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

YES 0.8140000000000001 H-Termination proof of /home/matraf/haskell/eval_FullyBlown_Fast/List.hs
H-Termination of the given Haskell-Program with start terms could successfully be proven:

↳ HASKELL

  ↳ IPR

mainModule List

((partition :: (a -> Bool) -> [a] -> ([a],[a])) :: (a -> Bool) -> [a] -> ([a],[a]))

module List where

import qualified Maybe
import qualified Prelude

partition :: (a -> Bool) -> [a] -> ([a],[a])

partition

p xs

foldr (select p) ([],[]) xs

select

p x ~((ts,fs))

p x

(x : ts,fs)

otherwise

(ts,x : fs)

module Maybe where

import qualified List
import qualified Prelude

IrrPat Reductions:
The variables of the following irrefutable Pattern

~(ts,fs)

are replaced by calls to these functions

select0 (ts,fs) = ts

select1 (ts,fs) = fs

↳ HASKELL

  ↳ IPR

    ↳ HASKELL

      ↳ BR

mainModule List

((partition :: (a -> Bool) -> [a] -> ([a],[a])) :: (a -> Bool) -> [a] -> ([a],[a]))

module List where

import qualified Maybe
import qualified Prelude

partition :: (a -> Bool) -> [a] -> ([a],[a])

partition

p xs

foldr (select p) ([],[]) xs

select

p x vw

p x

(x : select0 vw,select1 vw)

otherwise

(select0 vw,x : select1 vw)

select0

(ts,fs)

select1

(ts,fs)

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ IPR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

((partition :: (a -> Bool) -> [a] -> ([a],[a])) :: (a -> Bool) -> [a] -> ([a],[a]))

module List where

import qualified Maybe
import qualified Prelude

partition :: (a -> Bool) -> [a] -> ([a],[a])

partition

p xs

foldr (select p) ([],[]) xs

select

p x vw

p x

(x : select0 vw,select1 vw)

otherwise

(select0 vw,x : select1 vw)

select0

(ts,fs)

select1

(ts,fs)

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

select p x vw

| p x

= (x : select0 vw,select1 vw)

| otherwise

= (select0 vw,x : select1 vw)

is transformed to

select p x vw = select4 p x vw

select2 p x vw True = (select0 vw,x : select1 vw)

select3 p x vw True = (x : select0 vw,select1 vw)

select3 p x vw False = select2 p x vw otherwise

select4 p x vw = select3 p x vw (p x)

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ IPR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule List

(partition :: (a -> Bool) -> [a] -> ([a],[a]))

module List where

import qualified Maybe
import qualified Prelude

partition :: (a -> Bool) -> [a] -> ([a],[a])

partition

p xs

foldr (select p) ([],[]) xs

select

p x vw

select4 p x vw

select0

(ts,fs)

select1

(ts,fs)

select2

p x vw True

(select0 vw,x : select1 vw)

select3	p x vw True	=	(x : select0 vw,select1 vw)
select3	p x vw False	=	select2 p x vw otherwise

select4

p x vw

select3 p x vw (p x)

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="partition",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="partition vz3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="partition vz3 vz4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="foldr (select vz3) ([],[]) vz4",fontsize=16,color="burlywood",shape="triangle"];33[label="vz4/vz40 : vz41",fontsize=10,color="white",style="solid",shape="box"];5 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 6[label="",style="solid", color="burlywood", weight=3];
34[label="vz4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 34[label="",style="solid", color="burlywood", weight=9];
34 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="foldr (select vz3) ([],[]) (vz40 : vz41)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldr (select vz3) ([],[]) []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8 -> 10[label="",style="dashed", color="red", weight=0];
8[label="select vz3 vz40 (foldr (select vz3) ([],[]) vz41)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="([],[])",fontsize=16,color="green",shape="box"];11 -> 5[label="",style="dashed", color="red", weight=0];
11[label="foldr (select vz3) ([],[]) vz41",fontsize=16,color="magenta"];11 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="select vz3 vz40 vz5",fontsize=16,color="black",shape="triangle"];10 -> 13[label="",style="solid", color="black", weight=3];
12[label="vz41",fontsize=16,color="green",shape="box"];13[label="select4 vz3 vz40 vz5",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14 -> 15[label="",style="dashed", color="red", weight=0];
14[label="select3 vz3 vz40 vz5 (vz3 vz40)",fontsize=16,color="magenta"];14 -> 16[label="",style="dashed", color="magenta", weight=3];
16[label="vz3 vz40",fontsize=16,color="green",shape="box"];16 -> 20[label="",style="dashed", color="green", weight=3];
15[label="select3 vz3 vz40 vz5 vz6",fontsize=16,color="burlywood",shape="triangle"];38[label="vz6/False",fontsize=10,color="white",style="solid",shape="box"];15 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 18[label="",style="solid", color="burlywood", weight=3];
39[label="vz6/True",fontsize=10,color="white",style="solid",shape="box"];15 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 19[label="",style="solid", color="burlywood", weight=3];
20[label="vz40",fontsize=16,color="green",shape="box"];18[label="select3 vz3 vz40 vz5 False",fontsize=16,color="black",shape="box"];18 -> 21[label="",style="solid", color="black", weight=3];
19[label="select3 vz3 vz40 vz5 True",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
21[label="select2 vz3 vz40 vz5 otherwise",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="(vz40 : select0 vz5,select1 vz5)",fontsize=16,color="green",shape="box"];22 -> 24[label="",style="dashed", color="green", weight=3];
22 -> 25[label="",style="dashed", color="green", weight=3];
23[label="select2 vz3 vz40 vz5 True",fontsize=16,color="black",shape="box"];23 -> 26[label="",style="solid", color="black", weight=3];
24[label="select0 vz5",fontsize=16,color="burlywood",shape="triangle"];40[label="vz5/(vz50,vz51)",fontsize=10,color="white",style="solid",shape="box"];24 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 27[label="",style="solid", color="burlywood", weight=3];
25[label="select1 vz5",fontsize=16,color="burlywood",shape="triangle"];41[label="vz5/(vz50,vz51)",fontsize=10,color="white",style="solid",shape="box"];25 -> 41[label="",style="solid", color="burlywood", weight=9];
41 -> 28[label="",style="solid", color="burlywood", weight=3];
26[label="(select0 vz5,vz40 : select1 vz5)",fontsize=16,color="green",shape="box"];26 -> 29[label="",style="dashed", color="green", weight=3];
26 -> 30[label="",style="dashed", color="green", weight=3];
27[label="select0 (vz50,vz51)",fontsize=16,color="black",shape="box"];27 -> 31[label="",style="solid", color="black", weight=3];
28[label="select1 (vz50,vz51)",fontsize=16,color="black",shape="box"];28 -> 32[label="",style="solid", color="black", weight=3];
29 -> 24[label="",style="dashed", color="red", weight=0];
29[label="select0 vz5",fontsize=16,color="magenta"];30 -> 25[label="",style="dashed", color="red", weight=0];
30[label="select1 vz5",fontsize=16,color="magenta"];31[label="vz50",fontsize=16,color="green",shape="box"];32[label="vz51",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IPR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_foldr(vz3, :(vz40, vz41), ba) → new_foldr(vz3, vz41, ba)

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(vz3, :(vz40, vz41), ba) → new_foldr(vz3, vz41, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3

select3	p x vw True	= (x : select0 vw,select1 vw)
select3	p x vw False	= select2 p x vw otherwise