H-Termination proof of /home/matraf/haskell/eval_FullyBlown

YES 0.846 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ BR

mainModule List

((genericLength :: [a] -> Int) :: [a] -> Int)

module List where

import qualified Maybe
import qualified Prelude

genericLength :: Num b => [a] -> b

genericLength	[]	=	0
genericLength	(_ : l)	=	1 + genericLength l

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

((genericLength :: [a] -> Int) :: [a] -> Int)

module List where

import qualified Maybe
import qualified Prelude

genericLength :: Num b => [a] -> b

genericLength	[]	=	0
genericLength	(vw : l)	=	1 + genericLength l

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

mainModule List

((genericLength :: [a] -> Int) :: [a] -> Int)

module List where

import qualified Maybe
import qualified Prelude

genericLength :: Num b => [a] -> b

genericLength	[]	=	0
genericLength	(vw : l)	=	1 + genericLength l

module Maybe where

import qualified List
import qualified Prelude

Num Reduction: All numbers are transformed to thier corresponding representation with Pos, Neg, Succ and Zero.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

mainModule List

(genericLength :: [a] -> Int)

module List where

import qualified Maybe
import qualified Prelude

genericLength :: Num a => [b] -> a

genericLength	[]	=	fromInt (Pos Zero)
genericLength	(vw : l)	=	fromInt (Pos (Succ Zero)) + genericLength l

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="genericLength",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="genericLength vz3",fontsize=16,color="burlywood",shape="triangle"];36[label="vz3/vz30 : vz31",fontsize=10,color="white",style="solid",shape="box"];3 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 4[label="",style="solid", color="burlywood", weight=3];
37[label="vz3/[]",fontsize=10,color="white",style="solid",shape="box"];3 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 5[label="",style="solid", color="burlywood", weight=3];
4[label="genericLength (vz30 : vz31)",fontsize=16,color="black",shape="box"];4 -> 6[label="",style="solid", color="black", weight=3];
5[label="genericLength []",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6 -> 8[label="",style="dashed", color="red", weight=0];
6[label="fromInt (Pos (Succ Zero)) + genericLength vz31",fontsize=16,color="magenta"];6 -> 9[label="",style="dashed", color="magenta", weight=3];
7[label="fromInt (Pos Zero)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
9 -> 3[label="",style="dashed", color="red", weight=0];
9[label="genericLength vz31",fontsize=16,color="magenta"];9 -> 11[label="",style="dashed", color="magenta", weight=3];
8[label="fromInt (Pos (Succ Zero)) + vz4",fontsize=16,color="black",shape="triangle"];8 -> 12[label="",style="solid", color="black", weight=3];
10[label="Pos Zero",fontsize=16,color="green",shape="box"];11[label="vz31",fontsize=16,color="green",shape="box"];12[label="primPlusInt (fromInt (Pos (Succ Zero))) vz4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="primPlusInt (Pos (Succ Zero)) vz4",fontsize=16,color="burlywood",shape="box"];40[label="vz4/Pos vz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 14[label="",style="solid", color="burlywood", weight=3];
41[label="vz4/Neg vz40",fontsize=10,color="white",style="solid",shape="box"];13 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 15[label="",style="solid", color="burlywood", weight=3];
14[label="primPlusInt (Pos (Succ Zero)) (Pos vz40)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15[label="primPlusInt (Pos (Succ Zero)) (Neg vz40)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="Pos (primPlusNat (Succ Zero) vz40)",fontsize=16,color="green",shape="box"];16 -> 18[label="",style="dashed", color="green", weight=3];
17[label="primMinusNat (Succ Zero) vz40",fontsize=16,color="burlywood",shape="box"];42[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];17 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 19[label="",style="solid", color="burlywood", weight=3];
43[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];17 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 20[label="",style="solid", color="burlywood", weight=3];
18[label="primPlusNat (Succ Zero) vz40",fontsize=16,color="burlywood",shape="box"];44[label="vz40/Succ vz400",fontsize=10,color="white",style="solid",shape="box"];18 -> 44[label="",style="solid", color="burlywood", weight=9];
44 -> 21[label="",style="solid", color="burlywood", weight=3];
45[label="vz40/Zero",fontsize=10,color="white",style="solid",shape="box"];18 -> 45[label="",style="solid", color="burlywood", weight=9];
45 -> 22[label="",style="solid", color="burlywood", weight=3];
19[label="primMinusNat (Succ Zero) (Succ vz400)",fontsize=16,color="black",shape="box"];19 -> 23[label="",style="solid", color="black", weight=3];
20[label="primMinusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];20 -> 24[label="",style="solid", color="black", weight=3];
21[label="primPlusNat (Succ Zero) (Succ vz400)",fontsize=16,color="black",shape="box"];21 -> 25[label="",style="solid", color="black", weight=3];
22[label="primPlusNat (Succ Zero) Zero",fontsize=16,color="black",shape="box"];22 -> 26[label="",style="solid", color="black", weight=3];
23[label="primMinusNat Zero vz400",fontsize=16,color="burlywood",shape="box"];46[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];23 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 27[label="",style="solid", color="burlywood", weight=3];
47[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];23 -> 47[label="",style="solid", color="burlywood", weight=9];
47 -> 28[label="",style="solid", color="burlywood", weight=3];
24[label="Pos (Succ Zero)",fontsize=16,color="green",shape="box"];25[label="Succ (Succ (primPlusNat Zero vz400))",fontsize=16,color="green",shape="box"];25 -> 29[label="",style="dashed", color="green", weight=3];
26[label="Succ Zero",fontsize=16,color="green",shape="box"];27[label="primMinusNat Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];27 -> 30[label="",style="solid", color="black", weight=3];
28[label="primMinusNat Zero Zero",fontsize=16,color="black",shape="box"];28 -> 31[label="",style="solid", color="black", weight=3];
29[label="primPlusNat Zero vz400",fontsize=16,color="burlywood",shape="box"];48[label="vz400/Succ vz4000",fontsize=10,color="white",style="solid",shape="box"];29 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 32[label="",style="solid", color="burlywood", weight=3];
49[label="vz400/Zero",fontsize=10,color="white",style="solid",shape="box"];29 -> 49[label="",style="solid", color="burlywood", weight=9];
49 -> 33[label="",style="solid", color="burlywood", weight=3];
30[label="Neg (Succ vz4000)",fontsize=16,color="green",shape="box"];31[label="Pos Zero",fontsize=16,color="green",shape="box"];32[label="primPlusNat Zero (Succ vz4000)",fontsize=16,color="black",shape="box"];32 -> 34[label="",style="solid", color="black", weight=3];
33[label="primPlusNat Zero Zero",fontsize=16,color="black",shape="box"];33 -> 35[label="",style="solid", color="black", weight=3];
34[label="Succ vz4000",fontsize=16,color="green",shape="box"];35[label="Zero",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ NumRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

Q DP problem:
The TRS P consists of the following rules:

new_genericLength(:(vz30, vz31), ba) → new_genericLength(vz31, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the subterm criterion [20] together with the size-change analysis [32] 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_genericLength(:(vz30, vz31), ba) → new_genericLength(vz31, ba)
The graph contains the following edges 1 > 1, 2 >= 2