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

YES 0.8230000000000001 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

  ↳ IFR

mainModule List

((deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]) :: (a -> a -> Bool) -> a -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	_ _ []	=	[]
deleteBy	eq x (y : ys)	=	if x `eq` y then ys else y : deleteBy eq x ys

module Maybe where

import qualified List
import qualified Prelude

If Reductions:
The following If expression

if eq x y then ys else y : deleteBy eq x ys

is transformed to

deleteBy0 ys y eq x True = ys

deleteBy0 ys y eq x False = y : deleteBy eq x ys

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

mainModule List

((deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]) :: (a -> a -> Bool) -> a -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	_ _ []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

((deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]) :: (a -> a -> Bool) -> a -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	vw vx []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

module Maybe where

import qualified List
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

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

mainModule List

(deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a])

module List where

import qualified Maybe
import qualified Prelude

deleteBy :: (a -> a -> Bool) -> a -> [a] -> [a]

deleteBy	vw vx []	=	[]
deleteBy	eq x (y : ys)	=	deleteBy0 ys y eq x (x `eq` y)

deleteBy0	ys y eq x True	=	ys
deleteBy0	ys y eq x False	=	y : deleteBy eq x ys

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="deleteBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="deleteBy wu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="deleteBy wu3 wu4",fontsize=16,color="grey",shape="box"];4 -> 5[label="",style="dashed", color="grey", weight=3];
5[label="deleteBy wu3 wu4 wu5",fontsize=16,color="burlywood",shape="triangle"];22[label="wu5/wu50 : wu51",fontsize=10,color="white",style="solid",shape="box"];5 -> 22[label="",style="solid", color="burlywood", weight=9];
22 -> 6[label="",style="solid", color="burlywood", weight=3];
23[label="wu5/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 23[label="",style="solid", color="burlywood", weight=9];
23 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="deleteBy wu3 wu4 (wu50 : wu51)",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="deleteBy wu3 wu4 []",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="deleteBy0 wu51 wu50 wu3 wu4 (wu3 wu4 wu50)",fontsize=16,color="magenta"];8 -> 11[label="",style="dashed", color="magenta", weight=3];
9[label="[]",fontsize=16,color="green",shape="box"];11[label="wu3 wu4 wu50",fontsize=16,color="green",shape="box"];11 -> 16[label="",style="dashed", color="green", weight=3];
11 -> 17[label="",style="dashed", color="green", weight=3];
10[label="deleteBy0 wu51 wu50 wu3 wu4 wu6",fontsize=16,color="burlywood",shape="triangle"];25[label="wu6/False",fontsize=10,color="white",style="solid",shape="box"];10 -> 25[label="",style="solid", color="burlywood", weight=9];
25 -> 14[label="",style="solid", color="burlywood", weight=3];
26[label="wu6/True",fontsize=10,color="white",style="solid",shape="box"];10 -> 26[label="",style="solid", color="burlywood", weight=9];
26 -> 15[label="",style="solid", color="burlywood", weight=3];
16[label="wu4",fontsize=16,color="green",shape="box"];17[label="wu50",fontsize=16,color="green",shape="box"];14[label="deleteBy0 wu51 wu50 wu3 wu4 False",fontsize=16,color="black",shape="box"];14 -> 18[label="",style="solid", color="black", weight=3];
15[label="deleteBy0 wu51 wu50 wu3 wu4 True",fontsize=16,color="black",shape="box"];15 -> 19[label="",style="solid", color="black", weight=3];
18[label="wu50 : deleteBy wu3 wu4 wu51",fontsize=16,color="green",shape="box"];18 -> 20[label="",style="dashed", color="green", weight=3];
19[label="wu51",fontsize=16,color="green",shape="box"];20 -> 5[label="",style="dashed", color="red", weight=0];
20[label="deleteBy wu3 wu4 wu51",fontsize=16,color="magenta"];20 -> 21[label="",style="dashed", color="magenta", weight=3];
21[label="wu51",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ IFR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ QDPSizeChangeProof

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

new_deleteBy0(wu51, wu50, wu3, wu4, ba) → new_deleteBy(wu3, wu4, wu51, ba)
new_deleteBy(wu3, wu4, :(wu50, wu51), ba) → new_deleteBy0(wu51, wu50, wu3, wu4, 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_deleteBy(wu3, wu4, :(wu50, wu51), ba) → new_deleteBy0(wu51, wu50, wu3, wu4, ba)
The graph contains the following edges 3 > 1, 3 > 2, 1 >= 3, 2 >= 4, 4 >= 5
new_deleteBy0(wu51, wu50, wu3, wu4, ba) → new_deleteBy(wu3, wu4, wu51, ba)
The graph contains the following edges 3 >= 1, 4 >= 2, 1 >= 3, 5 >= 4

deleteBy0	ys y eq x True	= ys
deleteBy0	ys y eq x False	= y : deleteBy eq x ys