/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.hs /export/starexec/sandbox2/output/output_files


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


YES
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) AND
    (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (12) YES


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

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

data MyBool = MyTrue  | MyFalse ;

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

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

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
elemMyInt = pt any esEsMyInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

map :: (a  ->  b)  ->  List a  ->  List b;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

pt :: (b  ->  a)  ->  (c  ->  b)  ->  c  ->  a;
pt f g x = f (g x);

}

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

(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 MyBool = MyTrue  | MyFalse ;

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

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

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
elemMyInt = pt any esEsMyInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

foldr :: (a  ->  b  ->  b)  ->  b  ->  List a  ->  b;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

map :: (b  ->  a)  ->  List b  ->  List a;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

pt :: (a  ->  c)  ->  (b  ->  a)  ->  b  ->  c;
pt f g x = f (g x);

}

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

(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 MyBool = MyTrue  | MyFalse ;

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

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

any :: (a  ->  MyBool)  ->  List a  ->  MyBool;
any p = pt or (map p);

elemMyInt :: MyInt  ->  List MyInt  ->  MyBool;
elemMyInt = pt any esEsMyInt;

esEsMyInt :: MyInt  ->  MyInt  ->  MyBool;
esEsMyInt = primEqInt;

foldr :: (b  ->  a  ->  a)  ->  a  ->  List b  ->  a;
foldr f z Nil = z;
foldr f z (Cons x xs) = f x (foldr f z xs);

map :: (b  ->  a)  ->  List b  ->  List a;
map f Nil = Nil;
map f (Cons x xs) = Cons (f x) (map f xs);

or :: List MyBool  ->  MyBool;
or = foldr pePe MyFalse;

pePe :: MyBool  ->  MyBool  ->  MyBool;
pePe MyFalse x = x;
pePe MyTrue x = MyTrue;

primEqInt :: MyInt  ->  MyInt  ->  MyBool;
primEqInt (Main.Pos (Main.Succ x)) (Main.Pos (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Neg (Main.Succ x)) (Main.Neg (Main.Succ y)) = primEqNat x y;
primEqInt (Main.Pos Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt (Main.Neg Main.Zero) (Main.Neg Main.Zero) = MyTrue;
primEqInt (Main.Pos Main.Zero) (Main.Pos Main.Zero) = MyTrue;
primEqInt vv vw = MyFalse;

primEqNat :: Main.Nat  ->  Main.Nat  ->  MyBool;
primEqNat Main.Zero Main.Zero = MyTrue;
primEqNat Main.Zero (Main.Succ y) = MyFalse;
primEqNat (Main.Succ x) Main.Zero = MyFalse;
primEqNat (Main.Succ x) (Main.Succ y) = primEqNat x y;

pt :: (c  ->  a)  ->  (b  ->  c)  ->  b  ->  a;
pt f g x = f (g x);

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="elemMyInt",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="elemMyInt vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="elemMyInt vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="pt any esEsMyInt vz3 vz4",fontsize=16,color="black",shape="box"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="any (esEsMyInt vz3) vz4",fontsize=16,color="black",shape="box"];6 -> 7[label="",style="solid", color="black", weight=3];
7[label="pt or (map (esEsMyInt vz3)) vz4",fontsize=16,color="black",shape="box"];7 -> 8[label="",style="solid", color="black", weight=3];
8[label="or (map (esEsMyInt vz3) vz4)",fontsize=16,color="black",shape="box"];8 -> 9[label="",style="solid", color="black", weight=3];
9[label="foldr pePe MyFalse (map (esEsMyInt vz3) vz4)",fontsize=16,color="burlywood",shape="triangle"];76[label="vz4/Cons vz40 vz41",fontsize=10,color="white",style="solid",shape="box"];9 -> 76[label="",style="solid", color="burlywood", weight=9];
76 -> 10[label="",style="solid", color="burlywood", weight=3];
77[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];9 -> 77[label="",style="solid", color="burlywood", weight=9];
77 -> 11[label="",style="solid", color="burlywood", weight=3];
10[label="foldr pePe MyFalse (map (esEsMyInt vz3) (Cons vz40 vz41))",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="foldr pePe MyFalse (map (esEsMyInt vz3) Nil)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldr pePe MyFalse (Cons (esEsMyInt vz3 vz40) (map (esEsMyInt vz3) vz41))",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13[label="foldr pePe MyFalse Nil",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 16[label="",style="dashed", color="red", weight=0];
14[label="pePe (esEsMyInt vz3 vz40) (foldr pePe MyFalse (map (esEsMyInt vz3) vz41))",fontsize=16,color="magenta"];14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="MyFalse",fontsize=16,color="green",shape="box"];17 -> 9[label="",style="dashed", color="red", weight=0];
17[label="foldr pePe MyFalse (map (esEsMyInt vz3) vz41)",fontsize=16,color="magenta"];17 -> 18[label="",style="dashed", color="magenta", weight=3];
16[label="pePe (esEsMyInt vz3 vz40) vz5",fontsize=16,color="black",shape="triangle"];16 -> 19[label="",style="solid", color="black", weight=3];
18[label="vz41",fontsize=16,color="green",shape="box"];19[label="pePe (primEqInt vz3 vz40) vz5",fontsize=16,color="burlywood",shape="box"];78[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 78[label="",style="solid", color="burlywood", weight=9];
78 -> 20[label="",style="solid", color="burlywood", weight=3];
79[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];19 -> 79[label="",style="solid", color="burlywood", weight=9];
79 -> 21[label="",style="solid", color="burlywood", weight=3];
20[label="pePe (primEqInt (Pos vz30) vz40) vz5",fontsize=16,color="burlywood",shape="box"];80[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];20 -> 80[label="",style="solid", color="burlywood", weight=9];
80 -> 22[label="",style="solid", color="burlywood", weight=3];
81[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];20 -> 81[label="",style="solid", color="burlywood", weight=9];
81 -> 23[label="",style="solid", color="burlywood", weight=3];
21[label="pePe (primEqInt (Neg vz30) vz40) vz5",fontsize=16,color="burlywood",shape="box"];82[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];21 -> 82[label="",style="solid", color="burlywood", weight=9];
82 -> 24[label="",style="solid", color="burlywood", weight=3];
83[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 83[label="",style="solid", color="burlywood", weight=9];
83 -> 25[label="",style="solid", color="burlywood", weight=3];
22[label="pePe (primEqInt (Pos (Succ vz300)) vz40) vz5",fontsize=16,color="burlywood",shape="box"];84[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 84[label="",style="solid", color="burlywood", weight=9];
84 -> 26[label="",style="solid", color="burlywood", weight=3];
85[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];22 -> 85[label="",style="solid", color="burlywood", weight=9];
85 -> 27[label="",style="solid", color="burlywood", weight=3];
23[label="pePe (primEqInt (Pos Zero) vz40) vz5",fontsize=16,color="burlywood",shape="box"];86[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 86[label="",style="solid", color="burlywood", weight=9];
86 -> 28[label="",style="solid", color="burlywood", weight=3];
87[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];23 -> 87[label="",style="solid", color="burlywood", weight=9];
87 -> 29[label="",style="solid", color="burlywood", weight=3];
24[label="pePe (primEqInt (Neg (Succ vz300)) vz40) vz5",fontsize=16,color="burlywood",shape="box"];88[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 88[label="",style="solid", color="burlywood", weight=9];
88 -> 30[label="",style="solid", color="burlywood", weight=3];
89[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];24 -> 89[label="",style="solid", color="burlywood", weight=9];
89 -> 31[label="",style="solid", color="burlywood", weight=3];
25[label="pePe (primEqInt (Neg Zero) vz40) vz5",fontsize=16,color="burlywood",shape="box"];90[label="vz40/Pos vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 90[label="",style="solid", color="burlywood", weight=9];
90 -> 32[label="",style="solid", color="burlywood", weight=3];
91[label="vz40/Neg vz400",fontsize=10,color="white",style="solid",shape="box"];25 -> 91[label="",style="solid", color="burlywood", weight=9];
91 -> 33[label="",style="solid", color="burlywood", weight=3];
26[label="pePe (primEqInt (Pos (Succ vz300)) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];92[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];26 -> 92[label="",style="solid", color="burlywood", weight=9];
92 -> 34[label="",style="solid", color="burlywood", weight=3];
93[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 93[label="",style="solid", color="burlywood", weight=9];
93 -> 35[label="",style="solid", color="burlywood", weight=3];
27[label="pePe (primEqInt (Pos (Succ vz300)) (Neg vz400)) vz5",fontsize=16,color="black",shape="box"];27 -> 36[label="",style="solid", color="black", weight=3];
28[label="pePe (primEqInt (Pos Zero) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];94[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];28 -> 94[label="",style="solid", color="burlywood", weight=9];
94 -> 37[label="",style="solid", color="burlywood", weight=3];
95[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];28 -> 95[label="",style="solid", color="burlywood", weight=9];
95 -> 38[label="",style="solid", color="burlywood", weight=3];
29[label="pePe (primEqInt (Pos Zero) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];96[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 96[label="",style="solid", color="burlywood", weight=9];
96 -> 39[label="",style="solid", color="burlywood", weight=3];
97[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 97[label="",style="solid", color="burlywood", weight=9];
97 -> 40[label="",style="solid", color="burlywood", weight=3];
30[label="pePe (primEqInt (Neg (Succ vz300)) (Pos vz400)) vz5",fontsize=16,color="black",shape="box"];30 -> 41[label="",style="solid", color="black", weight=3];
31[label="pePe (primEqInt (Neg (Succ vz300)) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];98[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];31 -> 98[label="",style="solid", color="burlywood", weight=9];
98 -> 42[label="",style="solid", color="burlywood", weight=3];
99[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];31 -> 99[label="",style="solid", color="burlywood", weight=9];
99 -> 43[label="",style="solid", color="burlywood", weight=3];
32[label="pePe (primEqInt (Neg Zero) (Pos vz400)) vz5",fontsize=16,color="burlywood",shape="box"];100[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];32 -> 100[label="",style="solid", color="burlywood", weight=9];
100 -> 44[label="",style="solid", color="burlywood", weight=3];
101[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];32 -> 101[label="",style="solid", color="burlywood", weight=9];
101 -> 45[label="",style="solid", color="burlywood", weight=3];
33[label="pePe (primEqInt (Neg Zero) (Neg vz400)) vz5",fontsize=16,color="burlywood",shape="box"];102[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];33 -> 102[label="",style="solid", color="burlywood", weight=9];
102 -> 46[label="",style="solid", color="burlywood", weight=3];
103[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];33 -> 103[label="",style="solid", color="burlywood", weight=9];
103 -> 47[label="",style="solid", color="burlywood", weight=3];
34[label="pePe (primEqInt (Pos (Succ vz300)) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="pePe (primEqInt (Pos (Succ vz300)) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="pePe MyFalse vz5",fontsize=16,color="black",shape="triangle"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="pePe (primEqInt (Pos Zero) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="pePe (primEqInt (Pos Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="pePe (primEqInt (Pos Zero) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="pePe (primEqInt (Pos Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
41 -> 36[label="",style="dashed", color="red", weight=0];
41[label="pePe MyFalse vz5",fontsize=16,color="magenta"];42[label="pePe (primEqInt (Neg (Succ vz300)) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];42 -> 55[label="",style="solid", color="black", weight=3];
43[label="pePe (primEqInt (Neg (Succ vz300)) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];43 -> 56[label="",style="solid", color="black", weight=3];
44[label="pePe (primEqInt (Neg Zero) (Pos (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];44 -> 57[label="",style="solid", color="black", weight=3];
45[label="pePe (primEqInt (Neg Zero) (Pos Zero)) vz5",fontsize=16,color="black",shape="box"];45 -> 58[label="",style="solid", color="black", weight=3];
46[label="pePe (primEqInt (Neg Zero) (Neg (Succ vz4000))) vz5",fontsize=16,color="black",shape="box"];46 -> 59[label="",style="solid", color="black", weight=3];
47[label="pePe (primEqInt (Neg Zero) (Neg Zero)) vz5",fontsize=16,color="black",shape="box"];47 -> 60[label="",style="solid", color="black", weight=3];
48[label="pePe (primEqNat vz300 vz4000) vz5",fontsize=16,color="burlywood",shape="triangle"];104[label="vz300/Succ vz3000",fontsize=10,color="white",style="solid",shape="box"];48 -> 104[label="",style="solid", color="burlywood", weight=9];
104 -> 61[label="",style="solid", color="burlywood", weight=3];
105[label="vz300/Zero",fontsize=10,color="white",style="solid",shape="box"];48 -> 105[label="",style="solid", color="burlywood", weight=9];
105 -> 62[label="",style="solid", color="burlywood", weight=3];
49 -> 36[label="",style="dashed", color="red", weight=0];
49[label="pePe MyFalse vz5",fontsize=16,color="magenta"];50[label="vz5",fontsize=16,color="green",shape="box"];51 -> 36[label="",style="dashed", color="red", weight=0];
51[label="pePe MyFalse vz5",fontsize=16,color="magenta"];52[label="pePe MyTrue vz5",fontsize=16,color="black",shape="triangle"];52 -> 63[label="",style="solid", color="black", weight=3];
53 -> 36[label="",style="dashed", color="red", weight=0];
53[label="pePe MyFalse vz5",fontsize=16,color="magenta"];54 -> 52[label="",style="dashed", color="red", weight=0];
54[label="pePe MyTrue vz5",fontsize=16,color="magenta"];55 -> 48[label="",style="dashed", color="red", weight=0];
55[label="pePe (primEqNat vz300 vz4000) vz5",fontsize=16,color="magenta"];55 -> 64[label="",style="dashed", color="magenta", weight=3];
55 -> 65[label="",style="dashed", color="magenta", weight=3];
56 -> 36[label="",style="dashed", color="red", weight=0];
56[label="pePe MyFalse vz5",fontsize=16,color="magenta"];57 -> 36[label="",style="dashed", color="red", weight=0];
57[label="pePe MyFalse vz5",fontsize=16,color="magenta"];58 -> 52[label="",style="dashed", color="red", weight=0];
58[label="pePe MyTrue vz5",fontsize=16,color="magenta"];59 -> 36[label="",style="dashed", color="red", weight=0];
59[label="pePe MyFalse vz5",fontsize=16,color="magenta"];60 -> 52[label="",style="dashed", color="red", weight=0];
60[label="pePe MyTrue vz5",fontsize=16,color="magenta"];61[label="pePe (primEqNat (Succ vz3000) vz4000) vz5",fontsize=16,color="burlywood",shape="box"];106[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];61 -> 106[label="",style="solid", color="burlywood", weight=9];
106 -> 66[label="",style="solid", color="burlywood", weight=3];
107[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];61 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 67[label="",style="solid", color="burlywood", weight=3];
62[label="pePe (primEqNat Zero vz4000) vz5",fontsize=16,color="burlywood",shape="box"];108[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];62 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 68[label="",style="solid", color="burlywood", weight=3];
109[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];62 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 69[label="",style="solid", color="burlywood", weight=3];
63[label="MyTrue",fontsize=16,color="green",shape="box"];64[label="vz4000",fontsize=16,color="green",shape="box"];65[label="vz300",fontsize=16,color="green",shape="box"];66[label="pePe (primEqNat (Succ vz3000) (Succ vz40000)) vz5",fontsize=16,color="black",shape="box"];66 -> 70[label="",style="solid", color="black", weight=3];
67[label="pePe (primEqNat (Succ vz3000) Zero) vz5",fontsize=16,color="black",shape="box"];67 -> 71[label="",style="solid", color="black", weight=3];
68[label="pePe (primEqNat Zero (Succ vz40000)) vz5",fontsize=16,color="black",shape="box"];68 -> 72[label="",style="solid", color="black", weight=3];
69[label="pePe (primEqNat Zero Zero) vz5",fontsize=16,color="black",shape="box"];69 -> 73[label="",style="solid", color="black", weight=3];
70 -> 48[label="",style="dashed", color="red", weight=0];
70[label="pePe (primEqNat vz3000 vz40000) vz5",fontsize=16,color="magenta"];70 -> 74[label="",style="dashed", color="magenta", weight=3];
70 -> 75[label="",style="dashed", color="magenta", weight=3];
71 -> 36[label="",style="dashed", color="red", weight=0];
71[label="pePe MyFalse vz5",fontsize=16,color="magenta"];72 -> 36[label="",style="dashed", color="red", weight=0];
72[label="pePe MyFalse vz5",fontsize=16,color="magenta"];73 -> 52[label="",style="dashed", color="red", weight=0];
73[label="pePe MyTrue vz5",fontsize=16,color="magenta"];74[label="vz40000",fontsize=16,color="green",shape="box"];75[label="vz3000",fontsize=16,color="green",shape="box"];}

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(6)
Complex Obligation (AND)

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(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
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(8) 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_foldr(vz3, Cons(vz40, vz41)) -> new_foldr(vz3, vz41)
The graph contains the following edges 1 >= 1, 2 > 2


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(9)
YES

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(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_pePe(Main.Succ(vz3000), Main.Succ(vz40000), vz5) -> new_pePe(vz3000, vz40000, vz5)

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

(11) 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_pePe(Main.Succ(vz3000), Main.Succ(vz40000), vz5) -> new_pePe(vz3000, vz40000, vz5)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3


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(12)
YES