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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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


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

(0)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

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 b  ->  a  ->  a;
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;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

sumFloat :: List Float  ->  Float;
sumFloat = foldl' psFloat (fromIntFloat (Main.Pos Main.Zero));

}

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

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

(2)
Obligation:
mainModule Main 
module Main where {
import qualified Prelude;
data Float = Float MyInt MyInt ;

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' :: (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;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

sumFloat :: List Float  ->  Float;
sumFloat = foldl' psFloat (fromIntFloat (Main.Pos Main.Zero));

}

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

(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 Float = Float MyInt MyInt ;

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 b  ->  a  ->  a;
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;

fromIntFloat :: MyInt  ->  Float;
fromIntFloat = primIntToFloat;

primIntToFloat :: MyInt  ->  Float;
primIntToFloat x = Float x (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;

primMulInt :: MyInt  ->  MyInt  ->  MyInt;
primMulInt (Main.Pos x) (Main.Pos y) = Main.Pos (primMulNat x y);
primMulInt (Main.Pos x) (Main.Neg y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Pos y) = Main.Neg (primMulNat x y);
primMulInt (Main.Neg x) (Main.Neg y) = Main.Pos (primMulNat x y);

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

primPlusFloat :: Float  ->  Float  ->  Float;
primPlusFloat (Float x1 x2) (Float y1 y2) = Float (psMyInt x1 y1) (srMyInt x2 y2);

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));

psFloat :: Float  ->  Float  ->  Float;
psFloat = primPlusFloat;

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

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

srMyInt :: MyInt  ->  MyInt  ->  MyInt;
srMyInt = primMulInt;

sumFloat :: List Float  ->  Float;
sumFloat = foldl' psFloat (fromIntFloat (Main.Pos Main.Zero));

}

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

(5) Narrow (SOUND)
Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="sumFloat",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="sumFloat vx3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldl' psFloat (fromIntFloat (Pos Zero)) vx3",fontsize=16,color="burlywood",shape="box"];107[label="vx3/Cons vx30 vx31",fontsize=10,color="white",style="solid",shape="box"];4 -> 107[label="",style="solid", color="burlywood", weight=9];
107 -> 5[label="",style="solid", color="burlywood", weight=3];
108[label="vx3/Nil",fontsize=10,color="white",style="solid",shape="box"];4 -> 108[label="",style="solid", color="burlywood", weight=9];
108 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldl' psFloat (fromIntFloat (Pos Zero)) (Cons vx30 vx31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldl' psFloat (fromIntFloat (Pos Zero)) Nil",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="dsEm (foldl' psFloat) (psFloat (fromIntFloat (Pos Zero)) vx30) vx31",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="fromIntFloat (Pos Zero)",fontsize=16,color="black",shape="triangle"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="seq (psFloat (fromIntFloat (Pos Zero)) vx30) (foldl' psFloat (psFloat (fromIntFloat (Pos Zero)) vx30)) vx31",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
9 -> 13[label="",style="dashed", color="magenta", weight=3];
10[label="primIntToFloat (Pos Zero)",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
12 -> 8[label="",style="dashed", color="red", weight=0];
12[label="fromIntFloat (Pos Zero)",fontsize=16,color="magenta"];13 -> 8[label="",style="dashed", color="red", weight=0];
13[label="fromIntFloat (Pos Zero)",fontsize=16,color="magenta"];11[label="seq (psFloat vx4 vx30) (foldl' psFloat (psFloat vx5 vx30)) vx31",fontsize=16,color="black",shape="triangle"];11 -> 15[label="",style="solid", color="black", weight=3];
14[label="Float (Pos Zero) (Pos (Succ Zero))",fontsize=16,color="green",shape="box"];15[label="enforceWHNF (WHNF (psFloat vx4 vx30)) (foldl' psFloat (psFloat vx5 vx30)) vx31",fontsize=16,color="black",shape="box"];15 -> 16[label="",style="solid", color="black", weight=3];
16[label="foldl' psFloat (psFloat vx5 vx30) vx31",fontsize=16,color="burlywood",shape="box"];109[label="vx31/Cons vx310 vx311",fontsize=10,color="white",style="solid",shape="box"];16 -> 109[label="",style="solid", color="burlywood", weight=9];
109 -> 17[label="",style="solid", color="burlywood", weight=3];
110[label="vx31/Nil",fontsize=10,color="white",style="solid",shape="box"];16 -> 110[label="",style="solid", color="burlywood", weight=9];
110 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="foldl' psFloat (psFloat vx5 vx30) (Cons vx310 vx311)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="foldl' psFloat (psFloat vx5 vx30) Nil",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="dsEm (foldl' psFloat) (psFloat (psFloat vx5 vx30) vx310) vx311",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="psFloat vx5 vx30",fontsize=16,color="black",shape="triangle"];20 -> 22[label="",style="solid", color="black", weight=3];
21 -> 11[label="",style="dashed", color="red", weight=0];
21[label="seq (psFloat (psFloat vx5 vx30) vx310) (foldl' psFloat (psFloat (psFloat vx5 vx30) vx310)) vx311",fontsize=16,color="magenta"];21 -> 23[label="",style="dashed", color="magenta", weight=3];
21 -> 24[label="",style="dashed", color="magenta", weight=3];
21 -> 25[label="",style="dashed", color="magenta", weight=3];
21 -> 26[label="",style="dashed", color="magenta", weight=3];
22[label="primPlusFloat vx5 vx30",fontsize=16,color="burlywood",shape="box"];111[label="vx5/Float vx50 vx51",fontsize=10,color="white",style="solid",shape="box"];22 -> 111[label="",style="solid", color="burlywood", weight=9];
111 -> 27[label="",style="solid", color="burlywood", weight=3];
23 -> 20[label="",style="dashed", color="red", weight=0];
23[label="psFloat vx5 vx30",fontsize=16,color="magenta"];24 -> 20[label="",style="dashed", color="red", weight=0];
24[label="psFloat vx5 vx30",fontsize=16,color="magenta"];25[label="vx310",fontsize=16,color="green",shape="box"];26[label="vx311",fontsize=16,color="green",shape="box"];27[label="primPlusFloat (Float vx50 vx51) vx30",fontsize=16,color="burlywood",shape="box"];112[label="vx30/Float vx300 vx301",fontsize=10,color="white",style="solid",shape="box"];27 -> 112[label="",style="solid", color="burlywood", weight=9];
112 -> 28[label="",style="solid", color="burlywood", weight=3];
28[label="primPlusFloat (Float vx50 vx51) (Float vx300 vx301)",fontsize=16,color="black",shape="box"];28 -> 29[label="",style="solid", color="black", weight=3];
29[label="Float (psMyInt vx50 vx300) (srMyInt vx51 vx301)",fontsize=16,color="green",shape="box"];29 -> 30[label="",style="dashed", color="green", weight=3];
29 -> 31[label="",style="dashed", color="green", weight=3];
30[label="psMyInt vx50 vx300",fontsize=16,color="black",shape="box"];30 -> 32[label="",style="solid", color="black", weight=3];
31[label="srMyInt vx51 vx301",fontsize=16,color="black",shape="box"];31 -> 33[label="",style="solid", color="black", weight=3];
32[label="primPlusInt vx50 vx300",fontsize=16,color="burlywood",shape="box"];113[label="vx50/Pos vx500",fontsize=10,color="white",style="solid",shape="box"];32 -> 113[label="",style="solid", color="burlywood", weight=9];
113 -> 34[label="",style="solid", color="burlywood", weight=3];
114[label="vx50/Neg vx500",fontsize=10,color="white",style="solid",shape="box"];32 -> 114[label="",style="solid", color="burlywood", weight=9];
114 -> 35[label="",style="solid", color="burlywood", weight=3];
33[label="primMulInt vx51 vx301",fontsize=16,color="burlywood",shape="box"];115[label="vx51/Pos vx510",fontsize=10,color="white",style="solid",shape="box"];33 -> 115[label="",style="solid", color="burlywood", weight=9];
115 -> 36[label="",style="solid", color="burlywood", weight=3];
116[label="vx51/Neg vx510",fontsize=10,color="white",style="solid",shape="box"];33 -> 116[label="",style="solid", color="burlywood", weight=9];
116 -> 37[label="",style="solid", color="burlywood", weight=3];
34[label="primPlusInt (Pos vx500) vx300",fontsize=16,color="burlywood",shape="box"];117[label="vx300/Pos vx3000",fontsize=10,color="white",style="solid",shape="box"];34 -> 117[label="",style="solid", color="burlywood", weight=9];
117 -> 38[label="",style="solid", color="burlywood", weight=3];
118[label="vx300/Neg vx3000",fontsize=10,color="white",style="solid",shape="box"];34 -> 118[label="",style="solid", color="burlywood", weight=9];
118 -> 39[label="",style="solid", color="burlywood", weight=3];
35[label="primPlusInt (Neg vx500) vx300",fontsize=16,color="burlywood",shape="box"];119[label="vx300/Pos vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 119[label="",style="solid", color="burlywood", weight=9];
119 -> 40[label="",style="solid", color="burlywood", weight=3];
120[label="vx300/Neg vx3000",fontsize=10,color="white",style="solid",shape="box"];35 -> 120[label="",style="solid", color="burlywood", weight=9];
120 -> 41[label="",style="solid", color="burlywood", weight=3];
36[label="primMulInt (Pos vx510) vx301",fontsize=16,color="burlywood",shape="box"];121[label="vx301/Pos vx3010",fontsize=10,color="white",style="solid",shape="box"];36 -> 121[label="",style="solid", color="burlywood", weight=9];
121 -> 42[label="",style="solid", color="burlywood", weight=3];
122[label="vx301/Neg vx3010",fontsize=10,color="white",style="solid",shape="box"];36 -> 122[label="",style="solid", color="burlywood", weight=9];
122 -> 43[label="",style="solid", color="burlywood", weight=3];
37[label="primMulInt (Neg vx510) vx301",fontsize=16,color="burlywood",shape="box"];123[label="vx301/Pos vx3010",fontsize=10,color="white",style="solid",shape="box"];37 -> 123[label="",style="solid", color="burlywood", weight=9];
123 -> 44[label="",style="solid", color="burlywood", weight=3];
124[label="vx301/Neg vx3010",fontsize=10,color="white",style="solid",shape="box"];37 -> 124[label="",style="solid", color="burlywood", weight=9];
124 -> 45[label="",style="solid", color="burlywood", weight=3];
38[label="primPlusInt (Pos vx500) (Pos vx3000)",fontsize=16,color="black",shape="box"];38 -> 46[label="",style="solid", color="black", weight=3];
39[label="primPlusInt (Pos vx500) (Neg vx3000)",fontsize=16,color="black",shape="box"];39 -> 47[label="",style="solid", color="black", weight=3];
40[label="primPlusInt (Neg vx500) (Pos vx3000)",fontsize=16,color="black",shape="box"];40 -> 48[label="",style="solid", color="black", weight=3];
41[label="primPlusInt (Neg vx500) (Neg vx3000)",fontsize=16,color="black",shape="box"];41 -> 49[label="",style="solid", color="black", weight=3];
42[label="primMulInt (Pos vx510) (Pos vx3010)",fontsize=16,color="black",shape="box"];42 -> 50[label="",style="solid", color="black", weight=3];
43[label="primMulInt (Pos vx510) (Neg vx3010)",fontsize=16,color="black",shape="box"];43 -> 51[label="",style="solid", color="black", weight=3];
44[label="primMulInt (Neg vx510) (Pos vx3010)",fontsize=16,color="black",shape="box"];44 -> 52[label="",style="solid", color="black", weight=3];
45[label="primMulInt (Neg vx510) (Neg vx3010)",fontsize=16,color="black",shape="box"];45 -> 53[label="",style="solid", color="black", weight=3];
46[label="Pos (primPlusNat vx500 vx3000)",fontsize=16,color="green",shape="box"];46 -> 54[label="",style="dashed", color="green", weight=3];
47[label="primMinusNat vx500 vx3000",fontsize=16,color="burlywood",shape="triangle"];125[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];47 -> 125[label="",style="solid", color="burlywood", weight=9];
125 -> 55[label="",style="solid", color="burlywood", weight=3];
126[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];47 -> 126[label="",style="solid", color="burlywood", weight=9];
126 -> 56[label="",style="solid", color="burlywood", weight=3];
48 -> 47[label="",style="dashed", color="red", weight=0];
48[label="primMinusNat vx3000 vx500",fontsize=16,color="magenta"];48 -> 57[label="",style="dashed", color="magenta", weight=3];
48 -> 58[label="",style="dashed", color="magenta", weight=3];
49[label="Neg (primPlusNat vx500 vx3000)",fontsize=16,color="green",shape="box"];49 -> 59[label="",style="dashed", color="green", weight=3];
50[label="Pos (primMulNat vx510 vx3010)",fontsize=16,color="green",shape="box"];50 -> 60[label="",style="dashed", color="green", weight=3];
51[label="Neg (primMulNat vx510 vx3010)",fontsize=16,color="green",shape="box"];51 -> 61[label="",style="dashed", color="green", weight=3];
52[label="Neg (primMulNat vx510 vx3010)",fontsize=16,color="green",shape="box"];52 -> 62[label="",style="dashed", color="green", weight=3];
53[label="Pos (primMulNat vx510 vx3010)",fontsize=16,color="green",shape="box"];53 -> 63[label="",style="dashed", color="green", weight=3];
54[label="primPlusNat vx500 vx3000",fontsize=16,color="burlywood",shape="triangle"];127[label="vx500/Succ vx5000",fontsize=10,color="white",style="solid",shape="box"];54 -> 127[label="",style="solid", color="burlywood", weight=9];
127 -> 64[label="",style="solid", color="burlywood", weight=3];
128[label="vx500/Zero",fontsize=10,color="white",style="solid",shape="box"];54 -> 128[label="",style="solid", color="burlywood", weight=9];
128 -> 65[label="",style="solid", color="burlywood", weight=3];
55[label="primMinusNat (Succ vx5000) vx3000",fontsize=16,color="burlywood",shape="box"];129[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];55 -> 129[label="",style="solid", color="burlywood", weight=9];
129 -> 66[label="",style="solid", color="burlywood", weight=3];
130[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];55 -> 130[label="",style="solid", color="burlywood", weight=9];
130 -> 67[label="",style="solid", color="burlywood", weight=3];
56[label="primMinusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];131[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];56 -> 131[label="",style="solid", color="burlywood", weight=9];
131 -> 68[label="",style="solid", color="burlywood", weight=3];
132[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];56 -> 132[label="",style="solid", color="burlywood", weight=9];
132 -> 69[label="",style="solid", color="burlywood", weight=3];
57[label="vx500",fontsize=16,color="green",shape="box"];58[label="vx3000",fontsize=16,color="green",shape="box"];59 -> 54[label="",style="dashed", color="red", weight=0];
59[label="primPlusNat vx500 vx3000",fontsize=16,color="magenta"];59 -> 70[label="",style="dashed", color="magenta", weight=3];
59 -> 71[label="",style="dashed", color="magenta", weight=3];
60[label="primMulNat vx510 vx3010",fontsize=16,color="burlywood",shape="triangle"];133[label="vx510/Succ vx5100",fontsize=10,color="white",style="solid",shape="box"];60 -> 133[label="",style="solid", color="burlywood", weight=9];
133 -> 72[label="",style="solid", color="burlywood", weight=3];
134[label="vx510/Zero",fontsize=10,color="white",style="solid",shape="box"];60 -> 134[label="",style="solid", color="burlywood", weight=9];
134 -> 73[label="",style="solid", color="burlywood", weight=3];
61 -> 60[label="",style="dashed", color="red", weight=0];
61[label="primMulNat vx510 vx3010",fontsize=16,color="magenta"];61 -> 74[label="",style="dashed", color="magenta", weight=3];
62 -> 60[label="",style="dashed", color="red", weight=0];
62[label="primMulNat vx510 vx3010",fontsize=16,color="magenta"];62 -> 75[label="",style="dashed", color="magenta", weight=3];
63 -> 60[label="",style="dashed", color="red", weight=0];
63[label="primMulNat vx510 vx3010",fontsize=16,color="magenta"];63 -> 76[label="",style="dashed", color="magenta", weight=3];
63 -> 77[label="",style="dashed", color="magenta", weight=3];
64[label="primPlusNat (Succ vx5000) vx3000",fontsize=16,color="burlywood",shape="box"];135[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];64 -> 135[label="",style="solid", color="burlywood", weight=9];
135 -> 78[label="",style="solid", color="burlywood", weight=3];
136[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];64 -> 136[label="",style="solid", color="burlywood", weight=9];
136 -> 79[label="",style="solid", color="burlywood", weight=3];
65[label="primPlusNat Zero vx3000",fontsize=16,color="burlywood",shape="box"];137[label="vx3000/Succ vx30000",fontsize=10,color="white",style="solid",shape="box"];65 -> 137[label="",style="solid", color="burlywood", weight=9];
137 -> 80[label="",style="solid", color="burlywood", weight=3];
138[label="vx3000/Zero",fontsize=10,color="white",style="solid",shape="box"];65 -> 138[label="",style="solid", color="burlywood", weight=9];
138 -> 81[label="",style="solid", color="burlywood", weight=3];
66[label="primMinusNat (Succ vx5000) (Succ vx30000)",fontsize=16,color="black",shape="box"];66 -> 82[label="",style="solid", color="black", weight=3];
67[label="primMinusNat (Succ vx5000) Zero",fontsize=16,color="black",shape="box"];67 -> 83[label="",style="solid", color="black", weight=3];
68[label="primMinusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];68 -> 84[label="",style="solid", color="black", weight=3];
69[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];69 -> 85[label="",style="solid", color="black", weight=3];
70[label="vx500",fontsize=16,color="green",shape="box"];71[label="vx3000",fontsize=16,color="green",shape="box"];72[label="primMulNat (Succ vx5100) vx3010",fontsize=16,color="burlywood",shape="box"];139[label="vx3010/Succ vx30100",fontsize=10,color="white",style="solid",shape="box"];72 -> 139[label="",style="solid", color="burlywood", weight=9];
139 -> 86[label="",style="solid", color="burlywood", weight=3];
140[label="vx3010/Zero",fontsize=10,color="white",style="solid",shape="box"];72 -> 140[label="",style="solid", color="burlywood", weight=9];
140 -> 87[label="",style="solid", color="burlywood", weight=3];
73[label="primMulNat Zero vx3010",fontsize=16,color="burlywood",shape="box"];141[label="vx3010/Succ vx30100",fontsize=10,color="white",style="solid",shape="box"];73 -> 141[label="",style="solid", color="burlywood", weight=9];
141 -> 88[label="",style="solid", color="burlywood", weight=3];
142[label="vx3010/Zero",fontsize=10,color="white",style="solid",shape="box"];73 -> 142[label="",style="solid", color="burlywood", weight=9];
142 -> 89[label="",style="solid", color="burlywood", weight=3];
74[label="vx3010",fontsize=16,color="green",shape="box"];75[label="vx510",fontsize=16,color="green",shape="box"];76[label="vx510",fontsize=16,color="green",shape="box"];77[label="vx3010",fontsize=16,color="green",shape="box"];78[label="primPlusNat (Succ vx5000) (Succ vx30000)",fontsize=16,color="black",shape="box"];78 -> 90[label="",style="solid", color="black", weight=3];
79[label="primPlusNat (Succ vx5000) Zero",fontsize=16,color="black",shape="box"];79 -> 91[label="",style="solid", color="black", weight=3];
80[label="primPlusNat Zero (Succ vx30000)",fontsize=16,color="black",shape="box"];80 -> 92[label="",style="solid", color="black", weight=3];
81[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];81 -> 93[label="",style="solid", color="black", weight=3];
82 -> 47[label="",style="dashed", color="red", weight=0];
82[label="primMinusNat vx5000 vx30000",fontsize=16,color="magenta"];82 -> 94[label="",style="dashed", color="magenta", weight=3];
82 -> 95[label="",style="dashed", color="magenta", weight=3];
83[label="Pos (Succ vx5000)",fontsize=16,color="green",shape="box"];84[label="Neg (Succ vx30000)",fontsize=16,color="green",shape="box"];85[label="Pos Zero",fontsize=16,color="green",shape="box"];86[label="primMulNat (Succ vx5100) (Succ vx30100)",fontsize=16,color="black",shape="box"];86 -> 96[label="",style="solid", color="black", weight=3];
87[label="primMulNat (Succ vx5100) Zero",fontsize=16,color="black",shape="box"];87 -> 97[label="",style="solid", color="black", weight=3];
88[label="primMulNat Zero (Succ vx30100)",fontsize=16,color="black",shape="box"];88 -> 98[label="",style="solid", color="black", weight=3];
89[label="primMulNat Zero Zero",fontsize=16,color="black",shape="box"];89 -> 99[label="",style="solid", color="black", weight=3];
90[label="Succ (Succ (primPlusNat vx5000 vx30000))",fontsize=16,color="green",shape="box"];90 -> 100[label="",style="dashed", color="green", weight=3];
91[label="Succ vx5000",fontsize=16,color="green",shape="box"];92[label="Succ vx30000",fontsize=16,color="green",shape="box"];93[label="Zero",fontsize=16,color="green",shape="box"];94[label="vx30000",fontsize=16,color="green",shape="box"];95[label="vx5000",fontsize=16,color="green",shape="box"];96 -> 54[label="",style="dashed", color="red", weight=0];
96[label="primPlusNat (primMulNat vx5100 (Succ vx30100)) (Succ vx30100)",fontsize=16,color="magenta"];96 -> 101[label="",style="dashed", color="magenta", weight=3];
96 -> 102[label="",style="dashed", color="magenta", weight=3];
97[label="Zero",fontsize=16,color="green",shape="box"];98[label="Zero",fontsize=16,color="green",shape="box"];99[label="Zero",fontsize=16,color="green",shape="box"];100 -> 54[label="",style="dashed", color="red", weight=0];
100[label="primPlusNat vx5000 vx30000",fontsize=16,color="magenta"];100 -> 103[label="",style="dashed", color="magenta", weight=3];
100 -> 104[label="",style="dashed", color="magenta", weight=3];
101 -> 60[label="",style="dashed", color="red", weight=0];
101[label="primMulNat vx5100 (Succ vx30100)",fontsize=16,color="magenta"];101 -> 105[label="",style="dashed", color="magenta", weight=3];
101 -> 106[label="",style="dashed", color="magenta", weight=3];
102[label="Succ vx30100",fontsize=16,color="green",shape="box"];103[label="vx5000",fontsize=16,color="green",shape="box"];104[label="vx30000",fontsize=16,color="green",shape="box"];105[label="vx5100",fontsize=16,color="green",shape="box"];106[label="Succ vx30100",fontsize=16,color="green",shape="box"];}

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

(6)
Complex Obligation (AND)

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

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

   new_primMulNat(Main.Succ(vx5100), Main.Succ(vx30100)) -> new_primMulNat(vx5100, Main.Succ(vx30100))

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

(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_primMulNat(Main.Succ(vx5100), Main.Succ(vx30100)) -> new_primMulNat(vx5100, Main.Succ(vx30100))
The graph contains the following edges 1 > 1, 2 >= 2


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

(9)
YES

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

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

   new_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_psFloat(vx5, vx30), vx310, new_psFloat(vx5, vx30), vx311)

The TRS R consists of the following rules:

   new_primMinusNat0(Main.Zero, Main.Zero) -> Main.Pos(Main.Zero)
   new_primPlusInt(Main.Pos(vx500), Main.Pos(vx3000)) -> Main.Pos(new_primPlusNat0(vx500, vx3000))
   new_primMulInt(Main.Neg(vx510), Main.Neg(vx3010)) -> Main.Pos(new_primMulNat0(vx510, vx3010))
   new_primMinusNat0(Main.Succ(vx5000), Main.Succ(vx30000)) -> new_primMinusNat0(vx5000, vx30000)
   new_primMulInt(Main.Pos(vx510), Main.Pos(vx3010)) -> Main.Pos(new_primMulNat0(vx510, vx3010))
   new_primMulNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_psFloat(Float(vx50, vx51), Float(vx300, vx301)) -> Float(new_primPlusInt(vx50, vx300), new_primMulInt(vx51, vx301))
   new_primPlusNat0(Main.Succ(vx5000), Main.Succ(vx30000)) -> Main.Succ(Main.Succ(new_primPlusNat0(vx5000, vx30000)))
   new_primPlusNat0(Main.Zero, Main.Zero) -> Main.Zero
   new_primMinusNat0(Main.Zero, Main.Succ(vx30000)) -> Main.Neg(Main.Succ(vx30000))
   new_primPlusNat0(Main.Succ(vx5000), Main.Zero) -> Main.Succ(vx5000)
   new_primPlusNat0(Main.Zero, Main.Succ(vx30000)) -> Main.Succ(vx30000)
   new_primMulNat0(Main.Succ(vx5100), Main.Zero) -> Main.Zero
   new_primMulNat0(Main.Zero, Main.Succ(vx30100)) -> Main.Zero
   new_primPlusInt(Main.Pos(vx500), Main.Neg(vx3000)) -> new_primMinusNat0(vx500, vx3000)
   new_primPlusInt(Main.Neg(vx500), Main.Pos(vx3000)) -> new_primMinusNat0(vx3000, vx500)
   new_primMulInt(Main.Pos(vx510), Main.Neg(vx3010)) -> Main.Neg(new_primMulNat0(vx510, vx3010))
   new_primMulInt(Main.Neg(vx510), Main.Pos(vx3010)) -> Main.Neg(new_primMulNat0(vx510, vx3010))
   new_primPlusInt(Main.Neg(vx500), Main.Neg(vx3000)) -> Main.Neg(new_primPlusNat0(vx500, vx3000))
   new_primMinusNat0(Main.Succ(vx5000), Main.Zero) -> Main.Pos(Main.Succ(vx5000))
   new_primMulNat0(Main.Succ(vx5100), Main.Succ(vx30100)) -> new_primPlusNat0(new_primMulNat0(vx5100, Main.Succ(vx30100)), Main.Succ(vx30100))

The set Q consists of the following terms:

   new_primMulNat0(Main.Zero, Main.Succ(x0))
   new_primMulNat0(Main.Succ(x0), Main.Zero)
   new_primMulInt(Main.Neg(x0), Main.Neg(x1))
   new_primPlusNat0(Main.Zero, Main.Zero)
   new_primPlusNat0(Main.Zero, Main.Succ(x0))
   new_primPlusInt(Main.Neg(x0), Main.Neg(x1))
   new_primMulInt(Main.Pos(x0), Main.Pos(x1))
   new_primPlusNat0(Main.Succ(x0), Main.Zero)
   new_primPlusInt(Main.Pos(x0), Main.Pos(x1))
   new_primPlusNat0(Main.Succ(x0), Main.Succ(x1))
   new_primMulInt(Main.Pos(x0), Main.Neg(x1))
   new_primMulInt(Main.Neg(x0), Main.Pos(x1))
   new_primMulNat0(Main.Succ(x0), Main.Succ(x1))
   new_primPlusInt(Main.Pos(x0), Main.Neg(x1))
   new_primPlusInt(Main.Neg(x0), Main.Pos(x1))
   new_primMinusNat0(Main.Zero, Main.Zero)
   new_primMinusNat0(Main.Succ(x0), Main.Zero)
   new_primMinusNat0(Main.Succ(x0), Main.Succ(x1))
   new_primMinusNat0(Main.Zero, Main.Succ(x0))
   new_psFloat(Float(x0, x1), Float(x2, x3))
   new_primMulNat0(Main.Zero, Main.Zero)

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_seq(vx4, vx30, vx5, Cons(vx310, vx311)) -> new_seq(new_psFloat(vx5, vx30), vx310, new_psFloat(vx5, vx30), vx311)
The graph contains the following edges 4 > 2, 4 > 4


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

(12)
YES

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

(13)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primMinusNat(Main.Succ(vx5000), Main.Succ(vx30000)) -> new_primMinusNat(vx5000, vx30000)

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

(14) 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_primMinusNat(Main.Succ(vx5000), Main.Succ(vx30000)) -> new_primMinusNat(vx5000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2


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

(15)
YES

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

(16)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   new_primPlusNat(Main.Succ(vx5000), Main.Succ(vx30000)) -> new_primPlusNat(vx5000, vx30000)

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

(17) 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_primPlusNat(Main.Succ(vx5000), Main.Succ(vx30000)) -> new_primPlusNat(vx5000, vx30000)
The graph contains the following edges 1 > 1, 2 > 2


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

(18)
YES