.05	
9.05/4.05	primEvenNat :: Main.Nat  ->  MyBool;
9.05/4.05	primEvenNat Main.Zero = MyTrue;
9.05/4.05	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.05/4.05	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.05/4.05	
9.05/4.05	}
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(1) BR (EQUIVALENT)
9.05/4.05	Replaced joker patterns by fresh variables and removed binding patterns.
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(2)
9.05/4.05	Obligation:
9.05/4.05	mainModule Main 
9.05/4.05	module Main where {
9.05/4.05	import qualified Prelude;
9.05/4.05	data MyBool = MyTrue  | MyFalse ;
9.05/4.05	
9.05/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.05/4.05	
9.05/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.05/4.05	
9.05/4.05	evenMyInt :: MyInt  ->  MyBool;
9.05/4.05	evenMyInt = primEvenInt;
9.05/4.05	
9.05/4.05	primEvenInt :: MyInt  ->  MyBool;
9.05/4.05	primEvenInt (Main.Pos x) = primEvenNat x;
9.05/4.05	primEvenInt (Main.Neg x) = primEvenNat x;
9.05/4.05	
9.05/4.05	primEvenNat :: Main.Nat  ->  MyBool;
9.05/4.05	primEvenNat Main.Zero = MyTrue;
9.05/4.05	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.05/4.05	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.05/4.05	
9.05/4.05	}
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(3) COR (EQUIVALENT)
9.05/4.05	Cond Reductions:
9.05/4.05	The following Function with conditions
9.05/4.05	"undefined |Falseundefined;
9.05/4.05	"
9.05/4.05	is transformed to
9.05/4.05	"undefined  = undefined1;
9.05/4.05	"
9.05/4.05	"undefined0 True = undefined;
9.05/4.05	"
9.05/4.05	"undefined1  = undefined0 False;
9.05/4.05	"
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(4)
9.05/4.05	Obligation:
9.05/4.05	mainModule Main 
9.05/4.05	module Main where {
9.05/4.05	import qualified Prelude;
9.05/4.05	data MyBool = MyTrue  | MyFalse ;
9.05/4.05	
9.05/4.05	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.05/4.05	
9.05/4.05	data Main.Nat = Succ Main.Nat  | Zero ;
9.05/4.05	
9.05/4.05	evenMyInt :: MyInt  ->  MyBool;
9.05/4.05	evenMyInt = primEvenInt;
9.05/4.05	
9.05/4.05	primEvenInt :: MyInt  ->  MyBool;
9.05/4.05	primEvenInt (Main.Pos x) = primEvenNat x;
9.05/4.05	primEvenInt (Main.Neg x) = primEvenNat x;
9.05/4.05	
9.05/4.05	primEvenNat :: Main.Nat  ->  MyBool;
9.05/4.05	primEvenNat Main.Zero = MyTrue;
9.05/4.05	primEvenNat (Main.Succ Main.Zero) = MyFalse;
9.05/4.05	primEvenNat (Main.Succ (Main.Succ x)) = primEvenNat x;
9.05/4.05	
9.05/4.05	}
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(5) Narrow (SOUND)
9.05/4.05	Haskell To QDPs
9.05/4.05	
9.05/4.05	digraph dp_graph {
9.05/4.05	node [outthreshold=100, inthreshold=100];1[label="evenMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.05/4.05	3[label="evenMyInt vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
9.05/4.05	4[label="primEvenInt vx3",fontsize=16,color="burlywood",shape="box"];18[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];4 -> 18[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	18 -> 5[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	19[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];4 -> 19[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	19 -> 6[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	5[label="primEvenInt (Pos vx30)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
9.05/4.05	6[label="primEvenInt (Neg vx30)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
9.05/4.05	7[label="primEvenNat vx30",fontsize=16,color="burlywood",shape="triangle"];20[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];7 -> 20[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	20 -> 9[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	21[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];7 -> 21[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	21 -> 10[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	8 -> 7[label="",style="dashed", color="red", weight=0];
9.05/4.05	8[label="primEvenNat vx30",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9.05/4.05	9[label="primEvenNat (Succ vx300)",fontsize=16,color="burlywood",shape="box"];22[label="vx300/Succ vx3000",fontsize=10,color="white",style="solid",shape="box"];9 -> 22[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	22 -> 12[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	23[label="vx300/Zero",fontsize=10,color="white",style="solid",shape="box"];9 -> 23[label="",style="solid", color="burlywood", weight=9];
9.05/4.05	23 -> 13[label="",style="solid", color="burlywood", weight=3];
9.05/4.05	10[label="primEvenNat Zero",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
9.05/4.05	11[label="vx30",fontsize=16,color="green",shape="box"];12[label="primEvenNat (Succ (Succ vx3000))",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
9.05/4.05	13[label="primEvenNat (Succ Zero)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
9.05/4.05	14[label="MyTrue",fontsize=16,color="green",shape="box"];15 -> 7[label="",style="dashed", color="red", weight=0];
9.05/4.05	15[label="primEvenNat vx3000",fontsize=16,color="magenta"];15 -> 17[label="",style="dashed", color="magenta", weight=3];
9.05/4.05	16[label="MyFalse",fontsize=16,color="green",shape="box"];17[label="vx3000",fontsize=16,color="green",shape="box"];}
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(6)
9.05/4.05	Obligation:
9.05/4.05	Q DP problem:
9.05/4.05	The TRS P consists of the following rules:
9.05/4.05	
9.05/4.05	   new_primEvenNat(Main.Succ(Main.Succ(vx3000))) -> new_primEvenNat(vx3000)
9.05/4.05	
9.05/4.05	R is empty.
9.05/4.05	Q is empty.
9.05/4.05	We have to consider all minimal (P,Q,R)-chains.
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(7) QDPSizeChangeProof (EQUIVALENT)
9.05/4.05	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. 
9.05/4.05	
9.05/4.05	From the DPs we obtained the following set of size-change graphs:
9.05/4.05	*new_primEvenNat(Main.Succ(Main.Succ(vx3000))) -> new_primEvenNat(vx3000)
9.05/4.05	The graph contains the following edges 1 > 1
9.05/4.05	
9.05/4.05	
9.05/4.05	----------------------------------------
9.05/4.05	
9.05/4.05	(8)
9.05/4.05	YES
9.30/4.10	EOF