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

YES 1.157 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

  ↳ BR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

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

elem_by	_ _ []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

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

nubBy

eq l

nubBy' l []

where

nubBy'

[] _

[]

nubBy'

(y : ys) xs

elem_by eq y xs

nubBy' ys xs

otherwise

y : nubBy' ys (y : xs)

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

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

elem_by	vx vy []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

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

nubBy

eq l

nubBy' l []

where

nubBy'

[] vw

[]

nubBy'

(y : ys) xs

elem_by eq y xs

nubBy' ys xs

otherwise

y : nubBy' ys (y : xs)

module Maybe where

import qualified List
import qualified Prelude

Cond Reductions:
The following Function with conditions

nubBy' [] vw = []

nubBy' (y : ys) xs

| elem_by eq y xs

= nubBy' ys xs

| otherwise

= y : nubBy' ys (y : xs)

is transformed to

nubBy' [] vw = nubBy'3 [] vw

nubBy' (y : ys) xs = nubBy'2 (y : ys) xs

nubBy'1 y ys xs True = nubBy' ys xs

nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise

nubBy'0 y ys xs True = y : nubBy' ys (y : xs)

nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3 [] vw = []

nubBy'3 wx wy = nubBy'2 wx wy

The following Function with conditions

undefined

| False

= undefined

is transformed to

undefined = undefined1

undefined0 True = undefined

undefined1 = undefined0 False

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

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

elem_by	vx vy []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

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

nubBy

eq l

nubBy' l []

where

nubBy'	[] vw	=	nubBy'3 [] vw
nubBy'	(y : ys) xs	=	nubBy'2 (y : ys) xs

nubBy'0

y ys xs True

y : nubBy' ys (y : xs)

nubBy'1	y ys xs True	=	nubBy' ys xs
nubBy'1	y ys xs False	=	nubBy'0 y ys xs otherwise

nubBy'2

(y : ys) xs

nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3	[] vw	=	[]
nubBy'3	wx wy	=	nubBy'2 wx wy

module Maybe where

import qualified List
import qualified Prelude

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

nubBy' l []

where

nubBy' [] vw = nubBy'3 [] vw

nubBy' (y : ys) xs = nubBy'2 (y : ys) xs

nubBy'0 y ys xs True = y : nubBy' ys (y : xs)

nubBy'1 y ys xs True = nubBy' ys xs

nubBy'1 y ys xs False = nubBy'0 y ys xs otherwise

nubBy'2 (y : ys) xs = nubBy'1 y ys xs (elem_by eq y xs)

nubBy'3 [] vw = []

nubBy'3 wx wy = nubBy'2 wx wy

are unpacked to the following functions on top level

nubByNubBy'2 wz (y : ys) xs = nubByNubBy'1 wz y ys xs (elem_by wz y xs)

nubByNubBy'1 wz y ys xs True = nubByNubBy' wz ys xs

nubByNubBy'1 wz y ys xs False = nubByNubBy'0 wz y ys xs otherwise

nubByNubBy'3 wz [] vw = []

nubByNubBy'3 wz wx wy = nubByNubBy'2 wz wx wy

nubByNubBy' wz [] vw = nubByNubBy'3 wz [] vw

nubByNubBy' wz (y : ys) xs = nubByNubBy'2 wz (y : ys) xs

nubByNubBy'0 wz y ys xs True = y : nubByNubBy' wz ys (y : xs)

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

mainModule List

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

module List where

import qualified Maybe
import qualified Prelude

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

elem_by	vx vy []	=	False
elem_by	eq y (x : xs)	=	x `eq` y \|\| elem_by eq y xs

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

nubBy

eq l

nubByNubBy' eq l []

nubByNubBy'	wz [] vw	=	nubByNubBy'3 wz [] vw
nubByNubBy'	wz (y : ys) xs	=	nubByNubBy'2 wz (y : ys) xs

nubByNubBy'0

wz y ys xs True

y : nubByNubBy' wz ys (y : xs)

nubByNubBy'1	wz y ys xs True	=	nubByNubBy' wz ys xs
nubByNubBy'1	wz y ys xs False	=	nubByNubBy'0 wz y ys xs otherwise

nubByNubBy'2

wz (y : ys) xs

nubByNubBy'1 wz y ys xs (elem_by wz y xs)

nubByNubBy'3	wz [] vw	=	[]
nubByNubBy'3	wz wx wy	=	nubByNubBy'2 wz wx wy

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="nubBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="nubBy xu3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="nubBy xu3 xu4",fontsize=16,color="black",shape="triangle"];4 -> 5[label="",style="solid", color="black", weight=3];
5[label="nubByNubBy' xu3 xu4 []",fontsize=16,color="burlywood",shape="box"];493[label="xu4/xu40 : xu41",fontsize=10,color="white",style="solid",shape="box"];5 -> 493[label="",style="solid", color="burlywood", weight=9];
493 -> 6[label="",style="solid", color="burlywood", weight=3];
494[label="xu4/[]",fontsize=10,color="white",style="solid",shape="box"];5 -> 494[label="",style="solid", color="burlywood", weight=9];
494 -> 7[label="",style="solid", color="burlywood", weight=3];
6[label="nubByNubBy' xu3 (xu40 : xu41) []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="nubByNubBy' xu3 [] []",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="nubByNubBy'2 xu3 (xu40 : xu41) []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="nubByNubBy'3 xu3 [] []",fontsize=16,color="black",shape="box"];9 -> 11[label="",style="solid", color="black", weight=3];
10[label="nubByNubBy'1 xu3 xu40 xu41 [] (elem_by xu3 xu40 [])",fontsize=16,color="black",shape="box"];10 -> 12[label="",style="solid", color="black", weight=3];
11[label="[]",fontsize=16,color="green",shape="box"];12[label="nubByNubBy'1 xu3 xu40 xu41 [] False",fontsize=16,color="black",shape="box"];12 -> 13[label="",style="solid", color="black", weight=3];
13[label="nubByNubBy'0 xu3 xu40 xu41 [] otherwise",fontsize=16,color="black",shape="box"];13 -> 14[label="",style="solid", color="black", weight=3];
14[label="nubByNubBy'0 xu3 xu40 xu41 [] True",fontsize=16,color="black",shape="box"];14 -> 15[label="",style="solid", color="black", weight=3];
15[label="xu40 : nubByNubBy' xu3 xu41 (xu40 : [])",fontsize=16,color="green",shape="box"];15 -> 16[label="",style="dashed", color="green", weight=3];
16[label="nubByNubBy' xu3 xu41 (xu40 : [])",fontsize=16,color="burlywood",shape="triangle"];495[label="xu41/xu410 : xu411",fontsize=10,color="white",style="solid",shape="box"];16 -> 495[label="",style="solid", color="burlywood", weight=9];
495 -> 17[label="",style="solid", color="burlywood", weight=3];
496[label="xu41/[]",fontsize=10,color="white",style="solid",shape="box"];16 -> 496[label="",style="solid", color="burlywood", weight=9];
496 -> 18[label="",style="solid", color="burlywood", weight=3];
17[label="nubByNubBy' xu3 (xu410 : xu411) (xu40 : [])",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="nubByNubBy' xu3 [] (xu40 : [])",fontsize=16,color="black",shape="box"];18 -> 20[label="",style="solid", color="black", weight=3];
19[label="nubByNubBy'2 xu3 (xu410 : xu411) (xu40 : [])",fontsize=16,color="black",shape="box"];19 -> 21[label="",style="solid", color="black", weight=3];
20[label="nubByNubBy'3 xu3 [] (xu40 : [])",fontsize=16,color="black",shape="box"];20 -> 22[label="",style="solid", color="black", weight=3];
21[label="nubByNubBy'1 xu3 xu410 xu411 (xu40 : []) (elem_by xu3 xu410 (xu40 : []))",fontsize=16,color="black",shape="box"];21 -> 23[label="",style="solid", color="black", weight=3];
22[label="[]",fontsize=16,color="green",shape="box"];23 -> 452[label="",style="dashed", color="red", weight=0];
23[label="nubByNubBy'1 xu3 xu410 xu411 (xu40 : []) (xu3 xu40 xu410 || elem_by xu3 xu410 [])",fontsize=16,color="magenta"];23 -> 453[label="",style="dashed", color="magenta", weight=3];
23 -> 454[label="",style="dashed", color="magenta", weight=3];
23 -> 455[label="",style="dashed", color="magenta", weight=3];
23 -> 456[label="",style="dashed", color="magenta", weight=3];
23 -> 457[label="",style="dashed", color="magenta", weight=3];
23 -> 458[label="",style="dashed", color="magenta", weight=3];
23 -> 459[label="",style="dashed", color="magenta", weight=3];
453[label="xu410",fontsize=16,color="green",shape="box"];454[label="xu3 xu40 xu410",fontsize=16,color="green",shape="box"];454 -> 461[label="",style="dashed", color="green", weight=3];
454 -> 462[label="",style="dashed", color="green", weight=3];
455[label="[]",fontsize=16,color="green",shape="box"];456[label="xu411",fontsize=16,color="green",shape="box"];457[label="xu40",fontsize=16,color="green",shape="box"];458[label="[]",fontsize=16,color="green",shape="box"];459[label="xu3",fontsize=16,color="green",shape="box"];452[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (xu39 || elem_by xu32 xu33 xu38)",fontsize=16,color="burlywood",shape="triangle"];498[label="xu39/False",fontsize=10,color="white",style="solid",shape="box"];452 -> 498[label="",style="solid", color="burlywood", weight=9];
498 -> 463[label="",style="solid", color="burlywood", weight=3];
499[label="xu39/True",fontsize=10,color="white",style="solid",shape="box"];452 -> 499[label="",style="solid", color="burlywood", weight=9];
499 -> 464[label="",style="solid", color="burlywood", weight=3];
461[label="xu40",fontsize=16,color="green",shape="box"];462[label="xu410",fontsize=16,color="green",shape="box"];463[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (False || elem_by xu32 xu33 xu38)",fontsize=16,color="black",shape="box"];463 -> 467[label="",style="solid", color="black", weight=3];
464[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (True || elem_by xu32 xu33 xu38)",fontsize=16,color="black",shape="box"];464 -> 468[label="",style="solid", color="black", weight=3];
467[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (elem_by xu32 xu33 xu38)",fontsize=16,color="burlywood",shape="triangle"];500[label="xu38/xu380 : xu381",fontsize=10,color="white",style="solid",shape="box"];467 -> 500[label="",style="solid", color="burlywood", weight=9];
500 -> 469[label="",style="solid", color="burlywood", weight=3];
501[label="xu38/[]",fontsize=10,color="white",style="solid",shape="box"];467 -> 501[label="",style="solid", color="burlywood", weight=9];
501 -> 470[label="",style="solid", color="burlywood", weight=3];
468[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) True",fontsize=16,color="black",shape="box"];468 -> 471[label="",style="solid", color="black", weight=3];
469[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (elem_by xu32 xu33 (xu380 : xu381))",fontsize=16,color="black",shape="box"];469 -> 472[label="",style="solid", color="black", weight=3];
470[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (elem_by xu32 xu33 [])",fontsize=16,color="black",shape="box"];470 -> 473[label="",style="solid", color="black", weight=3];
471[label="nubByNubBy' xu32 xu34 (xu35 : xu36)",fontsize=16,color="burlywood",shape="triangle"];502[label="xu34/xu340 : xu341",fontsize=10,color="white",style="solid",shape="box"];471 -> 502[label="",style="solid", color="burlywood", weight=9];
502 -> 474[label="",style="solid", color="burlywood", weight=3];
503[label="xu34/[]",fontsize=10,color="white",style="solid",shape="box"];471 -> 503[label="",style="solid", color="burlywood", weight=9];
503 -> 475[label="",style="solid", color="burlywood", weight=3];
472 -> 452[label="",style="dashed", color="red", weight=0];
472[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) (xu32 xu380 xu33 || elem_by xu32 xu33 xu381)",fontsize=16,color="magenta"];472 -> 476[label="",style="dashed", color="magenta", weight=3];
472 -> 477[label="",style="dashed", color="magenta", weight=3];
473[label="nubByNubBy'1 xu32 xu33 xu34 (xu35 : xu36) False",fontsize=16,color="black",shape="box"];473 -> 478[label="",style="solid", color="black", weight=3];
474[label="nubByNubBy' xu32 (xu340 : xu341) (xu35 : xu36)",fontsize=16,color="black",shape="box"];474 -> 479[label="",style="solid", color="black", weight=3];
475[label="nubByNubBy' xu32 [] (xu35 : xu36)",fontsize=16,color="black",shape="box"];475 -> 480[label="",style="solid", color="black", weight=3];
476[label="xu32 xu380 xu33",fontsize=16,color="green",shape="box"];476 -> 481[label="",style="dashed", color="green", weight=3];
476 -> 482[label="",style="dashed", color="green", weight=3];
477[label="xu381",fontsize=16,color="green",shape="box"];478[label="nubByNubBy'0 xu32 xu33 xu34 (xu35 : xu36) otherwise",fontsize=16,color="black",shape="box"];478 -> 483[label="",style="solid", color="black", weight=3];
479[label="nubByNubBy'2 xu32 (xu340 : xu341) (xu35 : xu36)",fontsize=16,color="black",shape="box"];479 -> 484[label="",style="solid", color="black", weight=3];
480[label="nubByNubBy'3 xu32 [] (xu35 : xu36)",fontsize=16,color="black",shape="box"];480 -> 485[label="",style="solid", color="black", weight=3];
481[label="xu380",fontsize=16,color="green",shape="box"];482[label="xu33",fontsize=16,color="green",shape="box"];483[label="nubByNubBy'0 xu32 xu33 xu34 (xu35 : xu36) True",fontsize=16,color="black",shape="box"];483 -> 486[label="",style="solid", color="black", weight=3];
484 -> 467[label="",style="dashed", color="red", weight=0];
484[label="nubByNubBy'1 xu32 xu340 xu341 (xu35 : xu36) (elem_by xu32 xu340 (xu35 : xu36))",fontsize=16,color="magenta"];484 -> 487[label="",style="dashed", color="magenta", weight=3];
484 -> 488[label="",style="dashed", color="magenta", weight=3];
484 -> 489[label="",style="dashed", color="magenta", weight=3];
485[label="[]",fontsize=16,color="green",shape="box"];486[label="xu33 : nubByNubBy' xu32 xu34 (xu33 : xu35 : xu36)",fontsize=16,color="green",shape="box"];486 -> 490[label="",style="dashed", color="green", weight=3];
487[label="xu340",fontsize=16,color="green",shape="box"];488[label="xu341",fontsize=16,color="green",shape="box"];489[label="xu35 : xu36",fontsize=16,color="green",shape="box"];490 -> 471[label="",style="dashed", color="red", weight=0];
490[label="nubByNubBy' xu32 xu34 (xu33 : xu35 : xu36)",fontsize=16,color="magenta"];490 -> 491[label="",style="dashed", color="magenta", weight=3];
490 -> 492[label="",style="dashed", color="magenta", weight=3];
491[label="xu35 : xu36",fontsize=16,color="green",shape="box"];492[label="xu33",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

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

new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, [], ba) → new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) → new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), 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 1 SCC with 1 less node.

↳ HASKELL

  ↳ BR

    ↳ HASKELL

      ↳ COR

        ↳ HASKELL

          ↳ LetRed

            ↳ HASKELL

              ↳ Narrow

                ↳ QDP

                  ↳ DependencyGraphProof

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
new_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) → new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), 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_nubByNubBy'10(xu32, xu33, :(xu340, xu341), xu35, xu36, xu38, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 3 > 2, 3 > 3, 4 >= 4, 5 >= 5, 7 >= 7
new_nubByNubBy'(xu32, :(xu340, xu341), xu35, xu36, ba) → new_nubByNubBy'1(xu32, xu340, xu341, xu35, xu36, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4, 4 >= 5, 5 >= 7
new_nubByNubBy'1(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 7 >= 7
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, :(xu380, xu381), ba) → new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, xu381, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3, 4 >= 4, 5 >= 5, 6 > 6, 7 >= 7
new_nubByNubBy'10(xu32, xu33, xu34, xu35, xu36, [], ba) → new_nubByNubBy'(xu32, xu34, xu33, :(xu35, xu36), ba)
The graph contains the following edges 1 >= 1, 3 >= 2, 2 >= 3, 7 >= 5

nubBy'	[] vw	= nubBy'3 [] vw
nubBy'	(y : ys) xs	= nubBy'2 (y : ys) xs

nubBy'1	y ys xs True	= nubBy' ys xs
nubBy'1	y ys xs False	= nubBy'0 y ys xs otherwise

nubByNubBy'1	wz y ys xs True	= nubByNubBy' wz ys xs
nubByNubBy'1	wz y ys xs False	= nubByNubBy'0 wz y ys xs otherwise

nubByNubBy'3	wz [] vw	= []
nubByNubBy'3	wz wx wy	= nubByNubBy'2 wz wx wy

nubByNubBy'	wz [] vw	= nubByNubBy'3 wz [] vw
nubByNubBy'	wz (y : ys) xs	= nubByNubBy'2 wz (y : ys) xs