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

YES 0.644 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

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

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\(a,b)~(as,bs)→(a : as,b : bs)

is transformed to

unzip0 (a,b) ~(as,bs) = (a : as,b : bs)

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

mainModule Main

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

module Main where

import qualified Prelude

IrrPat Reductions:
The variables of the following irrefutable Pattern

~(as,bs)

are replaced by calls to these functions

unzip00 (as,bs) = as

unzip01 (as,bs) = bs

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

mainModule Main

((unzip :: [(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

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

mainModule Main

((unzip :: [(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

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="unzip",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="unzip vy3",fontsize=16,color="black",shape="triangle"];3 -> 4[label="",style="solid", color="black", weight=3];
4[label="foldr unzip0 ([],[]) vy3",fontsize=16,color="burlywood",shape="triangle"];20[label="vy3/vy30 : vy31",fontsize=10,color="white",style="solid",shape="box"];4 -> 20[label="",style="solid", color="burlywood", weight=9];
20 -> 5[label="",style="solid", color="burlywood", weight=3];
21[label="vy3/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 21[label="",style="solid", color="burlywood", weight=9];
21 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="foldr unzip0 ([],[]) (vy30 : vy31)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="foldr unzip0 ([],[]) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7 -> 9[label="",style="dashed", color="red", weight=0];
7[label="unzip0 vy30 (foldr unzip0 ([],[]) vy31)",fontsize=16,color="magenta"];7 -> 10[label="",style="dashed", color="magenta", weight=3];
8[label="([],[])",fontsize=16,color="green",shape="box"];10 -> 4[label="",style="dashed", color="red", weight=0];
10[label="foldr unzip0 ([],[]) vy31",fontsize=16,color="magenta"];10 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="unzip0 vy30 vy4",fontsize=16,color="burlywood",shape="triangle"];24[label="vy30/(vy300,vy301)",fontsize=10,color="white",style="solid",shape="box"];9 -> 24[label="",style="solid", color="burlywood", weight=9];
24 -> 12[label="",style="solid", color="burlywood", weight=3];
11[label="vy31",fontsize=16,color="green",shape="box"];12[label="unzip0 (vy300,vy301) vy4",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="(vy300 : unzip00 vy4,vy301 : unzip01 vy4)",fontsize=16,color="green",shape="box"];13 -> 14[label="",style="dashed", color="green", weight=3];
13 -> 15[label="",style="dashed", color="green", weight=3];
14[label="unzip00 vy4",fontsize=16,color="burlywood",shape="box"];25[label="vy4/(vy40,vy41)",fontsize=10,color="white",style="solid",shape="box"];14 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 16[label="",style="solid", color="burlywood", weight=3];
15[label="unzip01 vy4",fontsize=16,color="burlywood",shape="box"];26[label="vy4/(vy40,vy41)",fontsize=10,color="white",style="solid",shape="box"];15 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 17[label="",style="solid", color="burlywood", weight=3];
16[label="unzip00 (vy40,vy41)",fontsize=16,color="black",shape="box"];16 -> 18[label="",style="solid", color="black", weight=3];
17[label="unzip01 (vy40,vy41)",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="vy40",fontsize=16,color="green",shape="box"];19[label="vy41",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ IPR

        ↳ HASKELL

          ↳ BR

            ↳ HASKELL

              ↳ COR

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_foldr(:(vy30, vy31), ba, bb) → new_foldr(vy31, 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_foldr(:(vy30, vy31), ba, bb) → new_foldr(vy31, ba, bb)
The graph contains the following edges 1 > 1, 2 >= 2, 3 >= 3