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

YES 0.664 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

((scanr :: (a -> b -> b) -> b -> [a] -> [b]) :: (a -> b -> b) -> b -> [a] -> [b])

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\qs→qs

is transformed to

qs0 qs = qs

The following Lambda expression

\(q : _)→q

is transformed to

q1 (q : _) = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

((scanr :: (a -> b -> b) -> b -> [a] -> [b]) :: (a -> b -> b) -> b -> [a] -> [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

((scanr :: (b -> a -> a) -> a -> [b] -> [a]) :: (b -> a -> a) -> 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

              ↳ LetRed

mainModule Main

((scanr :: (b -> a -> a) -> a -> [b] -> [a]) :: (b -> a -> a) -> a -> [b] -> [a])

module Main where

import qualified Prelude

Let/Where Reductions:
The bindings of the following Let/Where expression

f x q : qs

where

q = q1 vu40

q1 (q : vv) = q

qs = qs0 vu40

qs0 qs = qs

vu40 = scanr f q0 xs

are unpacked to the following functions on top level

scanrVu40 vy vz wu = scanr vy vz wu

scanrQs0 vy vz wu qs = qs

scanrQs vy vz wu = scanrQs0 vy vz wu (scanrVu40 vy vz wu)

scanrQ vy vz wu = scanrQ1 vy vz wu (scanrVu40 vy vz wu)

scanrQ1 vy vz wu (q : vv) = q

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

(scanr :: (a -> b -> b) -> b -> [a] -> [b])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanr",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanr wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanr wv3 wv4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="scanr wv3 wv4 wv5",fontsize=16,color="burlywood",shape="triangle"];27[label="wv5/wv50 : wv51",fontsize=10,color="white",style="solid",shape="box"];5 -> 27[label="",style="solid", color="burlywood", weight=9];
27 -> 6[label="",style="solid", color="burlywood", weight=3];
28[label="wv5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 28[label="",style="solid", color="burlywood", weight=9];
28 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="scanr wv3 wv4 (wv50 : wv51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="scanr wv3 wv4 []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="wv3 wv50 (scanrQ wv3 wv4 wv51) : scanrQs wv3 wv4 wv51",fontsize=16,color="green",shape="box"];8 -> 10[label="",style="dashed", color="green", weight=3];
8 -> 11[label="",style="dashed", color="green", weight=3];
9[label="wv4 : []",fontsize=16,color="green",shape="box"];10[label="wv3 wv50 (scanrQ wv3 wv4 wv51)",fontsize=16,color="green",shape="box"];10 -> 12[label="",style="dashed", color="green", weight=3];
10 -> 13[label="",style="dashed", color="green", weight=3];
11[label="scanrQs wv3 wv4 wv51",fontsize=16,color="black",shape="box"];11 -> 14[label="",style="solid", color="black", weight=3];
12[label="wv50",fontsize=16,color="green",shape="box"];13[label="scanrQ wv3 wv4 wv51",fontsize=16,color="black",shape="box"];13 -> 15[label="",style="solid", color="black", weight=3];
14[label="scanrQs0 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="black",shape="box"];14 -> 16[label="",style="solid", color="black", weight=3];
15 -> 19[label="",style="dashed", color="red", weight=0];
15[label="scanrQ1 wv3 wv4 wv51 (scanrVu40 wv3 wv4 wv51)",fontsize=16,color="magenta"];15 -> 20[label="",style="dashed", color="magenta", weight=3];
16[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="black",shape="triangle"];16 -> 18[label="",style="solid", color="black", weight=3];
20 -> 16[label="",style="dashed", color="red", weight=0];
20[label="scanrVu40 wv3 wv4 wv51",fontsize=16,color="magenta"];19[label="scanrQ1 wv3 wv4 wv51 wv6",fontsize=16,color="burlywood",shape="triangle"];31[label="wv6/wv60 : wv61",fontsize=10,color="white",style="solid",shape="box"];19 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 22[label="",style="solid", color="burlywood", weight=3];
32[label="wv6/[]",fontsize=10,color="white",style="solid",shape="box"];19 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 23[label="",style="solid", color="burlywood", weight=3];
18 -> 5[label="",style="dashed", color="red", weight=0];
18[label="scanr wv3 wv4 wv51",fontsize=16,color="magenta"];18 -> 24[label="",style="dashed", color="magenta", weight=3];
22[label="scanrQ1 wv3 wv4 wv51 (wv60 : wv61)",fontsize=16,color="black",shape="box"];22 -> 25[label="",style="solid", color="black", weight=3];
23[label="scanrQ1 wv3 wv4 wv51 []",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="wv51",fontsize=16,color="green",shape="box"];25[label="wv60",fontsize=16,color="green",shape="box"];26[label="error []",fontsize=16,color="red",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_scanr(wv3, wv4, :(wv50, wv51), ba, bb) → new_scanr(wv3, wv4, wv51, ba, bb)
new_scanr(wv3, wv4, :(wv50, wv51), ba, bb) → new_scanrVu40(wv3, wv4, wv51, ba, bb)
new_scanrVu40(wv3, wv4, wv51, ba, bb) → new_scanr(wv3, wv4, wv51, 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_scanr(wv3, wv4, :(wv50, wv51), ba, bb) → new_scanr(wv3, wv4, wv51, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
new_scanr(wv3, wv4, :(wv50, wv51), ba, bb) → new_scanrVu40(wv3, wv4, wv51, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 > 3, 4 >= 4, 5 >= 5
new_scanrVu40(wv3, wv4, wv51, ba, bb) → new_scanr(wv3, wv4, wv51, ba, bb)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5