/export/starexec/sandbox/solver/bin/starexec_run_standard /export/starexec/sandbox/benchmark/theBenchmark.hs /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox/benchmark/theBenchmark.hs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


H-Termination with start terms of the given HASKELL could be proven:

(0) HASKELL
(1) BR [EQUIVALENT, 0 ms]
(2) HASKELL
(3) COR [EQUIVALENT, 0 ms]
(4) HASKELL
(5) Narrow [SOUND, 0 ms]
(6) QDP
(7) QDPSizeChangeProof [EQUIVALENT, 0 ms]
(8) YES


----------------------------------------

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (a  ->  b  ->  a)  ->  a  ->  List b  ->  a;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

length :: List a  ->  MyInt;
length = foldl' length0 (Main.Pos Main.Zero);

length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

}

----------------------------------------

(1) BR (EQUIVALENT)
Replaced joker patterns by fresh variables and removed binding patterns.
----------------------------------------

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (b  ->  a)  ->  b  ->  a;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF a  ->  b  ->  b;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

length :: List a  ->  MyInt;
length = foldl' length0 (Main.Pos Main.Zero);

length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: a  ->  b  ->  b;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

}

----------------------------------------

(3) COR (EQUIVALENT)
Cond Reductions:
The following Function with conditions
"undefined |Falseundefined;
"
is transformed to
"undefined  = undefined1;
"
"undefined0 True = undefined;
"
"undefined1  = undefined0 False;
"

----------------------------------------

(4)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data List a = Cons a (List a)  | Nil ;

data MyInt = Pos Main.Nat  | Neg Main.Nat ;

data Main.Nat = Succ Main.Nat  | Zero ;

data Main.WHNF a = WHNF a ;

dsEm :: (a  ->  b)  ->  a  ->  b;
dsEm f x = Main.seq x (f x);

enforceWHNF :: Main.WHNF b  ->  a  ->  a;
enforceWHNF (Main.WHNF x) y = y;

foldl' :: (b  ->  a  ->  b)  ->  b  ->  List a  ->  b;
foldl' f a Nil = a;
foldl' f a (Cons x xs) = dsEm (foldl' f) (f a x) xs;

length :: List a  ->  MyInt;
length = foldl' length0 (Main.Pos Main.Zero);

length0 n vv = psMyInt n (Main.Pos (Main.Succ Main.Zero));

primMinusNat :: Main.Nat  ->  Main.Nat  ->  MyInt;
primMinusNat Main.Zero Main.Zero = Main.Pos Main.Zero;
primMinusNat Main.Zero (Main.Succ y) = Main.Neg (Main.Succ y);
primMinusNat (Main.Succ x) Main.Zero = Main.Pos (Main.Succ x);
primMinusNat (Main.Succ x) (Main.Succ y) = primMinusNat x y;

primPlusInt :: MyInt  ->  MyInt  ->  MyInt;
primPlusInt (Main.Pos x) (Main.Neg y) = primMinusNat x y;
primPlusInt (Main.Neg x) (Main.Pos y) = primMinusNat y x;
primPlusInt (Main.Neg x) (Main.Neg y) = Main.Neg (primPlusNat x y);
primPlusInt (Main.Pos x) (Main.Pos y) = Main.Pos (primPlusNat x y);

primPlusNat :: Main.Nat  ->  Main.Nat  ->  Main.Nat;
primPlusNat Main.Zero Main.Zero = Main.Zero;
primPlusNat Main.Zero (Main.Succ y) = Main.Succ y;
primPlusNat (Main.Succ x) Main.Zero = Main.Succ x;
primPlusNat (Main.Succ x) (Main.Succ y) = Main.Succ (Main.Succ (primPlusNat x y));

psMyInt :: MyInt  ->  MyInt  ->  MyInt;
psMyInt = primPlusInt;

seq :: b  ->  a  ->  a;
seq x y = Main.enforceWHNF (Main.WHNF x) y;

}

----------------------------------------

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="length",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="length vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' length0 (Pos Zero) vy3",fontsize=16,color="burlywood",shape="box"];60[label="vy3/Cons vy30 vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 60[label="",style="solid", color="burlywood", weight=9];
60 -> 5[label="",style="solid", color="burlywood", weight=3];
61[label="vy3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 61[label="",style="solid", color="burlywood", weight=9];
61 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl' length0 (Pos Zero) (Cons vy30 vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl' length0 (Pos Zero) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="dsEm (foldl' length0) (length0 (Pos Zero) vy30) vy31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="Pos Zero",fontsize=16,color="green",shape="box"];9 -> 18[label="",style="dashed", color="red", weight=0];
9[label="seq (length0 (Pos Zero) vy30) (foldl' length0 (length0 (Pos Zero) vy30)) vy31",fontsize=16,color="magenta"];9 -> 19[label="",style="dashed", color="magenta", weight=3];
9 -> 20[label="",style="dashed", color="magenta", weight=3];
9 -> 21[label="",style="dashed", color="magenta", weight=3];
9 -> 22[label="",style="dashed", color="magenta", weight=3];
19[label="Pos Zero",fontsize=16,color="green",shape="box"];20[label="vy31",fontsize=16,color="green",shape="box"];21[label="vy30",fontsize=16,color="green",shape="box"];22[label="Pos Zero",fontsize=16,color="green",shape="box"];18[label="seq (length0 vy4 vy310) (foldl' length0 (length0 vy5 vy310)) vy311",fontsize=16,color="black",shape="triangle"];18 -> 25[label="",style="solid", color="black", weight=3];
25[label="enforceWHNF (WHNF (length0 vy4 vy310)) (foldl' length0 (length0 vy5 vy310)) vy311",fontsize=16,color="black",shape="box"];25 -> 26[label="",style="solid", color="black", weight=3];
26[label="foldl' length0 (length0 vy5 vy310) vy311",fontsize=16,color="burlywood",shape="box"];62[label="vy311/Cons vy3110 vy3111",fontsize=10,color="white",style="solid",shape="box"];26 -> 62[label="",style="solid", color="burlywood", weight=9];
62 -> 27[label="",style="solid", color="burlywood", weight=3];
63[label="vy311/Nil",fontsize=10,color="white",style="solid",shape="box"];26 -> 63[label="",style="solid", color="burlywood", weight=9];
63 -> 28[label="",style="solid", color="burlywood", weight=3];
27[label="foldl' length0 (length0 vy5 vy310) (Cons vy3110 vy3111)",fontsize=16,color="black",shape="box"];27 -> 29[label="",style="solid", color="black", weight=3];
28[label="foldl' length0 (length0 vy5 vy310) Nil",fontsize=16,color="black",shape="box"];28 -> 30[label="",style="solid", color="black", weight=3];
29[label="dsEm (foldl' length0) (length0 (length0 vy5 vy310) vy3110) vy3111",fontsize=16,color="black",shape="box"];29 -> 31[label="",style="solid", color="black", weight=3];
30[label="length0 vy5 vy310",fontsize=16,color="black",shape="triangle"];30 -> 32[label="",style="solid", color="black", weight=3];
31 -> 18[label="",style="dashed", color="red", weight=0];
31[label="seq (length0 (length0 vy5 vy310) vy3110) (foldl' length0 (length0 (length0 vy5 vy310) vy3110)) vy3111",fontsize=16,color="magenta"];31 -> 33[label="",style="dashed", color="magenta", weight=3];
31 -> 34[label="",style="dashed", color="magenta", weight=3];
31 -> 35[label="",style="dashed", color="magenta", weight=3];
31 -> 36[label="",style="dashed", color="magenta", weight=3];
32[label="psMyInt vy5 (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];32 -> 37[label="",style="solid", color="black", weight=3];
33 -> 30[label="",style="dashed", color="red", weight=0];
33[label="length0 vy5 vy310",fontsize=16,color="magenta"];34[label="vy3111",fontsize=16,color="green",shape="box"];35[label="vy3110",fontsize=16,color="green",shape="box"];36 -> 30[label="",style="dashed", color="red", weight=0];
36[label="length0 vy5 vy310",fontsize=16,color="magenta"];37[label="primPlusInt vy5 (Pos (Succ Zero))",fontsize=16,color="burlywood",shape="box"];64[label="vy5/Pos vy50",fontsize=10,color="white",style="solid",shape="box"];37 -> 64[label="",style="solid", color="burlywood", weight=9];
64 -> 38[label="",style="solid", color="burlywood", weight=3];
65[label="vy5/Neg vy50",fontsize=10,color="white",style="solid",shape="box"];37 -> 65[label="",style="solid", color="burlywood", weight=9];
65 -> 39[label="",style="solid", color="burlywood", weight=3];
38[label="primPlusInt (Pos vy50) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];38 -> 40[label="",style="solid", color="black", weight=3];
39[label="primPlusInt (Neg vy50) (Pos (Succ Zero))",fontsize=16,color="black",shape="box"];39 -> 41[label="",style="solid", color="black", weight=3];
40[label="Pos (primPlusNat vy50 (Succ Zero))",fontsize=16,color="green",shape="box"];40 -> 42[label="",style="dashed", color="green", weight=3];
41[label="primMinusNat (Succ Zero) vy50",fontsize=16,color="burlywood",shape="box"];66[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];41 -> 66[label="",style="solid", color="burlywood", weight=9];
66 -> 43[label="",style="solid", color="burlywood", weight=3];
67[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];41 -> 67[label="",style="solid", color="burlywood", weight=9];
67 -> 44[label="",style="solid", color="burlywood", weight=3];
42[label="primPlusNat vy50 (Succ Zero)",fontsize=16,color="burlywood",shape="box"];68[label="vy50/Succ vy500",fontsize=10,color="white",style="solid",shape="box"];42 -> 68[label="",style="solid", color="burlywood", weight=9];
68 -> 45[label="",style="solid", color="burlywood", weight=3];
69[label="vy50/Zero",fontsize=10,color="white",style="solid",shape="box"];42 -> 69[label="",style="solid", color="burlywood", weight=9];
69 -> 46[label="",style="solid", color="burlywood", weight=3];
43[label="primMinusNat (Succ Zero) (Succ vy500)",fontsize=16,color="black",shape="box"];43 -> 47[label="",style="solid", color="black", weight=3];
44[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];44 -> 48[label="",style="solid", color="black", weight=3];
45[label="primPlusNat (Succ vy500) (Succ Zero)",fontsize=16,color="black",shape="box"];45 -> 49[label="",style="solid", color="black", weight=3];
46[label="primPlusNat Zero (Succ Zero)",fontsize=16,color="black",shape="box"];46 -> 50[label="",style="solid", color="black", weight=3];
47[label="primMinusNat Zero vy500",fontsize=16,color="burlywood",shape="box"];70[label="vy500/Succ vy5000",fontsize=10,color="white",style="solid",shape="box"];47 -> 70[label="",style="solid", color="burlywood", weight=9];
70 -> 51[label="",style="solid", color="burlywood", weight=3];
71[label="vy500/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 71[label="",style="solid", color="burlywood", weight=9];
71 -> 52[label="",style="solid", color="burlywood", weight=3];
48[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];49[label="Succ (Succ (primPlusNat vy500 Zero))",fontsize=16,color="green",shape="box"];49 -> 53[label="",style="dashed", color="green", weight=3];
50[label="Succ Zero",fontsize=16,color="green",shape="box"];51[label="primMinusNat Zero (Succ vy5000)",fontsize=16,color="black",shape="box"];51 -> 54[label="",style="solid", color="black", weight=3];
52[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];52 -> 55[label="",style="solid", color="black", weight=3];
53[label="primPlusNat vy500 Zero",fontsize=16,color="burlywood",shape="box"];72[label="vy500/Succ vy5000",fontsize=10,color="white",style="solid",shape="box"];53 -> 72[label="",style="solid", color="burlywood", weight=9];
72 -> 56[label="",style="solid", color="burlywood", weight=3];
73[label="vy500/Zero",fontsize=10,color="white",style="solid",shape="box"];53 -> 73[label="",style="solid", color="burlywood", weight=9];
73 -> 57[label="",style="solid", color="burlywood", weight=3];
54[label="Neg (Succ vy5000)",fontsize=16,color="green",shape="box"];55[label="Pos Zero",fontsize=16,color="green",shape="box"];56[label="primPlusNat (Succ vy5000) Zero",fontsize=16,color="black",shape="box"];56 -> 58[label="",style="solid", color="black", weight=3];
57[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];57 -> 59[label="",style="solid", color="black", weight=3];
58[label="Succ vy5000",fontsize=16,color="green",shape="box"];59[label="Zero",fontsize=16,color="green",shape="box"];}

----------------------------------------

(6)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_seq(vy4, vy310, vy5, Cons(vy3110, vy3111), h) -> new_seq(new_length0(vy5, vy310, h), vy3110, new_length0(vy5, vy310, h), vy3111, h)

The TRS R consists of the following rules:

   new_primPlusNat(Main.Zero) -> Main.Succ(Main.Zero)
   new_length0(Main.Neg(Main.Zero), vy310, h) -> Main.Pos(Main.Succ(Main.Zero))
   new_primPlusNat(Main.Succ(vy500)) -> Main.Succ(Main.Succ(new_primPlusNat0(vy500)))
   new_length0(Main.Neg(Main.Succ(Main.Zero)), vy310, h) -> Main.Pos(Main.Zero)
   new_length0(Main.Neg(Main.Succ(Main.Succ(vy5000))), vy310, h) -> Main.Neg(Main.Succ(vy5000))
   new_primPlusNat0(Main.Succ(vy5000)) -> Main.Succ(vy5000)
   new_length0(Main.Pos(vy50), vy310, h) -> Main.Pos(new_primPlusNat(vy50))
   new_primPlusNat0(Main.Zero) -> Main.Zero

The set Q consists of the following terms:

   new_primPlusNat0(Main.Succ(x0))
   new_primPlusNat0(Main.Zero)
   new_length0(Main.Neg(Main.Zero), x0, x1)
   new_length0(Main.Neg(Main.Succ(Main.Succ(x0))), x1, x2)
   new_length0(Main.Pos(x0), x1, x2)
   new_primPlusNat(Main.Succ(x0))
   new_primPlusNat(Main.Zero)
   new_length0(Main.Neg(Main.Succ(Main.Zero)), x0, x1)

We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(7) QDPSizeChangeProof (EQUIVALENT)
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. 

From the DPs we obtained the following set of size-change graphs:
*new_seq(vy4, vy310, vy5, Cons(vy3110, vy3111), h) -> new_seq(new_length0(vy5, vy310, h), vy3110, new_length0(vy5, vy310, h), vy3111, h)
The graph contains the following edges 4 > 2, 4 > 4, 5 >= 5


----------------------------------------

(8)
YES