/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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) DependencyGraphProof [EQUIVALENT, 0 ms]
(8) AND
    (9) QDP
        (10) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (17) YES
    (18) QDP
        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (20) YES


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

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

data Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

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

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyInt k x);

lookup3 k Nil = Main.Nothing;
lookup3 wu wv = lookup2 wu wv;

otherwise :: MyBool;
otherwise = 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;

}

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

(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 Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

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

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyInt k x);

lookup3 k Nil = Main.Nothing;
lookup3 wu wv = lookup2 wu wv;

otherwise :: MyBool;
otherwise = 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;

}

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

(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 Main.Maybe a = Nothing  | Just a ;

data MyBool = MyTrue  | MyFalse ;

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

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

data Tup2 b a = Tup2 b a ;

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

lookup k Nil = lookup3 k Nil;
lookup k (Cons (Tup2 x y) xys) = lookup2 k (Cons (Tup2 x y) xys);

lookup0 k x y xys MyTrue = lookup k xys;

lookup1 k x y xys MyTrue = Main.Just y;
lookup1 k x y xys MyFalse = lookup0 k x y xys otherwise;

lookup2 k (Cons (Tup2 x y) xys) = lookup1 k x y xys (esEsMyInt k x);

lookup3 k Nil = Main.Nothing;
lookup3 wu wv = lookup2 wu wv;

otherwise :: MyBool;
otherwise = 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;

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="lookup",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="lookup vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="lookup vz3 vz4",fontsize=16,color="burlywood",shape="triangle"];778[label="vz4/Cons vz40 vz41",fontsize=10,color="white",style="solid",shape="box"];4 -> 778[label="",style="solid", color="burlywood", weight=9];
778 -> 5[label="",style="solid", color="burlywood", weight=3];
779[label="vz4/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 779[label="",style="solid", color="burlywood", weight=9];
779 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="lookup vz3 (Cons vz40 vz41)",fontsize=16,color="burlywood",shape="box"];780[label="vz40/Tup2 vz400 vz401",fontsize=10,color="white",style="solid",shape="box"];5 -> 780[label="",style="solid", color="burlywood", weight=9];
780 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="lookup vz3 Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="lookup vz3 (Cons (Tup2 vz400 vz401) vz41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="lookup3 vz3 Nil",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="lookup2 vz3 (Cons (Tup2 vz400 vz401) vz41)",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="Nothing",fontsize=16,color="green",shape="box"];11[label="lookup1 vz3 vz400 vz401 vz41 (esEsMyInt vz3 vz400)",fontsize=16,color="black",shape="box"];11 -> 12[label="",style="solid", color="black", weight=3];
12[label="lookup1 vz3 vz400 vz401 vz41 (primEqInt vz3 vz400)",fontsize=16,color="burlywood",shape="box"];781[label="vz3/Pos vz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 781[label="",style="solid", color="burlywood", weight=9];
781 -> 13[label="",style="solid", color="burlywood", weight=3];
782[label="vz3/Neg vz30",fontsize=10,color="white",style="solid",shape="box"];12 -> 782[label="",style="solid", color="burlywood", weight=9];
782 -> 14[label="",style="solid", color="burlywood", weight=3];
13[label="lookup1 (Pos vz30) vz400 vz401 vz41 (primEqInt (Pos vz30) vz400)",fontsize=16,color="burlywood",shape="box"];783[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];13 -> 783[label="",style="solid", color="burlywood", weight=9];
783 -> 15[label="",style="solid", color="burlywood", weight=3];
784[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];13 -> 784[label="",style="solid", color="burlywood", weight=9];
784 -> 16[label="",style="solid", color="burlywood", weight=3];
14[label="lookup1 (Neg vz30) vz400 vz401 vz41 (primEqInt (Neg vz30) vz400)",fontsize=16,color="burlywood",shape="box"];785[label="vz30/Succ vz300",fontsize=10,color="white",style="solid",shape="box"];14 -> 785[label="",style="solid", color="burlywood", weight=9];
785 -> 17[label="",style="solid", color="burlywood", weight=3];
786[label="vz30/Zero",fontsize=10,color="white",style="solid",shape="box"];14 -> 786[label="",style="solid", color="burlywood", weight=9];
786 -> 18[label="",style="solid", color="burlywood", weight=3];
15[label="lookup1 (Pos (Succ vz300)) vz400 vz401 vz41 (primEqInt (Pos (Succ vz300)) vz400)",fontsize=16,color="burlywood",shape="box"];787[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 787[label="",style="solid", color="burlywood", weight=9];
787 -> 19[label="",style="solid", color="burlywood", weight=3];
788[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];15 -> 788[label="",style="solid", color="burlywood", weight=9];
788 -> 20[label="",style="solid", color="burlywood", weight=3];
16[label="lookup1 (Pos Zero) vz400 vz401 vz41 (primEqInt (Pos Zero) vz400)",fontsize=16,color="burlywood",shape="box"];789[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];16 -> 789[label="",style="solid", color="burlywood", weight=9];
789 -> 21[label="",style="solid", color="burlywood", weight=3];
790[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];16 -> 790[label="",style="solid", color="burlywood", weight=9];
790 -> 22[label="",style="solid", color="burlywood", weight=3];
17[label="lookup1 (Neg (Succ vz300)) vz400 vz401 vz41 (primEqInt (Neg (Succ vz300)) vz400)",fontsize=16,color="burlywood",shape="box"];791[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 791[label="",style="solid", color="burlywood", weight=9];
791 -> 23[label="",style="solid", color="burlywood", weight=3];
792[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];17 -> 792[label="",style="solid", color="burlywood", weight=9];
792 -> 24[label="",style="solid", color="burlywood", weight=3];
18[label="lookup1 (Neg Zero) vz400 vz401 vz41 (primEqInt (Neg Zero) vz400)",fontsize=16,color="burlywood",shape="box"];793[label="vz400/Pos vz4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 793[label="",style="solid", color="burlywood", weight=9];
793 -> 25[label="",style="solid", color="burlywood", weight=3];
794[label="vz400/Neg vz4000",fontsize=10,color="white",style="solid",shape="box"];18 -> 794[label="",style="solid", color="burlywood", weight=9];
794 -> 26[label="",style="solid", color="burlywood", weight=3];
19[label="lookup1 (Pos (Succ vz300)) (Pos vz4000) vz401 vz41 (primEqInt (Pos (Succ vz300)) (Pos vz4000))",fontsize=16,color="burlywood",shape="box"];795[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];19 -> 795[label="",style="solid", color="burlywood", weight=9];
795 -> 27[label="",style="solid", color="burlywood", weight=3];
796[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];19 -> 796[label="",style="solid", color="burlywood", weight=9];
796 -> 28[label="",style="solid", color="burlywood", weight=3];
20[label="lookup1 (Pos (Succ vz300)) (Neg vz4000) vz401 vz41 (primEqInt (Pos (Succ vz300)) (Neg vz4000))",fontsize=16,color="black",shape="box"];20 -> 29[label="",style="solid", color="black", weight=3];
21[label="lookup1 (Pos Zero) (Pos vz4000) vz401 vz41 (primEqInt (Pos Zero) (Pos vz4000))",fontsize=16,color="burlywood",shape="box"];797[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];21 -> 797[label="",style="solid", color="burlywood", weight=9];
797 -> 30[label="",style="solid", color="burlywood", weight=3];
798[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];21 -> 798[label="",style="solid", color="burlywood", weight=9];
798 -> 31[label="",style="solid", color="burlywood", weight=3];
22[label="lookup1 (Pos Zero) (Neg vz4000) vz401 vz41 (primEqInt (Pos Zero) (Neg vz4000))",fontsize=16,color="burlywood",shape="box"];799[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];22 -> 799[label="",style="solid", color="burlywood", weight=9];
799 -> 32[label="",style="solid", color="burlywood", weight=3];
800[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];22 -> 800[label="",style="solid", color="burlywood", weight=9];
800 -> 33[label="",style="solid", color="burlywood", weight=3];
23[label="lookup1 (Neg (Succ vz300)) (Pos vz4000) vz401 vz41 (primEqInt (Neg (Succ vz300)) (Pos vz4000))",fontsize=16,color="black",shape="box"];23 -> 34[label="",style="solid", color="black", weight=3];
24[label="lookup1 (Neg (Succ vz300)) (Neg vz4000) vz401 vz41 (primEqInt (Neg (Succ vz300)) (Neg vz4000))",fontsize=16,color="burlywood",shape="box"];801[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];24 -> 801[label="",style="solid", color="burlywood", weight=9];
801 -> 35[label="",style="solid", color="burlywood", weight=3];
802[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];24 -> 802[label="",style="solid", color="burlywood", weight=9];
802 -> 36[label="",style="solid", color="burlywood", weight=3];
25[label="lookup1 (Neg Zero) (Pos vz4000) vz401 vz41 (primEqInt (Neg Zero) (Pos vz4000))",fontsize=16,color="burlywood",shape="box"];803[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];25 -> 803[label="",style="solid", color="burlywood", weight=9];
803 -> 37[label="",style="solid", color="burlywood", weight=3];
804[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];25 -> 804[label="",style="solid", color="burlywood", weight=9];
804 -> 38[label="",style="solid", color="burlywood", weight=3];
26[label="lookup1 (Neg Zero) (Neg vz4000) vz401 vz41 (primEqInt (Neg Zero) (Neg vz4000))",fontsize=16,color="burlywood",shape="box"];805[label="vz4000/Succ vz40000",fontsize=10,color="white",style="solid",shape="box"];26 -> 805[label="",style="solid", color="burlywood", weight=9];
805 -> 39[label="",style="solid", color="burlywood", weight=3];
806[label="vz4000/Zero",fontsize=10,color="white",style="solid",shape="box"];26 -> 806[label="",style="solid", color="burlywood", weight=9];
806 -> 40[label="",style="solid", color="burlywood", weight=3];
27[label="lookup1 (Pos (Succ vz300)) (Pos (Succ vz40000)) vz401 vz41 (primEqInt (Pos (Succ vz300)) (Pos (Succ vz40000)))",fontsize=16,color="black",shape="box"];27 -> 41[label="",style="solid", color="black", weight=3];
28[label="lookup1 (Pos (Succ vz300)) (Pos Zero) vz401 vz41 (primEqInt (Pos (Succ vz300)) (Pos Zero))",fontsize=16,color="black",shape="box"];28 -> 42[label="",style="solid", color="black", weight=3];
29[label="lookup1 (Pos (Succ vz300)) (Neg vz4000) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];29 -> 43[label="",style="solid", color="black", weight=3];
30[label="lookup1 (Pos Zero) (Pos (Succ vz40000)) vz401 vz41 (primEqInt (Pos Zero) (Pos (Succ vz40000)))",fontsize=16,color="black",shape="box"];30 -> 44[label="",style="solid", color="black", weight=3];
31[label="lookup1 (Pos Zero) (Pos Zero) vz401 vz41 (primEqInt (Pos Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];31 -> 45[label="",style="solid", color="black", weight=3];
32[label="lookup1 (Pos Zero) (Neg (Succ vz40000)) vz401 vz41 (primEqInt (Pos Zero) (Neg (Succ vz40000)))",fontsize=16,color="black",shape="box"];32 -> 46[label="",style="solid", color="black", weight=3];
33[label="lookup1 (Pos Zero) (Neg Zero) vz401 vz41 (primEqInt (Pos Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];33 -> 47[label="",style="solid", color="black", weight=3];
34[label="lookup1 (Neg (Succ vz300)) (Pos vz4000) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];34 -> 48[label="",style="solid", color="black", weight=3];
35[label="lookup1 (Neg (Succ vz300)) (Neg (Succ vz40000)) vz401 vz41 (primEqInt (Neg (Succ vz300)) (Neg (Succ vz40000)))",fontsize=16,color="black",shape="box"];35 -> 49[label="",style="solid", color="black", weight=3];
36[label="lookup1 (Neg (Succ vz300)) (Neg Zero) vz401 vz41 (primEqInt (Neg (Succ vz300)) (Neg Zero))",fontsize=16,color="black",shape="box"];36 -> 50[label="",style="solid", color="black", weight=3];
37[label="lookup1 (Neg Zero) (Pos (Succ vz40000)) vz401 vz41 (primEqInt (Neg Zero) (Pos (Succ vz40000)))",fontsize=16,color="black",shape="box"];37 -> 51[label="",style="solid", color="black", weight=3];
38[label="lookup1 (Neg Zero) (Pos Zero) vz401 vz41 (primEqInt (Neg Zero) (Pos Zero))",fontsize=16,color="black",shape="box"];38 -> 52[label="",style="solid", color="black", weight=3];
39[label="lookup1 (Neg Zero) (Neg (Succ vz40000)) vz401 vz41 (primEqInt (Neg Zero) (Neg (Succ vz40000)))",fontsize=16,color="black",shape="box"];39 -> 53[label="",style="solid", color="black", weight=3];
40[label="lookup1 (Neg Zero) (Neg Zero) vz401 vz41 (primEqInt (Neg Zero) (Neg Zero))",fontsize=16,color="black",shape="box"];40 -> 54[label="",style="solid", color="black", weight=3];
41 -> 620[label="",style="dashed", color="red", weight=0];
41[label="lookup1 (Pos (Succ vz300)) (Pos (Succ vz40000)) vz401 vz41 (primEqNat vz300 vz40000)",fontsize=16,color="magenta"];41 -> 621[label="",style="dashed", color="magenta", weight=3];
41 -> 622[label="",style="dashed", color="magenta", weight=3];
41 -> 623[label="",style="dashed", color="magenta", weight=3];
41 -> 624[label="",style="dashed", color="magenta", weight=3];
41 -> 625[label="",style="dashed", color="magenta", weight=3];
41 -> 626[label="",style="dashed", color="magenta", weight=3];
42[label="lookup1 (Pos (Succ vz300)) (Pos Zero) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];42 -> 57[label="",style="solid", color="black", weight=3];
43[label="lookup0 (Pos (Succ vz300)) (Neg vz4000) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];43 -> 58[label="",style="solid", color="black", weight=3];
44[label="lookup1 (Pos Zero) (Pos (Succ vz40000)) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];44 -> 59[label="",style="solid", color="black", weight=3];
45[label="lookup1 (Pos Zero) (Pos Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];45 -> 60[label="",style="solid", color="black", weight=3];
46[label="lookup1 (Pos Zero) (Neg (Succ vz40000)) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];46 -> 61[label="",style="solid", color="black", weight=3];
47[label="lookup1 (Pos Zero) (Neg Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];47 -> 62[label="",style="solid", color="black", weight=3];
48[label="lookup0 (Neg (Succ vz300)) (Pos vz4000) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];48 -> 63[label="",style="solid", color="black", weight=3];
49 -> 683[label="",style="dashed", color="red", weight=0];
49[label="lookup1 (Neg (Succ vz300)) (Neg (Succ vz40000)) vz401 vz41 (primEqNat vz300 vz40000)",fontsize=16,color="magenta"];49 -> 684[label="",style="dashed", color="magenta", weight=3];
49 -> 685[label="",style="dashed", color="magenta", weight=3];
49 -> 686[label="",style="dashed", color="magenta", weight=3];
49 -> 687[label="",style="dashed", color="magenta", weight=3];
49 -> 688[label="",style="dashed", color="magenta", weight=3];
49 -> 689[label="",style="dashed", color="magenta", weight=3];
50[label="lookup1 (Neg (Succ vz300)) (Neg Zero) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];50 -> 66[label="",style="solid", color="black", weight=3];
51[label="lookup1 (Neg Zero) (Pos (Succ vz40000)) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];51 -> 67[label="",style="solid", color="black", weight=3];
52[label="lookup1 (Neg Zero) (Pos Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];52 -> 68[label="",style="solid", color="black", weight=3];
53[label="lookup1 (Neg Zero) (Neg (Succ vz40000)) vz401 vz41 MyFalse",fontsize=16,color="black",shape="box"];53 -> 69[label="",style="solid", color="black", weight=3];
54[label="lookup1 (Neg Zero) (Neg Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];54 -> 70[label="",style="solid", color="black", weight=3];
621[label="vz300",fontsize=16,color="green",shape="box"];622[label="vz300",fontsize=16,color="green",shape="box"];623[label="vz41",fontsize=16,color="green",shape="box"];624[label="vz40000",fontsize=16,color="green",shape="box"];625[label="vz40000",fontsize=16,color="green",shape="box"];626[label="vz401",fontsize=16,color="green",shape="box"];620[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat vz78 vz79)",fontsize=16,color="burlywood",shape="triangle"];807[label="vz78/Succ vz780",fontsize=10,color="white",style="solid",shape="box"];620 -> 807[label="",style="solid", color="burlywood", weight=9];
807 -> 681[label="",style="solid", color="burlywood", weight=3];
808[label="vz78/Zero",fontsize=10,color="white",style="solid",shape="box"];620 -> 808[label="",style="solid", color="burlywood", weight=9];
808 -> 682[label="",style="solid", color="burlywood", weight=3];
57[label="lookup0 (Pos (Succ vz300)) (Pos Zero) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];57 -> 75[label="",style="solid", color="black", weight=3];
58[label="lookup0 (Pos (Succ vz300)) (Neg vz4000) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];58 -> 76[label="",style="solid", color="black", weight=3];
59[label="lookup0 (Pos Zero) (Pos (Succ vz40000)) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];59 -> 77[label="",style="solid", color="black", weight=3];
60[label="Just vz401",fontsize=16,color="green",shape="box"];61[label="lookup0 (Pos Zero) (Neg (Succ vz40000)) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];61 -> 78[label="",style="solid", color="black", weight=3];
62[label="Just vz401",fontsize=16,color="green",shape="box"];63[label="lookup0 (Neg (Succ vz300)) (Pos vz4000) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];63 -> 79[label="",style="solid", color="black", weight=3];
684[label="vz300",fontsize=16,color="green",shape="box"];685[label="vz300",fontsize=16,color="green",shape="box"];686[label="vz40000",fontsize=16,color="green",shape="box"];687[label="vz401",fontsize=16,color="green",shape="box"];688[label="vz41",fontsize=16,color="green",shape="box"];689[label="vz40000",fontsize=16,color="green",shape="box"];683[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat vz85 vz86)",fontsize=16,color="burlywood",shape="triangle"];809[label="vz85/Succ vz850",fontsize=10,color="white",style="solid",shape="box"];683 -> 809[label="",style="solid", color="burlywood", weight=9];
809 -> 744[label="",style="solid", color="burlywood", weight=3];
810[label="vz85/Zero",fontsize=10,color="white",style="solid",shape="box"];683 -> 810[label="",style="solid", color="burlywood", weight=9];
810 -> 745[label="",style="solid", color="burlywood", weight=3];
66[label="lookup0 (Neg (Succ vz300)) (Neg Zero) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];66 -> 84[label="",style="solid", color="black", weight=3];
67[label="lookup0 (Neg Zero) (Pos (Succ vz40000)) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];67 -> 85[label="",style="solid", color="black", weight=3];
68[label="Just vz401",fontsize=16,color="green",shape="box"];69[label="lookup0 (Neg Zero) (Neg (Succ vz40000)) vz401 vz41 otherwise",fontsize=16,color="black",shape="box"];69 -> 86[label="",style="solid", color="black", weight=3];
70[label="Just vz401",fontsize=16,color="green",shape="box"];681[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat (Succ vz780) vz79)",fontsize=16,color="burlywood",shape="box"];811[label="vz79/Succ vz790",fontsize=10,color="white",style="solid",shape="box"];681 -> 811[label="",style="solid", color="burlywood", weight=9];
811 -> 746[label="",style="solid", color="burlywood", weight=3];
812[label="vz79/Zero",fontsize=10,color="white",style="solid",shape="box"];681 -> 812[label="",style="solid", color="burlywood", weight=9];
812 -> 747[label="",style="solid", color="burlywood", weight=3];
682[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat Zero vz79)",fontsize=16,color="burlywood",shape="box"];813[label="vz79/Succ vz790",fontsize=10,color="white",style="solid",shape="box"];682 -> 813[label="",style="solid", color="burlywood", weight=9];
813 -> 748[label="",style="solid", color="burlywood", weight=3];
814[label="vz79/Zero",fontsize=10,color="white",style="solid",shape="box"];682 -> 814[label="",style="solid", color="burlywood", weight=9];
814 -> 749[label="",style="solid", color="burlywood", weight=3];
75[label="lookup0 (Pos (Succ vz300)) (Pos Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];75 -> 91[label="",style="solid", color="black", weight=3];
76 -> 4[label="",style="dashed", color="red", weight=0];
76[label="lookup (Pos (Succ vz300)) vz41",fontsize=16,color="magenta"];76 -> 92[label="",style="dashed", color="magenta", weight=3];
76 -> 93[label="",style="dashed", color="magenta", weight=3];
77[label="lookup0 (Pos Zero) (Pos (Succ vz40000)) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];77 -> 94[label="",style="solid", color="black", weight=3];
78[label="lookup0 (Pos Zero) (Neg (Succ vz40000)) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];78 -> 95[label="",style="solid", color="black", weight=3];
79 -> 4[label="",style="dashed", color="red", weight=0];
79[label="lookup (Neg (Succ vz300)) vz41",fontsize=16,color="magenta"];79 -> 96[label="",style="dashed", color="magenta", weight=3];
79 -> 97[label="",style="dashed", color="magenta", weight=3];
744[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat (Succ vz850) vz86)",fontsize=16,color="burlywood",shape="box"];815[label="vz86/Succ vz860",fontsize=10,color="white",style="solid",shape="box"];744 -> 815[label="",style="solid", color="burlywood", weight=9];
815 -> 750[label="",style="solid", color="burlywood", weight=3];
816[label="vz86/Zero",fontsize=10,color="white",style="solid",shape="box"];744 -> 816[label="",style="solid", color="burlywood", weight=9];
816 -> 751[label="",style="solid", color="burlywood", weight=3];
745[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat Zero vz86)",fontsize=16,color="burlywood",shape="box"];817[label="vz86/Succ vz860",fontsize=10,color="white",style="solid",shape="box"];745 -> 817[label="",style="solid", color="burlywood", weight=9];
817 -> 752[label="",style="solid", color="burlywood", weight=3];
818[label="vz86/Zero",fontsize=10,color="white",style="solid",shape="box"];745 -> 818[label="",style="solid", color="burlywood", weight=9];
818 -> 753[label="",style="solid", color="burlywood", weight=3];
84[label="lookup0 (Neg (Succ vz300)) (Neg Zero) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];84 -> 102[label="",style="solid", color="black", weight=3];
85[label="lookup0 (Neg Zero) (Pos (Succ vz40000)) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];85 -> 103[label="",style="solid", color="black", weight=3];
86[label="lookup0 (Neg Zero) (Neg (Succ vz40000)) vz401 vz41 MyTrue",fontsize=16,color="black",shape="box"];86 -> 104[label="",style="solid", color="black", weight=3];
746[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat (Succ vz780) (Succ vz790))",fontsize=16,color="black",shape="box"];746 -> 754[label="",style="solid", color="black", weight=3];
747[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat (Succ vz780) Zero)",fontsize=16,color="black",shape="box"];747 -> 755[label="",style="solid", color="black", weight=3];
748[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat Zero (Succ vz790))",fontsize=16,color="black",shape="box"];748 -> 756[label="",style="solid", color="black", weight=3];
749[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];749 -> 757[label="",style="solid", color="black", weight=3];
91 -> 4[label="",style="dashed", color="red", weight=0];
91[label="lookup (Pos (Succ vz300)) vz41",fontsize=16,color="magenta"];91 -> 110[label="",style="dashed", color="magenta", weight=3];
91 -> 111[label="",style="dashed", color="magenta", weight=3];
92[label="vz41",fontsize=16,color="green",shape="box"];93[label="Pos (Succ vz300)",fontsize=16,color="green",shape="box"];94 -> 4[label="",style="dashed", color="red", weight=0];
94[label="lookup (Pos Zero) vz41",fontsize=16,color="magenta"];94 -> 112[label="",style="dashed", color="magenta", weight=3];
94 -> 113[label="",style="dashed", color="magenta", weight=3];
95 -> 4[label="",style="dashed", color="red", weight=0];
95[label="lookup (Pos Zero) vz41",fontsize=16,color="magenta"];95 -> 114[label="",style="dashed", color="magenta", weight=3];
95 -> 115[label="",style="dashed", color="magenta", weight=3];
96[label="vz41",fontsize=16,color="green",shape="box"];97[label="Neg (Succ vz300)",fontsize=16,color="green",shape="box"];750[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat (Succ vz850) (Succ vz860))",fontsize=16,color="black",shape="box"];750 -> 758[label="",style="solid", color="black", weight=3];
751[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat (Succ vz850) Zero)",fontsize=16,color="black",shape="box"];751 -> 759[label="",style="solid", color="black", weight=3];
752[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat Zero (Succ vz860))",fontsize=16,color="black",shape="box"];752 -> 760[label="",style="solid", color="black", weight=3];
753[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat Zero Zero)",fontsize=16,color="black",shape="box"];753 -> 761[label="",style="solid", color="black", weight=3];
102 -> 4[label="",style="dashed", color="red", weight=0];
102[label="lookup (Neg (Succ vz300)) vz41",fontsize=16,color="magenta"];102 -> 121[label="",style="dashed", color="magenta", weight=3];
102 -> 122[label="",style="dashed", color="magenta", weight=3];
103 -> 4[label="",style="dashed", color="red", weight=0];
103[label="lookup (Neg Zero) vz41",fontsize=16,color="magenta"];103 -> 123[label="",style="dashed", color="magenta", weight=3];
103 -> 124[label="",style="dashed", color="magenta", weight=3];
104 -> 4[label="",style="dashed", color="red", weight=0];
104[label="lookup (Neg Zero) vz41",fontsize=16,color="magenta"];104 -> 125[label="",style="dashed", color="magenta", weight=3];
104 -> 126[label="",style="dashed", color="magenta", weight=3];
754 -> 620[label="",style="dashed", color="red", weight=0];
754[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 (primEqNat vz780 vz790)",fontsize=16,color="magenta"];754 -> 762[label="",style="dashed", color="magenta", weight=3];
754 -> 763[label="",style="dashed", color="magenta", weight=3];
755[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 MyFalse",fontsize=16,color="black",shape="triangle"];755 -> 764[label="",style="solid", color="black", weight=3];
756 -> 755[label="",style="dashed", color="red", weight=0];
756[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 MyFalse",fontsize=16,color="magenta"];757[label="lookup1 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 MyTrue",fontsize=16,color="black",shape="box"];757 -> 765[label="",style="solid", color="black", weight=3];
110[label="vz41",fontsize=16,color="green",shape="box"];111[label="Pos (Succ vz300)",fontsize=16,color="green",shape="box"];112[label="vz41",fontsize=16,color="green",shape="box"];113[label="Pos Zero",fontsize=16,color="green",shape="box"];114[label="vz41",fontsize=16,color="green",shape="box"];115[label="Pos Zero",fontsize=16,color="green",shape="box"];758 -> 683[label="",style="dashed", color="red", weight=0];
758[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 (primEqNat vz850 vz860)",fontsize=16,color="magenta"];758 -> 766[label="",style="dashed", color="magenta", weight=3];
758 -> 767[label="",style="dashed", color="magenta", weight=3];
759[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 MyFalse",fontsize=16,color="black",shape="triangle"];759 -> 768[label="",style="solid", color="black", weight=3];
760 -> 759[label="",style="dashed", color="red", weight=0];
760[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 MyFalse",fontsize=16,color="magenta"];761[label="lookup1 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 MyTrue",fontsize=16,color="black",shape="box"];761 -> 769[label="",style="solid", color="black", weight=3];
121[label="vz41",fontsize=16,color="green",shape="box"];122[label="Neg (Succ vz300)",fontsize=16,color="green",shape="box"];123[label="vz41",fontsize=16,color="green",shape="box"];124[label="Neg Zero",fontsize=16,color="green",shape="box"];125[label="vz41",fontsize=16,color="green",shape="box"];126[label="Neg Zero",fontsize=16,color="green",shape="box"];762[label="vz780",fontsize=16,color="green",shape="box"];763[label="vz790",fontsize=16,color="green",shape="box"];764[label="lookup0 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 otherwise",fontsize=16,color="black",shape="box"];764 -> 770[label="",style="solid", color="black", weight=3];
765[label="Just vz76",fontsize=16,color="green",shape="box"];766[label="vz850",fontsize=16,color="green",shape="box"];767[label="vz860",fontsize=16,color="green",shape="box"];768[label="lookup0 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 otherwise",fontsize=16,color="black",shape="box"];768 -> 771[label="",style="solid", color="black", weight=3];
769[label="Just vz83",fontsize=16,color="green",shape="box"];770[label="lookup0 (Pos (Succ vz74)) (Pos (Succ vz75)) vz76 vz77 MyTrue",fontsize=16,color="black",shape="box"];770 -> 772[label="",style="solid", color="black", weight=3];
771[label="lookup0 (Neg (Succ vz81)) (Neg (Succ vz82)) vz83 vz84 MyTrue",fontsize=16,color="black",shape="box"];771 -> 773[label="",style="solid", color="black", weight=3];
772 -> 4[label="",style="dashed", color="red", weight=0];
772[label="lookup (Pos (Succ vz74)) vz77",fontsize=16,color="magenta"];772 -> 774[label="",style="dashed", color="magenta", weight=3];
772 -> 775[label="",style="dashed", color="magenta", weight=3];
773 -> 4[label="",style="dashed", color="red", weight=0];
773[label="lookup (Neg (Succ vz81)) vz84",fontsize=16,color="magenta"];773 -> 776[label="",style="dashed", color="magenta", weight=3];
773 -> 777[label="",style="dashed", color="magenta", weight=3];
774[label="vz77",fontsize=16,color="green",shape="box"];775[label="Pos (Succ vz74)",fontsize=16,color="green",shape="box"];776[label="vz84",fontsize=16,color="green",shape="box"];777[label="Neg (Succ vz81)",fontsize=16,color="green",shape="box"];}

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

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

   new_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)
   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
   new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Succ(vz790), h) -> new_lookup1(vz74, vz75, vz76, vz77, vz780, vz790, h)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup11(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
   new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Zero, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)
   new_lookup12(vz81, vz82, vz83, vz84, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)
   new_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Neg(vz4000), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
   new_lookup1(vz74, vz75, vz76, vz77, Main.Zero, Main.Succ(vz790), h) -> new_lookup10(vz74, vz75, vz76, vz77, h)
   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Pos(vz4000), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
   new_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup1(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Zero, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)
   new_lookup10(vz74, vz75, vz76, vz77, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)
   new_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Succ(vz860), bb) -> new_lookup11(vz81, vz82, vz83, vz84, vz850, vz860, bb)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Zero, Main.Succ(vz860), bb) -> new_lookup12(vz81, vz82, vz83, vz84, bb)

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

(7) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs.
----------------------------------------

(8)
Complex Obligation (AND)

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

(9)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)
   new_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)

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

(10) 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_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(Main.Neg(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Zero), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(11)
YES

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

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Zero, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Neg(vz4000), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
   new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup1(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
   new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Succ(vz790), h) -> new_lookup1(vz74, vz75, vz76, vz77, vz780, vz790, h)
   new_lookup1(vz74, vz75, vz76, vz77, Main.Zero, Main.Succ(vz790), h) -> new_lookup10(vz74, vz75, vz76, vz77, h)
   new_lookup10(vz74, vz75, vz76, vz77, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)

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

(13) 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_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup1(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4, 1 > 5, 2 > 6, 3 >= 7


*new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Succ(vz790), h) -> new_lookup1(vz74, vz75, vz76, vz77, vz780, vz790, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7


*new_lookup1(vz74, vz75, vz76, vz77, Main.Zero, Main.Succ(vz790), h) -> new_lookup10(vz74, vz75, vz76, vz77, h)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5


*new_lookup1(vz74, vz75, vz76, vz77, Main.Succ(vz780), Main.Zero, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)
The graph contains the following edges 4 >= 2, 7 >= 3


*new_lookup10(vz74, vz75, vz76, vz77, h) -> new_lookup(Main.Pos(Main.Succ(vz74)), vz77, h)
The graph contains the following edges 4 >= 2, 5 >= 3


*new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Pos(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(Main.Pos(Main.Succ(vz300)), Cons(Tup2(Main.Neg(vz4000), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Succ(vz300)), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(14)
YES

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

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup11(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Zero, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)
   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
   new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Pos(vz4000), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Succ(vz860), bb) -> new_lookup11(vz81, vz82, vz83, vz84, vz850, vz860, bb)
   new_lookup11(vz81, vz82, vz83, vz84, Main.Zero, Main.Succ(vz860), bb) -> new_lookup12(vz81, vz82, vz83, vz84, bb)
   new_lookup12(vz81, vz82, vz83, vz84, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)

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

(16) 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_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Zero, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)
The graph contains the following edges 4 >= 2, 7 >= 3


*new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup11(vz300, vz40000, vz401, vz41, vz300, vz40000, ba)
The graph contains the following edges 1 > 1, 2 > 2, 2 > 3, 2 > 4, 1 > 5, 2 > 6, 3 >= 7


*new_lookup11(vz81, vz82, vz83, vz84, Main.Succ(vz850), Main.Succ(vz860), bb) -> new_lookup11(vz81, vz82, vz83, vz84, vz850, vz860, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 > 5, 6 > 6, 7 >= 7


*new_lookup11(vz81, vz82, vz83, vz84, Main.Zero, Main.Succ(vz860), bb) -> new_lookup12(vz81, vz82, vz83, vz84, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 7 >= 5


*new_lookup12(vz81, vz82, vz83, vz84, bb) -> new_lookup(Main.Neg(Main.Succ(vz81)), vz84, bb)
The graph contains the following edges 4 >= 2, 5 >= 3


*new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Neg(Main.Zero), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(Main.Neg(Main.Succ(vz300)), Cons(Tup2(Main.Pos(vz4000), vz401), vz41), ba) -> new_lookup(Main.Neg(Main.Succ(vz300)), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(17)
YES

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

(18)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)
   new_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)

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

(19) 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_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Neg(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


*new_lookup(Main.Pos(Main.Zero), Cons(Tup2(Main.Pos(Main.Succ(vz40000)), vz401), vz41), ba) -> new_lookup(Main.Pos(Main.Zero), vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3


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

(20)
YES