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

YES 0.678 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

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

module Main where

import qualified Prelude

Lambda Reductions:
The following Lambda expression

\(_,zs)→zs

is transformed to

zs0 (_,zs) = zs

The following Lambda expression

\(ys,_)→ys

is transformed to

ys0 (ys,_) = ys

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

mainModule Main

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

module Main where

import qualified Prelude

Replaced joker patterns by fresh variables and removed binding patterns.
Binding Reductions:
The bind variable of the following binding Pattern

xs@(vw : vx)

is replaced by the following term

vw : vx

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

mainModule Main

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

The following Function with conditions

span p [] = ([],[])

span p (vw : vx)

| p vw

= (vw : ys,zs)

| otherwise

= ([],vw : vx)

where

vu43 = span p vx

ys = ys0 vu43

ys0 (ys,vy) = ys

zs = zs0 vu43

zs0 (vz,zs) = zs

is transformed to

span p [] = span3 p []

span p (vw : vx) = span2 p (vw : vx)

span2 p (vw : vx) =

span1 p vw vx (p vw)

where

span0 p vw vx True = ([],vw : vx)

span1 p vw vx True = (vw : ys,zs)

span1 p vw vx False = span0 p vw vx otherwise

vu43 = span p vx

ys = ys0 vu43

ys0 (ys,vy) = ys

zs = zs0 vu43

zs0 (vz,zs) = zs

span3 p [] = ([],[])

span3 wx wy = span2 wx wy

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

mainModule Main

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

module Main where

import qualified Prelude

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

span1 p vw vx (p vw)

where

span0 p vw vx True = ([],vw : vx)

span1 p vw vx True = (vw : ys,zs)

span1 p vw vx False = span0 p vw vx otherwise

vu43 = span p vx

ys = ys0 vu43

ys0 (ys,vy) = ys

zs = zs0 vu43

zs0 (vz,zs) = zs

are unpacked to the following functions on top level

span2Ys wz xu = span2Ys0 wz xu (span2Vu43 wz xu)

span2Ys0 wz xu (ys,vy) = ys

span2Zs0 wz xu (vz,zs) = zs

span2Span1 wz xu p vw vx True = (vw : span2Ys wz xu,span2Zs wz xu)

span2Span1 wz xu p vw vx False = span2Span0 wz xu p vw vx otherwise

span2Zs wz xu = span2Zs0 wz xu (span2Vu43 wz xu)

span2Span0 wz xu p vw vx True = ([],vw : vx)

span2Vu43 wz xu = span wz xu

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

mainModule Main

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

module Main where

import qualified Prelude

Haskell To QDPs

digraph dp_graph {
node [outthreshold=100, inthreshold=100];1[label="span",fontsize=16,color="grey",shape="box"];1 -> 3[label="",style="dashed", color="grey", weight=3];
3[label="span xv3",fontsize=16,color="grey",shape="box"];3 -> 4[label="",style="dashed", color="grey", weight=3];
4[label="span xv3 xv4",fontsize=16,color="burlywood",shape="triangle"];39[label="xv4/xv40 : xv41",fontsize=10,color="white",style="solid",shape="box"];4 -> 39[label="",style="solid", color="burlywood", weight=9];
39 -> 5[label="",style="solid", color="burlywood", weight=3];
40[label="xv4/[]",fontsize=10,color="white",style="solid",shape="box"];4 -> 40[label="",style="solid", color="burlywood", weight=9];
40 -> 6[label="",style="solid", color="burlywood", weight=3];
5[label="span xv3 (xv40 : xv41)",fontsize=16,color="black",shape="box"];5 -> 7[label="",style="solid", color="black", weight=3];
6[label="span xv3 []",fontsize=16,color="black",shape="box"];6 -> 8[label="",style="solid", color="black", weight=3];
7[label="span2 xv3 (xv40 : xv41)",fontsize=16,color="black",shape="box"];7 -> 9[label="",style="solid", color="black", weight=3];
8[label="span3 xv3 []",fontsize=16,color="black",shape="box"];8 -> 10[label="",style="solid", color="black", weight=3];
9 -> 11[label="",style="dashed", color="red", weight=0];
9[label="span2Span1 xv3 xv41 xv3 xv40 xv41 (xv3 xv40)",fontsize=16,color="magenta"];9 -> 12[label="",style="dashed", color="magenta", weight=3];
10[label="([],[])",fontsize=16,color="green",shape="box"];12[label="xv3 xv40",fontsize=16,color="green",shape="box"];12 -> 16[label="",style="dashed", color="green", weight=3];
11[label="span2Span1 xv3 xv41 xv3 xv40 xv41 xv5",fontsize=16,color="burlywood",shape="triangle"];42[label="xv5/False",fontsize=10,color="white",style="solid",shape="box"];11 -> 42[label="",style="solid", color="burlywood", weight=9];
42 -> 14[label="",style="solid", color="burlywood", weight=3];
43[label="xv5/True",fontsize=10,color="white",style="solid",shape="box"];11 -> 43[label="",style="solid", color="burlywood", weight=9];
43 -> 15[label="",style="solid", color="burlywood", weight=3];
16[label="xv40",fontsize=16,color="green",shape="box"];14[label="span2Span1 xv3 xv41 xv3 xv40 xv41 False",fontsize=16,color="black",shape="box"];14 -> 17[label="",style="solid", color="black", weight=3];
15[label="span2Span1 xv3 xv41 xv3 xv40 xv41 True",fontsize=16,color="black",shape="box"];15 -> 18[label="",style="solid", color="black", weight=3];
17[label="span2Span0 xv3 xv41 xv3 xv40 xv41 otherwise",fontsize=16,color="black",shape="box"];17 -> 19[label="",style="solid", color="black", weight=3];
18[label="(xv40 : span2Ys xv3 xv41,span2Zs xv3 xv41)",fontsize=16,color="green",shape="box"];18 -> 20[label="",style="dashed", color="green", weight=3];
18 -> 21[label="",style="dashed", color="green", weight=3];
19[label="span2Span0 xv3 xv41 xv3 xv40 xv41 True",fontsize=16,color="black",shape="box"];19 -> 22[label="",style="solid", color="black", weight=3];
20[label="span2Ys xv3 xv41",fontsize=16,color="black",shape="box"];20 -> 23[label="",style="solid", color="black", weight=3];
21[label="span2Zs xv3 xv41",fontsize=16,color="black",shape="box"];21 -> 24[label="",style="solid", color="black", weight=3];
22[label="([],xv40 : xv41)",fontsize=16,color="green",shape="box"];23 -> 27[label="",style="dashed", color="red", weight=0];
23[label="span2Ys0 xv3 xv41 (span2Vu43 xv3 xv41)",fontsize=16,color="magenta"];23 -> 28[label="",style="dashed", color="magenta", weight=3];
24 -> 32[label="",style="dashed", color="red", weight=0];
24[label="span2Zs0 xv3 xv41 (span2Vu43 xv3 xv41)",fontsize=16,color="magenta"];24 -> 33[label="",style="dashed", color="magenta", weight=3];
28[label="span2Vu43 xv3 xv41",fontsize=16,color="black",shape="triangle"];28 -> 30[label="",style="solid", color="black", weight=3];
27[label="span2Ys0 xv3 xv41 xv6",fontsize=16,color="burlywood",shape="triangle"];46[label="xv6/(xv60,xv61)",fontsize=10,color="white",style="solid",shape="box"];27 -> 46[label="",style="solid", color="burlywood", weight=9];
46 -> 31[label="",style="solid", color="burlywood", weight=3];
33 -> 28[label="",style="dashed", color="red", weight=0];
33[label="span2Vu43 xv3 xv41",fontsize=16,color="magenta"];32[label="span2Zs0 xv3 xv41 xv7",fontsize=16,color="burlywood",shape="triangle"];48[label="xv7/(xv70,xv71)",fontsize=10,color="white",style="solid",shape="box"];32 -> 48[label="",style="solid", color="burlywood", weight=9];
48 -> 35[label="",style="solid", color="burlywood", weight=3];
30 -> 4[label="",style="dashed", color="red", weight=0];
30[label="span xv3 xv41",fontsize=16,color="magenta"];30 -> 36[label="",style="dashed", color="magenta", weight=3];
31[label="span2Ys0 xv3 xv41 (xv60,xv61)",fontsize=16,color="black",shape="box"];31 -> 37[label="",style="solid", color="black", weight=3];
35[label="span2Zs0 xv3 xv41 (xv70,xv71)",fontsize=16,color="black",shape="box"];35 -> 38[label="",style="solid", color="black", weight=3];
36[label="xv41",fontsize=16,color="green",shape="box"];37[label="xv60",fontsize=16,color="green",shape="box"];38[label="xv71",fontsize=16,color="green",shape="box"];}

↳ HASKELL

  ↳ LR

    ↳ HASKELL

      ↳ BR

        ↳ HASKELL

          ↳ COR

            ↳ HASKELL

              ↳ LetRed

                ↳ HASKELL

                  ↳ Narrow

                    ↳ QDP

                      ↳ QDPSizeChangeProof

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

new_span(xv3, :(xv40, xv41), ba) → new_span2Span1(xv3, xv41, xv40, ba)
new_span2Vu43(xv3, xv41, ba) → new_span(xv3, xv41, ba)
new_span2Span1(xv3, xv41, xv40, ba) → new_span2Vu43(xv3, xv41, ba)
new_span2Span1(xv3, xv41, xv40, ba) → new_span(xv3, xv41, 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_span2Span1(xv3, xv41, xv40, ba) → new_span(xv3, xv41, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
new_span2Span1(xv3, xv41, xv40, ba) → new_span2Vu43(xv3, xv41, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 4 >= 3
new_span2Vu43(xv3, xv41, ba) → new_span(xv3, xv41, ba)
The graph contains the following edges 1 >= 1, 2 >= 2, 3 >= 3
new_span(xv3, :(xv40, xv41), ba) → new_span2Span1(xv3, xv41, xv40, ba)
The graph contains the following edges 1 >= 1, 2 > 2, 2 > 3, 3 >= 4

span2Span1	wz xu p vw vx True	= (vw : span2Ys wz xu,span2Zs wz xu)
span2Span1	wz xu p vw vx False	= span2Span0 wz xu p vw vx otherwise