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

YES 0.706 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/empty.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ LR

mainModule Main

((zip :: [b] -> [a] -> [(b,a)]) :: [b] -> [a] -> [(b,a)])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\ab→(a,b)

is transformed to

zip0 a b = (a,b)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((zip :: [b] -> [a] -> [(b,a)]) :: [b] -> [a] -> [(b,a)])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((zip :: [b] -> [a] -> [(b,a)]) :: [b] -> [a] -> [(b,a)])

module Main where

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

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule Main

(zip :: [b] -> [a] -> [(b,a)])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zip",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zip wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zip wu3 wu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="zipWith zip0 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];18[label="wu3/wu30 : wu31",fontsize=10,color="white",style="solid",shape="box"];5 -> 18[label="",style="solid", color="burlywood", weight=9];
18 -> 6[label="",style="solid", color="burlywood", weight=3];
19[label="wu3/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 19[label="",style="solid", color="burlywood", weight=9];
19 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="zipWith zip0 (wu30 : wu31) wu4",fontsize=16,color="burlywood",shape="box"];20[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];6 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 8[label="",style="solid", color="burlywood", weight=3];
21[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 9[label="",style="solid", color="burlywood", weight=3];
7[label="zipWith zip0 [] wu4",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="zipWith zip0 (wu30 : wu31) (wu40 : wu41)",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="zipWith zip0 (wu30 : wu31) []",fontsize=16,color="black",shape="box"];9 -> 12[label="",style="solid", color="black", weight=3];
10[label="[]",fontsize=16,color="green",shape="box"];11[label="zip0 wu30 wu40 : zipWith zip0 wu31 wu41",fontsize=16,color="green",shape="box"];11 -> 13[label="",style="dashed", color="green", weight=3];
11 -> 14[label="",style="dashed", color="green", weight=3];
12[label="[]",fontsize=16,color="green",shape="box"];13[label="zip0 wu30 wu40",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14 -> 5[label="",style="dashed", color="red", weight=0];
14[label="zipWith zip0 wu31 wu41",fontsize=16,color="magenta"];14 -> 16[label="",style="dashed", color="magenta", weight=3];
14 -> 17[label="",style="dashed", color="magenta", weight=3];
15[label="(wu30,wu40)",fontsize=16,color="green",shape="box"];16[label="wu31",fontsize=16,color="green",shape="box"];17[label="wu41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_zipWith(:(wu30, wu31), :(wu40, wu41), ba, bb) → new_zipWith(wu31, wu41, ba, bb)

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_zipWith(:(wu30, wu31), :(wu40, wu41), ba, bb) → new_zipWith(wu31, wu41, ba, bb)
The graph contains the following edges 1 > 1, 2 > 2, 3 >= 3, 4 >= 4