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

YES 1.27 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

((scanr1 :: (a -> a -> a) -> [a] -> [a]) :: (a -> a -> a) -> [a] -> [a])

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

((scanr1 :: (a -> a -> a) -> [a] -> [a]) :: (a -> a -> a) -> [a] -> [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

((scanr1 :: (a -> a -> a) -> [a] -> [a]) :: (a -> a -> a) -> [a] -> [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

((scanr1 :: (a -> a -> a) -> [a] -> [a]) :: (a -> a -> a) -> [a] -> [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 vu41

q1 (q : vv) = q

qs = qs0 vu41

qs0 qs = qs

vu41 = scanr1 f xs

are unpacked to the following functions on top level

scanr1Q vy vz = scanr1Q1 vy vz (scanr1Vu41 vy vz)

scanr1Qs0 vy vz qs = qs

scanr1Vu41 vy vz = scanr1 vy vz

scanr1Q1 vy vz (q : vv) = q

scanr1Qs vy vz = scanr1Qs0 vy vz (scanr1Vu41 vy vz)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

(scanr1 :: (a -> a -> a) -> [a] -> [a])

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="scanr1",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="scanr1 wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="scanr1 wu3 wu4",fontsize=16,color="burlywood",shape="triangle"];29[label="wu4/wu40 : wu41",fontsize=10,color="white",style="solid",shape="box"];4 -> 29[label="",style="solid", color="burlywood", weight=9];
29 -> 5[label="",style="solid", color="burlywood", weight=3];
30[label="wu4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="scanr1 wu3 (wu40 : wu41)",fontsize=16,color="burlywood",shape="box"];31[label="wu41/wu410 : wu411",fontsize=10,color="white",style="solid",shape="box"];5 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 7[label="",style="solid", color="burlywood", weight=3];
32[label="wu41/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 8[label="",style="solid", color="burlywood", weight=3];
6[label="scanr1 wu3 []",fontsize=16,color="black",shape="box"];6 -> 9[label="",style="solid", color="black", weight=3];
7[label="scanr1 wu3 (wu40 : wu410 : wu411)",fontsize=16,color="black",shape="box"];7 -> 10[label="",style="solid", color="black", weight=3];
8[label="scanr1 wu3 (wu40 : [])",fontsize=16,color="black",shape="box"];8 -> 11[label="",style="solid", color="black", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];10[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411)) : scanr1Qs wu3 (wu410 : wu411)",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="wu40 : []",fontsize=16,color="green",shape="box"];12[label="wu3 wu40 (scanr1Q wu3 (wu410 : wu411))",fontsize=16,color="green",shape="box"];12 -> 14[label="",style="dashed", color="green", weight=3];
12 -> 15[label="",style="dashed", color="green", weight=3];
13[label="scanr1Qs wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];13 -> 16[label="",style="solid", color="black", weight=3];
14[label="wu40",fontsize=16,color="green",shape="box"];15[label="scanr1Q wu3 (wu410 : wu411)",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="scanr1Qs0 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17 -> 21[label="",style="dashed", color="red", weight=0];
17[label="scanr1Q1 wu3 (wu410 : wu411) (scanr1Vu41 wu3 (wu410 : wu411))",fontsize=16,color="magenta"];17 -> 22[label="",style="dashed", color="magenta", weight=3];
18[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="black",shape="triangle"];18 -> 20[label="",style="solid", color="black", weight=3];
22 -> 18[label="",style="dashed", color="red", weight=0];
22[label="scanr1Vu41 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];21[label="scanr1Q1 wu3 (wu410 : wu411) wu5",fontsize=16,color="burlywood",shape="triangle"];35[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];21 -> 35[label="",style="solid", color="burlywood", weight=9];
35 -> 24[label="",style="solid", color="burlywood", weight=3];
36[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];21 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 25[label="",style="solid", color="burlywood", weight=3];
20 -> 4[label="",style="dashed", color="red", weight=0];
20[label="scanr1 wu3 (wu410 : wu411)",fontsize=16,color="magenta"];20 -> 26[label="",style="dashed", color="magenta", weight=3];
24[label="scanr1Q1 wu3 (wu410 : wu411) (wu50 : wu51)",fontsize=16,color="black",shape="box"];24 -> 27[label="",style="solid", color="black", weight=3];
25[label="scanr1Q1 wu3 (wu410 : wu411) []",fontsize=16,color="black",shape="box"];25 -> 28[label="",style="solid", color="black", weight=3];
26[label="wu410 : wu411",fontsize=16,color="green",shape="box"];27[label="wu50",fontsize=16,color="green",shape="box"];28[label="error []",fontsize=16,color="red",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ RuleRemovalProof

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

new_scanr1(wu3, :(wu40, :(wu410, wu411)), ba) → new_scanr1(wu3, :(wu410, wu411), ba)
new_scanr1Vu41(wu3, wu410, wu411, ba) → new_scanr1(wu3, :(wu410, wu411), ba)
new_scanr1(wu3, :(wu40, :(wu410, wu411)), ba) → new_scanr1Vu41(wu3, wu410, wu411, ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
By using the rule removal processor [15] with the following polynomial ordering [25], at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.
Strictly oriented dependency pairs:

new_scanr1(wu3, :(wu40, :(wu410, wu411)), ba) → new_scanr1(wu3, :(wu410, wu411), ba)
new_scanr1(wu3, :(wu40, :(wu410, wu411)), ba) → new_scanr1Vu41(wu3, wu410, wu411, ba)

Used ordering: POLO with Polynomial interpretation [25]:

POL(:(x₁, x₂)) = 2 + 2·x₁ + x₂
POL(new_scanr1(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(new_scanr1Vu41(x₁, x₂, x₃, x₄)) = 2 + x₁ + 2·x₂ + x₃ + x₄

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ RuleRemovalProof

                        ↳ QDP

                          ↳ DependencyGraphProof

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

new_scanr1Vu41(wu3, wu410, wu411, ba) → new_scanr1(wu3, :(wu410, wu411), ba)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
The approximation of the Dependency Graph [15,17,22] contains 0 SCCs with 1 less node.