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

YES 0.637 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

  ↳ BR

mainModule Main

((zipWith3 :: (b -> a -> d -> c) -> [b] -> [a] -> [d] -> [c]) :: (b -> a -> d -> c) -> [b] -> [a] -> [d] -> [c])

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule Main

((zipWith3 :: (d -> c -> a -> b) -> [d] -> [c] -> [a] -> [b]) :: (d -> c -> a -> b) -> [d] -> [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

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

mainModule Main

(zipWith3 :: (c -> a -> d -> b) -> [c] -> [a] -> [d] -> [b])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="zipWith3",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="zipWith3 wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="zipWith3 wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="zipWith3 wv3 wv4 wv5",fontsize=16,color="grey",shape="box"];5 -> 6[label="",style="dashed", color="grey", weight=3];
6[label="zipWith3 wv3 wv4 wv5 wv6",fontsize=16,color="burlywood",shape="triangle"];25[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];6 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 7[label="",style="solid", color="burlywood", weight=3];
26[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];6 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 8[label="",style="solid", color="burlywood", weight=3];
7[label="zipWith3 wv3 (wv40 : wv41) wv5 wv6",fontsize=16,color="burlywood",shape="box"];27[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];7 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 9[label="",style="solid", color="burlywood", weight=3];
28[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];7 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 10[label="",style="solid", color="burlywood", weight=3];
8[label="zipWith3 wv3 [] wv5 wv6",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="zipWith3 wv3 (wv40 : wv41) (wv50 : wv51) wv6",fontsize=16,color="burlywood",shape="box"];29[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];9 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 12[label="",style="solid", color="burlywood", weight=3];
30[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 13[label="",style="solid", color="burlywood", weight=3];
10[label="zipWith3 wv3 (wv40 : wv41) [] wv6",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 wv3 (wv40 : wv41) (wv50 : wv51) (wv60 : wv61)",fontsize=16,color="black",shape="box"];12 -> 15[label="",style="solid", color="black", weight=3];
13[label="zipWith3 wv3 (wv40 : wv41) (wv50 : wv51) []",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="wv3 wv40 wv50 wv60 : zipWith3 wv3 wv41 wv51 wv61",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="wv3 wv40 wv50 wv60",fontsize=16,color="green",shape="box"];17 -> 19[label="",style="dashed", color="green", weight=3];
17 -> 20[label="",style="dashed", color="green", weight=3];
17 -> 21[label="",style="dashed", color="green", weight=3];
18 -> 6[label="",style="dashed", color="red", weight=0];
18[label="zipWith3 wv3 wv41 wv51 wv61",fontsize=16,color="magenta"];18 -> 22[label="",style="dashed", color="magenta", weight=3];
18 -> 23[label="",style="dashed", color="magenta", weight=3];
18 -> 24[label="",style="dashed", color="magenta", weight=3];
19[label="wv40",fontsize=16,color="green",shape="box"];20[label="wv50",fontsize=16,color="green",shape="box"];21[label="wv60",fontsize=16,color="green",shape="box"];22[label="wv61",fontsize=16,color="green",shape="box"];23[label="wv51",fontsize=16,color="green",shape="box"];24[label="wv41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ Narrow

            ↳ QDP

              ↳ QDPSizeChangeProof

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

new_zipWith3(wv3, :(wv40, wv41), :(wv50, wv51), :(wv60, wv61), ba, bb, bc, bd) → new_zipWith3(wv3, wv41, wv51, wv61, ba, bb, bc, bd)

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(wv3, :(wv40, wv41), :(wv50, wv51), :(wv60, wv61), ba, bb, bc, bd) → new_zipWith3(wv3, wv41, wv51, wv61, ba, bb, bc, bd)
The graph contains the following edges 1 >= 1, 2 > 2, 3 > 3, 4 > 4, 5 >= 5, 6 >= 6, 7 >= 7, 8 >= 8