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

YES 0.635 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

((zip3 :: [b] -> [a] -> [c] -> [(b,a,c)]) :: [b] -> [a] -> [c] -> [(b,a,c)])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\abc→(a,b,c)

is transformed to

zip30 a b c = (a,b,c)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((zip3 :: [a] -> [c] -> [b] -> [(a,c,b)]) :: [a] -> [c] -> [b] -> [(a,c,b)])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

((zip3 :: [c] -> [a] -> [b] -> [(c,a,b)]) :: [c] -> [a] -> [b] -> [(c,a,b)])

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

(zip3 :: [c] -> [a] -> [b] -> [(c,a,b)])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zip3",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zip3 wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zip3 wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="zip3 wv3 wv4 wv5",fontsize=16,color="black",shape="triangle"];5 -> 6[label="",style="solid", color="black", weight=3];
6[label="zipWith3 zip30 wv3 wv4 wv5",fontsize=16,color="burlywood",shape="triangle"];23[label="wv3/wv30 : wv31",fontsize=10,color="white",style="solid",shape="box"];6 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 7[label="",style="solid", color="burlywood", weight=3];
24[label="wv3/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="zipWith3 zip30 (wv30 : wv31) wv4 wv5",fontsize=16,color="burlywood",shape="box"];25[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];7 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 9[label="",style="solid", color="burlywood", weight=3];
26[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="zipWith3 zip30 [] wv4 wv5",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="zipWith3 zip30 (wv30 : wv31) (wv40 : wv41) wv5",fontsize=16,color="burlywood",shape="box"];27[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];9 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 12[label="",style="solid", color="burlywood", weight=3];
28[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="zipWith3 zip30 (wv30 : wv31) [] wv5",fontsize=16,color="black",shape="box"];10 -> 14[label="",style="solid", color="black", weight=3];
11[label="[]",fontsize=16,color="green",shape="box"];12[label="zipWith3 zip30 (wv30 : wv31) (wv40 : wv41) (wv50 : wv51)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="zipWith3 zip30 (wv30 : wv31) (wv40 : wv41) []",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="[]",fontsize=16,color="green",shape="box"];15[label="zip30 wv30 wv40 wv50 : zipWith3 zip30 wv31 wv41 wv51",fontsize=16,color="green",shape="box"];15 -> 17[label="",style="dashed", color="green", weight=3];
15 -> 18[label="",style="dashed", color="green", weight=3];
16[label="[]",fontsize=16,color="green",shape="box"];17[label="zip30 wv30 wv40 wv50",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18 -> 6[label="",style="dashed", color="red", weight=0];
18[label="zipWith3 zip30 wv31 wv41 wv51",fontsize=16,color="magenta"];18 -> 20[label="",style="dashed", color="magenta", weight=3];
18 -> 21[label="",style="dashed", color="magenta", weight=3];
18 -> 22[label="",style="dashed", color="magenta", weight=3];
19[label="(wv30,wv40,wv50)",fontsize=16,color="green",shape="box"];20[label="wv51",fontsize=16,color="green",shape="box"];21[label="wv31",fontsize=16,color="green",shape="box"];22[label="wv41",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_zipWith3(:(wv30, wv31), :(wv40, wv41), :(wv50, wv51), ba, bb, bc) → new_zipWith3(wv31, wv41, wv51, ba, bb, bc)

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_zipWith3(:(wv30, wv31), :(wv40, wv41), :(wv50, wv51), ba, bb, bc) → new_zipWith3(wv31, wv41, wv51, ba, bb, bc)
The graph contains the following edges 1 > 1, 2 > 2, 3 > 3, 4 >= 4, 5 >= 5, 6 >= 6