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

YES 0.834 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

  ↳ CR

mainModule List

((maximumBy :: (a -> a -> Ordering) -> [a] -> a) :: (a -> a -> Ordering) -> [a] -> a)

module List where

import qualified Maybe
import qualified Prelude

maximumBy :: (a -> a -> Ordering) -> [a] -> a

maximumBy

_ []

error []

maximumBy

cmp xs

foldl1 max xs

where

max

x y

case	cmp x y of
	GT	->	x
	_	->	y

module Maybe where

import qualified List
import qualified Prelude

Case Reductions:
The following Case expression

case cmp x y of
GT → x

_ → y

is transformed to

max0 x y GT = x

max0 x y _ = y

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

mainModule List

((maximumBy :: (a -> a -> Ordering) -> [a] -> a) :: (a -> a -> Ordering) -> [a] -> a)

module List where

import qualified Maybe
import qualified Prelude

maximumBy :: (a -> a -> Ordering) -> [a] -> a

maximumBy

_ []

error []

maximumBy

cmp xs

foldl1 max xs

where

max

x y

max0 x y (cmp x y)

max0	x y GT	=	x
max0	x y _	=	y

module Maybe where

import qualified List
import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule List

((maximumBy :: (a -> a -> Ordering) -> [a] -> a) :: (a -> a -> Ordering) -> [a] -> a)

module List where

import qualified Maybe
import qualified Prelude

maximumBy :: (a -> a -> Ordering) -> [a] -> a

maximumBy

vw []

error []

maximumBy

cmp xs

foldl1 max xs

where

max

x y

max0 x y (cmp x y)

max0	x y GT	=	x
max0	x y vx	=	y

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

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule List

((maximumBy :: (a -> a -> Ordering) -> [a] -> a) :: (a -> a -> Ordering) -> [a] -> a)

module List where

import qualified Maybe
import qualified Prelude

maximumBy :: (a -> a -> Ordering) -> [a] -> a

maximumBy

vw []

error []

maximumBy

cmp xs

foldl1 max xs

where

max

x y

max0 x y (cmp x y)

max0	x y GT	=	x
max0	x y vx	=	y

module Maybe where

import qualified List
import qualified Prelude

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

foldl1 max xs

where

max x y = max0 x y (cmp x y)

max0 x y GT = x

max0 x y vx = y

are unpacked to the following functions on top level

maximumByMax0 wu x y GT = x

maximumByMax0 wu x y vx = y

maximumByMax wu x y = maximumByMax0 wu x y (wu x y)

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule List

(maximumBy :: (a -> a -> Ordering) -> [a] -> a)

module List where

import qualified Maybe
import qualified Prelude

maximumBy :: (a -> a -> Ordering) -> [a] -> a

maximumBy	vw []	=	error []
maximumBy	cmp xs	=	foldl1 (maximumByMax cmp) xs

maximumByMax

wu x y

maximumByMax0 wu x y (wu x y)

maximumByMax0	wu x y GT	=	x
maximumByMax0	wu x y vx	=	y

module Maybe where

import qualified List
import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="maximumBy",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="maximumBy wv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="maximumBy wv3 wv4",fontsize=16,color="burlywood",shape="triangle"];30[label="wv4/wv40 : wv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 30[label="",style="solid", color="burlywood", weight=9];
30 -> 5[label="",style="solid", color="burlywood", weight=3];
31[label="wv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 31[label="",style="solid", color="burlywood", weight=9];
31 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="maximumBy wv3 (wv40 : wv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="maximumBy wv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="foldl1 (maximumByMax wv3) (wv40 : wv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="error []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9[label="foldl (maximumByMax wv3) wv40 wv41",fontsize=16,color="burlywood",shape="triangle"];32[label="wv41/wv410 : wv411",fontsize=10,color="white",style="solid",shape="box"];9 -> 32[label="",style="solid", color="burlywood", weight=9];
32 -> 11[label="",style="solid", color="burlywood", weight=3];
33[label="wv41/[]",fontsize=10,color="white",style="solid",shape="box"];9 -> 33[label="",style="solid", color="burlywood", weight=9];
33 -> 12[label="",style="solid", color="burlywood", weight=3];
10[label="error []",fontsize=16,color="red",shape="box"];11[label="foldl (maximumByMax wv3) wv40 (wv410 : wv411)",fontsize=16,color="black",shape="box"];11 -> 13[label="",style="solid", color="black", weight=3];
12[label="foldl (maximumByMax wv3) wv40 []",fontsize=16,color="black",shape="box"];12 -> 14[label="",style="solid", color="black", weight=3];
13 -> 9[label="",style="dashed", color="red", weight=0];
13[label="foldl (maximumByMax wv3) (maximumByMax wv3 wv40 wv410) wv411",fontsize=16,color="magenta"];13 -> 15[label="",style="dashed", color="magenta", weight=3];
13 -> 16[label="",style="dashed", color="magenta", weight=3];
14[label="wv40",fontsize=16,color="green",shape="box"];15[label="maximumByMax wv3 wv40 wv410",fontsize=16,color="black",shape="box"];15 -> 17[label="",style="solid", color="black", weight=3];
16[label="wv411",fontsize=16,color="green",shape="box"];17 -> 18[label="",style="dashed", color="red", weight=0];
17[label="maximumByMax0 wv3 wv40 wv410 (wv3 wv40 wv410)",fontsize=16,color="magenta"];17 -> 19[label="",style="dashed", color="magenta", weight=3];
19[label="wv3 wv40 wv410",fontsize=16,color="green",shape="box"];19 -> 25[label="",style="dashed", color="green", weight=3];
19 -> 26[label="",style="dashed", color="green", weight=3];
18[label="maximumByMax0 wv3 wv40 wv410 wv5",fontsize=16,color="burlywood",shape="triangle"];36[label="wv5/LT",fontsize=10,color="white",style="solid",shape="box"];18 -> 36[label="",style="solid", color="burlywood", weight=9];
36 -> 22[label="",style="solid", color="burlywood", weight=3];
37[label="wv5/EQ",fontsize=10,color="white",style="solid",shape="box"];18 -> 37[label="",style="solid", color="burlywood", weight=9];
37 -> 23[label="",style="solid", color="burlywood", weight=3];
38[label="wv5/GT",fontsize=10,color="white",style="solid",shape="box"];18 -> 38[label="",style="solid", color="burlywood", weight=9];
38 -> 24[label="",style="solid", color="burlywood", weight=3];
25[label="wv40",fontsize=16,color="green",shape="box"];26[label="wv410",fontsize=16,color="green",shape="box"];22[label="maximumByMax0 wv3 wv40 wv410 LT",fontsize=16,color="black",shape="box"];22 -> 27[label="",style="solid", color="black", weight=3];
23[label="maximumByMax0 wv3 wv40 wv410 EQ",fontsize=16,color="black",shape="box"];23 -> 28[label="",style="solid", color="black", weight=3];
24[label="maximumByMax0 wv3 wv40 wv410 GT",fontsize=16,color="black",shape="box"];24 -> 29[label="",style="solid", color="black", weight=3];
27[label="wv410",fontsize=16,color="green",shape="box"];28[label="wv410",fontsize=16,color="green",shape="box"];29[label="wv40",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ CR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_foldl(wv3, :(wv410, wv411), ba) → new_foldl(wv3, wv411, 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_foldl(wv3, :(wv410, wv411), ba) → new_foldl(wv3, wv411, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 3 >= 3