7.75/3.58	YES
9.41/4.06	proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
9.41/4.06	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
9.41/4.06	
9.41/4.06	
9.41/4.06	H-Termination with start terms of the given HASKELL could be proven:
9.41/4.06	
9.41/4.06	(0) HASKELL
9.41/4.06	(1) BR [EQUIVALENT, 0 ms]
9.41/4.06	(2) HASKELL
9.41/4.06	(3) COR [EQUIVALENT, 0 ms]
9.41/4.06	(4) HASKELL
9.41/4.06	(5) Narrow [SOUND, 0 ms]
9.41/4.06	(6) AND
9.41/4.06	    (7) QDP
9.41/4.06	        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.41/4.06	        (9) YES
9.41/4.06	    (10) QDP
9.41/4.06	        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
9.41/4.06	        (12) YES
9.41/4.06	
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(0)
9.41/4.06	Obligation:
9.41/4.06	mainModule Main 
9.41/4.06	module Main where {
9.41/4.06	import qualified Prelude;
9.41/4.06	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.41/4.06	
9.41/4.06	data Main.Nat = Succ Main.Nat  | Zero ;
9.41/4.06	
9.41/4.06	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	msMyInt = primMinusInt;
9.41/4.06	
9.41/4.06	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.41/4.06	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.41/4.06	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.41/4.06	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.41/4.06	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.41/4.06	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.41/4.06	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.41/4.06	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.41/4.06	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.41/4.06	
9.41/4.06	}
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(1) BR (EQUIVALENT)
9.41/4.06	Replaced joker patterns by fresh variables and removed binding patterns.
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(2)
9.41/4.06	Obligation:
9.41/4.06	mainModule Main 
9.41/4.06	module Main where {
9.41/4.06	import qualified Prelude;
9.41/4.06	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.41/4.06	
9.41/4.06	data Main.Nat = Succ Main.Nat  | Zero ;
9.41/4.06	
9.41/4.06	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	msMyInt = primMinusInt;
9.41/4.06	
9.41/4.06	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.41/4.06	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.41/4.06	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.41/4.06	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.41/4.06	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.41/4.06	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.41/4.06	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.41/4.06	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.41/4.06	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.41/4.06	
9.41/4.06	}
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(3) COR (EQUIVALENT)
9.41/4.06	Cond Reductions:
9.41/4.06	The following Function with conditions
9.41/4.06	"undefined |Falseundefined;
9.41/4.06	"
9.41/4.06	is transformed to
9.41/4.06	"undefined  = undefined1;
9.41/4.06	"
9.41/4.06	"undefined0 True = undefined;
9.41/4.06	"
9.41/4.06	"undefined1  = undefined0 False;
9.41/4.06	"
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(4)
9.41/4.06	Obligation:
9.41/4.06	mainModule Main 
9.41/4.06	module Main where {
9.41/4.06	import qualified Prelude;
9.41/4.06	data MyInt = Pos Main.Nat  | Neg Main.Nat ;
9.41/4.06	
9.41/4.06	data Main.Nat = Succ Main.Nat  | Zero ;
9.41/4.06	
9.41/4.06	msMyInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	msMyInt = primMinusInt;
9.41/4.06	
9.41/4.06	primMinusInt :: MyInt  ->  MyInt  ->  MyInt;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Neg y) = Main.Pos (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Pos y) = Main.Neg (primPlusNat x y);
9.41/4.06	primMinusInt (Main.Neg x) (Main.Neg y) = primMinusNat y x;
9.41/4.06	primMinusInt (Main.Pos x) (Main.Pos y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
9.41/4.06	primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
9.41/4.06	primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
9.41/4.06	primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
9.41/4.06	primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;
9.41/4.06	
9.41/4.06	primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
9.41/4.06	primPlusNat Main.Zero Main.Zero = Main.Zero;
9.41/4.06	primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
9.41/4.06	primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
9.41/4.06	primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));
9.41/4.06	
9.41/4.06	}
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(5) Narrow (SOUND)
9.41/4.06	Haskell To QDPs
9.41/4.06	
9.41/4.06	digraph dp_graph {
9.41/4.06	node [outthreshold=100, inthreshold=100];1[label="msMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
9.41/4.06	3[label="msMyInt vx3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
9.41/4.06	4[label="msMyInt vx3 vx4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
9.41/4.06	5[label="primMinusInt vx3 vx4",fontsize=16,color="burlywood",shape="box"];47[label="vx3/Pos vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 47[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	47 -> 6[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	48[label="vx3/Neg vx30",fontsize=10,color="white",style="solid",shape="box"];5 -> 48[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	48 -> 7[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	6[label="primMinusInt (Pos vx30) vx4",fontsize=16,color="burlywood",shape="box"];49[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];6 -> 49[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	49 -> 8[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	50[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];6 -> 50[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	50 -> 9[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	7[label="primMinusInt (Neg vx30) vx4",fontsize=16,color="burlywood",shape="box"];51[label="vx4/Pos vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 51[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	51 -> 10[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	52[label="vx4/Neg vx40",fontsize=10,color="white",style="solid",shape="box"];7 -> 52[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	52 -> 11[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	8[label="primMinusInt (Pos vx30) (Pos vx40)",fontsize=16,color="black",shape="box"];8 -> 12[label="",style="solid", color="black", weight=3];
9.41/4.06	9[label="primMinusInt (Pos vx30) (Neg vx40)",fontsize=16,color="black",shape="box"];9 -> 13[label="",style="solid", color="black", weight=3];
9.41/4.06	10[label="primMinusInt (Neg vx30) (Pos vx40)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
9.41/4.06	11[label="primMinusInt (Neg vx30) (Neg vx40)",fontsize=16,color="black",shape="box"];11 -> 15[label="",style="solid", color="black", weight=3];
9.41/4.06	12[label="primMinusNat vx30 vx40",fontsize=16,color="burlywood",shape="triangle"];53[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];12 -> 53[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	53 -> 16[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	54[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];12 -> 54[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	54 -> 17[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	13[label="Pos (primPlusNat vx30 vx40)",fontsize=16,color="green",shape="box"];13 -> 18[label="",style="dashed", color="green", weight=3];
9.41/4.06	14[label="Neg (primPlusNat vx30 vx40)",fontsize=16,color="green",shape="box"];14 -> 19[label="",style="dashed", color="green", weight=3];
9.41/4.06	15 -> 12[label="",style="dashed", color="red", weight=0];
9.41/4.06	15[label="primMinusNat vx40 vx30",fontsize=16,color="magenta"];15 -> 20[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	15 -> 21[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	16[label="primMinusNat (Succ vx300) vx40",fontsize=16,color="burlywood",shape="box"];55[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];16 -> 55[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	55 -> 22[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	56[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];16 -> 56[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	56 -> 23[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	17[label="primMinusNat Zero vx40",fontsize=16,color="burlywood",shape="box"];57[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];17 -> 57[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	57 -> 24[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	58[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 58[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	58 -> 25[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	18[label="primPlusNat vx30 vx40",fontsize=16,color="burlywood",shape="triangle"];59[label="vx30/Succ vx300",fontsize=10,color="white",style="solid",shape="box"];18 -> 59[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	59 -> 26[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	60[label="vx30/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 60[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	60 -> 27[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	19 -> 18[label="",style="dashed", color="red", weight=0];
9.41/4.06	19[label="primPlusNat vx30 vx40",fontsize=16,color="magenta"];19 -> 28[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	19 -> 29[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	20[label="vx30",fontsize=16,color="green",shape="box"];21[label="vx40",fontsize=16,color="green",shape="box"];22[label="primMinusNat (Succ vx300) (Succ vx400)",fontsize=16,color="black",shape="box"];22 -> 30[label="",style="solid", color="black", weight=3];
9.41/4.06	23[label="primMinusNat (Succ vx300) Zero",fontsize=16,color="black",shape="box"];23 -> 31[label="",style="solid", color="black", weight=3];
9.41/4.06	24[label="primMinusNat Zero (Succ vx400)",fontsize=16,color="black",shape="box"];24 -> 32[label="",style="solid", color="black", weight=3];
9.41/4.06	25[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];25 -> 33[label="",style="solid", color="black", weight=3];
9.41/4.06	26[label="primPlusNat (Succ vx300) vx40",fontsize=16,color="burlywood",shape="box"];61[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];26 -> 61[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	61 -> 34[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	62[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 62[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	62 -> 35[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	27[label="primPlusNat Zero vx40",fontsize=16,color="burlywood",shape="box"];63[label="vx40/Succ vx400",fontsize=10,color="white",style="solid",shape="box"];27 -> 63[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	63 -> 36[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	64[label="vx40/Zero",fontsize=10,color="white",style="solid",shape="box"];27 -> 64[label="",style="solid", color="burlywood", weight=9];
9.41/4.06	64 -> 37[label="",style="solid", color="burlywood", weight=3];
9.41/4.06	28[label="vx40",fontsize=16,color="green",shape="box"];29[label="vx30",fontsize=16,color="green",shape="box"];30 -> 12[label="",style="dashed", color="red", weight=0];
9.41/4.06	30[label="primMinusNat vx300 vx400",fontsize=16,color="magenta"];30 -> 38[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	30 -> 39[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	31[label="Pos (Succ vx300)",fontsize=16,color="green",shape="box"];32[label="Neg (Succ vx400)",fontsize=16,color="green",shape="box"];33[label="Pos Zero",fontsize=16,color="green",shape="box"];34[label="primPlusNat (Succ vx300) (Succ vx400)",fontsize=16,color="black",shape="box"];34 -> 40[label="",style="solid", color="black", weight=3];
9.41/4.06	35[label="primPlusNat (Succ vx300) Zero",fontsize=16,color="black",shape="box"];35 -> 41[label="",style="solid", color="black", weight=3];
9.41/4.06	36[label="primPlusNat Zero (Succ vx400)",fontsize=16,color="black",shape="box"];36 -> 42[label="",style="solid", color="black", weight=3];
9.41/4.06	37[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];37 -> 43[label="",style="solid", color="black", weight=3];
9.41/4.06	38[label="vx400",fontsize=16,color="green",shape="box"];39[label="vx300",fontsize=16,color="green",shape="box"];40[label="Succ (Succ (primPlusNat vx300 vx400))",fontsize=16,color="green",shape="box"];40 -> 44[label="",style="dashed", color="green", weight=3];
9.41/4.06	41[label="Succ vx300",fontsize=16,color="green",shape="box"];42[label="Succ vx400",fontsize=16,color="green",shape="box"];43[label="Zero",fontsize=16,color="green",shape="box"];44 -> 18[label="",style="dashed", color="red", weight=0];
9.41/4.06	44[label="primPlusNat vx300 vx400",fontsize=16,color="magenta"];44 -> 45[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	44 -> 46[label="",style="dashed", color="magenta", weight=3];
9.41/4.06	45[label="vx400",fontsize=16,color="green",shape="box"];46[label="vx300",fontsize=16,color="green",shape="box"];}
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(6)
9.41/4.06	Complex Obligation (AND)
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(7)
9.41/4.06	Obligation:
9.41/4.06	Q DP problem:
9.41/4.06	The TRS P consists of the following rules:
9.41/4.06	
9.41/4.06	   new_primPlusNat(Main.Succ(vx300), Main.Succ(vx400)) -> new_primPlusNat(vx300, vx400)
9.41/4.06	
9.41/4.06	R is empty.
9.41/4.06	Q is empty.
9.41/4.06	We have to consider all minimal (P,Q,R)-chains.
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(8) QDPSizeChangeProof (EQUIVALENT)
9.41/4.06	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. 
9.41/4.06	
9.41/4.06	From the DPs we obtained the following set of size-change graphs:
9.41/4.06	*new_primPlusNat(Main.Succ(vx300), Main.Succ(vx400)) -> new_primPlusNat(vx300, vx400)
9.41/4.06	The graph contains the following edges 1 > 1, 2 > 2
9.41/4.06	
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(9)
9.41/4.06	YES
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(10)
9.41/4.06	Obligation:
9.41/4.06	Q DP problem:
9.41/4.06	The TRS P consists of the following rules:
9.41/4.06	
9.41/4.06	   new_primMinusNat(Main.Succ(vx300), Main.Succ(vx400)) -> new_primMinusNat(vx300, vx400)
9.41/4.06	
9.41/4.06	R is empty.
9.41/4.06	Q is empty.
9.41/4.06	We have to consider all minimal (P,Q,R)-chains.
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(11) QDPSizeChangeProof (EQUIVALENT)
9.41/4.06	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. 
9.41/4.06	
9.41/4.06	From the DPs we obtained the following set of size-change graphs:
9.41/4.06	*new_primMinusNat(Main.Succ(vx300), Main.Succ(vx400)) -> new_primMinusNat(vx300, vx400)
9.41/4.06	The graph contains the following edges 1 > 1, 2 > 2
9.41/4.06	
9.41/4.06	
9.41/4.06	----------------------------------------
9.41/4.06	
9.41/4.06	(12)
9.41/4.06	YES
9.61/4.10	EOF