My posts count is stuck to want had two days ago has not moved today or yesterday .Who else this happening to.
giffgaff already know about this problem.
It will be corrected as soon as they know what's caused it.
It's happening to everyone! The problem has already been reported and is being worked on.
The post count issue is affecting all members. It is also best answers that are not being updated.
Giffgaff know about this issue and are currently trying their best to fix it.
It seems to be affecting everyone. But don't panic giffgaff will sort it.


Now with a stainless steel interior option, Purair LF Series Laminar Flow Hoods ensure easier work zone cleaning and optimal chemical resistance in laboratory, pharmaceutical and biotechnology applications.
All of our standard horizontal and vertical laminar flow hoods are available in stainless steel. Please specify your stainless interior when ordering.
View our full line of laminar flow and clean bench products, or contact us today for a customized quote based on your application.


Google's TensorFlow framework is the current leading software for implementing and experimenting with the algorithms that power AI and machine learning. Google deploys TensorFlow for many of its products, such as Translate and Maps.
We will embark on this journey by quickly wrapping up some important fundamental concepts, followed by a focus on TensorFlow to complete tasks in computer vision and natural language processing. You will be introduced to some important tips and tricks necessary for enhancing the efficiency of our models. We will highlight how TensorFlow is used in an advanced environment and brush through some of the unique concepts at the cutting edge of practical AI.
If you want to develop a solid foundation on using TensorFlow and continue your journey into advancing the state of the art in AI to create your own smart machine learning solutions, this course is for you.
This is an introductory course with a right balance between theory and practical implementation.
Some exciting tips & tricks for designing, training and evaluating the models.
Brandon McKinzie is an NLP engineer/researcher and lover of all things associated with machine learning, with a particular interest in deep learning for natural language processing. The author is extremely passionate about contributing to research and learning in general, and in his free time he's either working through textbooks, personal projects, or browsing blogs related to ML/AI.


The Mehrik Earring for The August 2015 Round of The Fantasy Collective is inspired by the rounds overall theme of "Anarchy". The actual definition is as follows, "a state of disorder due to absence or nonrecognition of authority." As you can well imagine, this can be interpreted in a variety of ways. I've decided to go with a very well known symbol of "Anarchy" and that would be the Skull and Crossbones. The Skull and crossbones have been a well established symbol of "Anarchy" throughout the ages. From Pirates to Bikers, the Skull and Crossbones represent their angst and aggression toward the accepted and organized rule of Government. In other words, it's a big "F@*%k You!" to the man!
The Plastik has been a staple in SL since 2006, making it one of the original iconic Second Life stores. When you think of The Plastik you automatically think of gorgeously textured skins, horns, make-ups, accessories, and clothing. The plastik does it all effortlessly transitioning between one trend and another. The Plastik's huge mainstore is literally a one stop shop for all your avatar's needs. With the rise of Second Life Shopping Events, The Plastik has been able to showcase her unique work in all sorts of different genres.
*If you're interested in seeing more, please visit the website: https://thebeautifulmachine.com/ for up to date information on new releases, events, and other fun stuff!
Sorcha Irelund-Store Manager & Event Coordinator. I'll do my very best to get back to you as soon as possible. Inquires about The Plastik joining your event should clearly state the event name in the title along with the events time line.
~ by Vae on August 24, 2015.


text:
Q: AS3 air multiple resolutions, assets and layout iphone and ipad app I am building an app in AS3/Air and I would like to target both iPhone and iPad resolutions. I understand the different aspect ratios between iPhone and iPad however the app I am building currently has different layout and slightly different content to fit the different screen sizes. I currently have 2 versions of the app already built, one for iPhone the other for iPad. All assets have been created with the target platform in mind but now, I would like to combine the 2 apps into a single one.
I am thinking I will rename each each screen file to iphone_login, ipad_menu, ipad_settings etc and include them all in the same build. Then during startup, check what device the user is on and set iphone_ or ipad_ and also set the resolution at this time too.
I prefer not to have black edges going from iphone resolution to ipad so my questions are:

*

*Is my approach a suitable one considering the outcome I would like?

*How do I determine what device a user is on to show the correct files, assets and resolution?

I understand the app size will increase at least double by adding 2 sets of assets and 2 sets of code files but considering the differences in design layout and content I don't see another solution, apart from keeping 2 apps.
Thanks :)

A: What's the problem? iPad and iPhone have different resolution and dpi combination, check them and identify current platform.
Get view you need by class like this:
public static const PAGE1:String = "page1";
public static const PAGE2:String = "page2";

private static var PHONE_VIEW_NAME_2_CLASS:Dictionary = new Dictionary();
private static var TABLET_VIEW_NAME_2_CLASS:Dictionary = new Dictionary();

public class ViewProvider
{
    {
        PHONE_VIEW_NAME_2_CLASS[ViewProvider.PAGE1] = Page1PhoneView;
        PHONE_VIEW_NAME_2_CLASS[ViewProvider.PAGE2] = Page2PhoneView;

        TABLET_VIEW_NAME_2_CLASS[ViewProvider.PAGE1] = Page1TabletView;
        TABLET_VIEW_NAME_2_CLASS[ViewProvider.PAGE2] = Page2TabletView;
    }
    public function ViewProvider()
    {
    }

    public static function isTablet():Boolean {
    ...analyze Capabilities.screenResolutionY, Capabilities.screenResolutionX and Capabilities.screenDPI
    }

    public static function getViewClass(name:String):Class
    {
        return isTablet() ? TABLET_VIEW_NAME_2_CLASS[name] : PHONE_VIEW_NAME_2_CLASS[name];
    }

}

And in your program
navigator.pushView(ViewProvider.getViewClass(ViewProvider.PAGE1))
All coordinated, paddings and another position numbers, font sizes etc correct with multiplier depending on runtime dpi by simular way...

A: I'm in the middle of a similar problem.
My solution is to have the images at the best resolution in a file pool, and then downscale them depending on the device when the app starts. You can do this also with non animated vector assets and put them into a bitmapData object.
Another option is to always have the asset pool with files at the maximum resolution needed loaded in the memory, and downscale them at runtime when they are needed. This works well if you will be using some asset in different places at different sizes, for example an icon.
As for the code, you should find a way to separate the code that manages data, the codes that manages the logic, and the code that "paints" the UI. This way you will be able to reuse most of the code in both versions, but only change the code that "paints" the UI. Check the MVC pattern for more info.

redpajama_set_name: RedPajamaStackExchange


New York Fashion Week proved an ideal time for Natalie Massenet to herald the arrival of Sarah Rutson, Net-a-Porter's new Vice President of Global Buying. There was a splashy party at Indochine, attended by a gang including Amber Le Bon, Karlie Kloss, Caroline Issa and Jenna Lyons. Tables were laden with crab cakes, spicy squid and glazed duck breast, while the gang quaffed back celebratory champagne quicker than you can say 'FASHION'. It wasn't a particularly late night - everyone had early morning shows to attend the next day.


Features padded back, adjustable shoulder straps, nylon hand strap, and multiple pockets. Measures 9-1/2" x 18-1/2" x 5".
Features multiple pockets with zippered closures and adjustable straps for extra security. Measures 13" x 34" x 7-1/2".


Exploring the onsen town near Tokyo of Yamanouchiwhere to next?
Home / Blog / Snow monkeys and steaming hot baths.
It's a bit surreal watching a troop of snow monkeys squabble over best position in their steaming hot bath, surrounded by snow-covered slopes. Like watching kids in a crowded swimming pool.
It's not quite the David Attenborough moment I was expecting, but according to No. 1 son, it's "so cool." Praise indeed.
This is our introduction to Japan's onsen tradition, in the hot springs region of Yamanouchi, an onsen town near Tokyo (well, 60 minutes away by bullet train). As we head into the nearby village of Shibu Onsen, we're about to learn a whole lot more.
Shibu Onsen is around 1300 years old, and wandering through its cobbled streets is almost like stepping back in time.
The narrow lanes are great fun to explore, and if you're staying at one of the local traditional inns or Ryoken, you're given a key to all the village's onsen (public baths).
It's strange to see people walking through the snow in their bathrobes and sandals, so warm from soaking in the hot springs they don't feel the cold.
However within a couple of days here, we were both sold on the onsen culture of bathing in public, even my normally cynical teenage son. On our first night here he booked himself into the private rooftop onsen.
Apparently all his anime heroes hang out at the onsen so he's already quite familiar with the routine – he even knows what to do with the modesty towel (it's placed on your head while you soak apparently).
Undress completely before you enter the bath area (no swimmers allowed).
Wash first, using lots of soap and your wash towel, while crouching or sitting on the small stools provided (don't stand).
Use the wash towel for modesty when walking, and ease yourself into the water (don't splash or dive).
Place the modesty towel on your head while in the water. Don't wring out your towel into the bath or let it touch the water.
Don't swim – the bath is meant for quiet soaking and contemplation.
On the culinary side, there are a couple of local delicacies worth sampling while you explore the onsen village: onsen eggs and snow monkey manjus.
Onsen eggs are soft boiled eggs, cooked slowly by hot springs water flowing over them, the result being eggs with a white as soft as the yolk (or yolk as firm as the white – whichever way you want to look at it).
Manjus are steamed buns with sweet or savory filling and Snow Monkey Manjus are a local specialty of Shibu Onsen and have a sweet red bean filling.
The meal we are served at our Ryoken is a ceremonious affair (again impressive even to No. 1 Son).
We're seated in a private dining room with tatami matting floor and rice paper walls. Small portions of beautifully prepared food are served in a never ending parade, covering practically the whole gamut of Japanese cuisine.
We're served sashimi, miso and tempura vegetables, given small burners to barbeque mushrooms and finely sliced beef and various dipping sauces for each.
This is apparently, the coolest meal EVER.
Ryokan Biyu no Yado is in the onsen village of Yudanaka. It's definitely worth staying here at least one night to explore the onsen village and nearby snow monkey park.
The spacious rooms are a mix of traditional Japanese and Western style and the many-course meals are an event in themselves. Be sure to book at least one session in the private rooftop onsen.
We flew Cathay Pacific, who I can highly recommend, even at the non-pointy end of the plane. Fantastic service on the ground and in the air.
The writer was a guest of Shiga Kogen Tourism Association and Yamanouchi Town Tourism Association. For further information on travel to Japan, contact Japan National Tourism Organization (JNTO).
Previous Article A high tea fit for princesses.
Hello there, I would adore seeing a monkey in a hot pool, and even more being in a hot pool myself NOT ALLOWED TO MOVE…. sounds like bliss. Is this far from Tokyo? Can you give the internal travel info… makes me feel as if I could plan a trip there. No young kids of mine would be invited!
Bliss indeed! It's in the Nagano prefecture, which is exactly 1 hour 25 minutes by bullet train from Tokyo Station, with regular shuttle bus service from Nagano Station to Yamanouchi and the ski fields. The hotel was absolutely beautiful – highly recommended!
Next time we are doing this!


A look at how teams fared Friday.
MADISON - De Pere's Nathan Balthazor went 2-0 in play at the WIAA boys individual state tennis tournament on Friday.
Balthazor defeated Nick Yang of Marquette University High School 6-2, 6-3 and beat top-seeded Will Tennison of Verona 2-6, 6-4, 6-3.
With the wins, Balthazor is guaranteed a top-four finish at the tournament.
Southwest's Johnny Zakowski went 1-1 with a 6-1, 6-0 win over Marquette University's Richard Balistreri. Zakowski was defeated by Neel Raut of Oshkosh West in the next round 6-7(2), 3-6).
Zakowski has the chance to compete for a chance to place fifth on Saturday.
West De Pere's Elijah Zifferblatt fell in his match 6-7(6), 2-6 against Colt Tegtmeier of Madison Memorial.
In doubles play, Southwest's team of Jayden Wartman and Ben Fredeen went 1-1 with a 6-0, 6-3 win, but fell in the next round.
The Southwest pair of Juan Pablo Cabazes and the Ashwaubenon duo of Kole Kolinski and Colin Chapman each lost their respective matches.


To be honest, I've been obsessed with menswear as early as middle school. I've always loved the sharp lines of a crisp tailored blazer, the classic leather of a men's messenger bag, and the fancy laces on a shiny pair of oxford loafers. Sadly, as a middle schooler, I didn't have a lot of courage to rock these pieces due to a longing to fit in with the rest of the students, who chose Abercrombie & Fitch fleeces and jeans to be their uniform. Now, as a working adult, I find that menswear actually plays really well into a woman's work wardrobe. So much that I often feel more confident being in a classic blazer and button up combo than a feminine dress and heels.
The suit combo trend has been another one of my obsessions lately. I've been scouring for a two-piece suit with a bold color or pattern for a while, but haven't found the right one in the right size (shopping for suits or any two-piece sets can be quite the challenge for petites). So to recreate this suit combo look, I merely took a classic black blazer from the back of my closet and paired it with my classic black cigarette trousers. Because the all-black combo can look a bit stuffy and overly-formal, I made sure to layer a casual T-shirt underneath it and to compliment the suit with a pair of Adidas Stan Smiths.
This was a joint shoot with Maya Deschamps of M.A.D Mgmt, a San Francisco-based style and beauty consulting company. Maya, who used to model in Paris and walk runways for fashion houses like Christian Dior, reached out to me via my post about the Glossier pop up, and I was super excited to meet up and collaborate with her on new style content instantly. If you're interested in more of her work, follow her on Instagram and check out the style consulting services at M.A.D Mgmt.
To downplay the formal vibe created by the black suit, I decided to layer a minimal white graphic underneath it. This T-shirt I purchased when I saw Emily King in her "You and I" tour back in 2017. The graphic in this tee is actually an outline of the photo in the cover of her album titled "The Switch," which I put on repeat throughout my senior year in college. I love how the tour tee is just a simple outline, and not gaudy or overly-colorful like many artist's fan merchandise turn out to be.
To complete the outfit, I added a feminine touch to balance out the menswear with some jewelry. I added a pair of dangly hoops, which helped add a little shine to monochromatic outfit. These hoops, which were found on Yesstyle, are also super minimal, which helped tie together the clean lines from the blazer and the minimal art in the graphic tee.
Location: These were taken on the corner of Alabama and 18th in the Mission, which is filled with colorful murals and beautiful walls. This area in the Mission is also home to some great dining, including Tartine Manufactory, Flour + Water, and Farmhouse Kitchen.
A huge thanks to Bianca for taking the photos and Maya for coordinating the shoot!


On a community, such a device is known as a bunch Along with server, the phrases serve and repair (as noun and as verb) are often used, although servicer and servant are not. Once I started my new place I did not have any bother working with the community operating system, but I used to be a bit baffled by the server hardware.
It can make sure that only files from a single laptop are located on one tape. The metadata database and the media supervisor signify two elements that tend to be hidden. Consumer computers are mainly the customers(users) of the community, as they request and obtain services from the servers.
Hub – Hub is a device that splits a network connection into a number of computer systems. An internet server farm is a assortment of networked web servers, every with entry to content material on the same site that functions as a cluster, conceptually. The drawback of this is that over time the backed up information can turn into distributed over numerous tapes.


text:
\section{Introduction}
Natural deduction systems, as proposed by Gentzen~\cite{gentzen1969} and further studied by Prawitz~\cite{prawitz1965}, is one of the most well known proof-theoretical frameworks. Part of its success is based on the fact that natural deduction rules present a simple characterization of logical constants, especially in the case of intuitionistic logic. However, there has been a lot of criticism on extensions of the intuitionistic set of rules in order to deal with classical logic. Indeed, most of such extensions add, to the usual introduction and elimination rules, extra rules governing  negation. As a consequence, several meta-logical properties, the most prominent one being {\em harmony}, are lost.\footnote{A logical connective is called {\em harmonious} in a certain proof system if there exists a certain balance between the rules defining it. For example, in natural deduction based systems, harmony is ensured when introduction/elimination rules do not contain insufficient/excessive amounts of information~\cite{DBLP:conf/ictac/Diaz-CaroD21}.}

In~\cite{DBLP:journals/Prawitz15}, Dag Prawitz proposed a
natural deduction {\em ecumenical system}, where classical logic and intuitionistic logic are codified in the same system. In this system,
the classical logician and the intuitionistic logician would share the universal quantifier, conjunction, negation and the constant for the absurd ($\forall,\wedge,\neg,\bot$), but they would each have their own existential quantifier, disjunction and implication, with different meanings ($\exists_j,\vee_j,\to_j$, where $j\in\{i,c\}$ for the intuitionistic and classical versions, respectively). Prawitz' main idea is that these different meanings are given by a semantical framework that can be accepted by both parties.

In his ecumenical system, Prawitz recovers the harmony of rules, but the rules for the classical operators do not satisfy {\em separability}~\cite{Murzi2018}. In fact, the classical rules are not {\em pure}, in the sense that
negation
is used in the definition of the introduction and elimination rules for the classical operators.

For example, the rules for $\vee_{c}$ are defined as

\begin{prooftree}
\AxiomC{$[\neg A,\neg B]$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$\vee_{c}$-int}
\UnaryInfC{$A \vee_{c} B$}
\DisplayProof
\qquad
\AxiomC{$A \vee_{c} B$}
\AxiomC{$\neg A$}
\AxiomC{$\neg B$}
\RightLabel{$\vee_{c}$-elim}
\TrinaryInfC{$\bot$}
\end{prooftree}
The situation is not different in the case of the definition of left and right rules for these classical operators in a {\em sequent calculus} codification, as presented in~\cite{DBLP:journals/synthese/PimentelPP21}. The rules for $\vee_{c}$, for example, are defined as
\begin{prooftree}
\AxiomC{$\Gamma , \neg A , \neg B \Rightarrow \bot$}
\RightLabel{$\vee_{c}$-R}
\UnaryInfC{$\Gamma \Rightarrow A \vee_{c} B$}
\DisplayProof
\qquad
\AxiomC{$\Gamma \Rightarrow \neg A$}
\AxiomC{$\Delta \Rightarrow \neg B$}
\RightLabel{$\vee_{c}$-L}
\BinaryInfC{$A \vee_{c} B, \Gamma ,\Delta \Rightarrow \bot$}
\end{prooftree}

There are many ways of proposing pure, harmonic natural deduction systems for (propositional) classical logic.
Indeed, Murzi~\cite{Murzi2018} proposes a new set of rules for classical logical operators based on absurdity as a punctuation mark, and higher-level rules~\cite{DBLP:journals/sLogica/Schroeder-Heister14}.
D'Agostino~\cite{DBLP:conf/birthday/DAgostino05}, on the other hand, brings a totally sifferent approach, presenting a theory of classical natural deduction that makes a distinction between operational rules, governing the use of logical operators, and structural rules dealing with the metaphysical assumptions governing the (classical) notions of truth and falsity, namely the principle of bivalence and the principle of non-contradiction.

A complete different approach is presented in~\cite{DBLP:conf/birthday/GabbayG05}, where Michael and Murdoch Gabbay present the natural deduction version of Dov Gabbay's {\em Restart} rule
\[
\infer[Restart]{B}{A}
\]
with the side-condition that, below every occurrence of $Restart$ from $A$ to $B$, there is (at least) one occurrence of $A$. The intended meaning is that $B$ is a new start to a line of reasoning concluding $A$. For example, in the derivation of the Peirce's Law
\begin{prooftree}
\AxiomC{$[(A \to B) \to A)]$}
\AxiomC{$[A]$}
\RightLabel{$Restart^\star$}
\UnaryInfC{$B$}
\RightLabel{$\to$-int}
\UnaryInfC{$A\to B$}
\RightLabel{$\to$-elim}
\BinaryInfC{$A^\dagger$}
\RightLabel{$\to$-int}
\UnaryInfC{$((A \to B) \to A) \to_{c} A)$}
\end{prooftree}
the restart at ${}^\star$ is justified at $\dagger$.

Similar to the Gabbays, Restall's $Alt$ rule~\cite{restall}
\[
\infer[Alt, \downarrow A]{B}{\deduce{A}{\deduce{}{\Pi}}}
\]
has the following interpretation: Having a proof of $A$, one can set $A$ aside and consider some alternative conclusion, $B$, while $A$ is added to the collection of alternatives current at this point of the proof. Hence, the proof of Peirce's Law would have the form
\begin{prooftree}
\AxiomC{$[(A \to B) \to A)]$}
\AxiomC{$[A]$}
\RightLabel{$Alt, \downarrow A$}
\UnaryInfC{$B$}
\RightLabel{$\to$-int}
\UnaryInfC{$A\to B$}
\RightLabel{$\to$-elim,$\uparrow A$}
\BinaryInfC{$A$}
\RightLabel{$\to$-int}
\UnaryInfC{$((A \to B) \to A) \to_{c} A)$}
\end{prooftree}

In this paper, we propose a different approach adapting, to the natural deduction framework,  Girard's mechanism of \textit{stoup}~\cite{DBLP:journals/mscs/Girard91}.  This will allow the definition of a pure harmonic natural deduction system $\mathcal{LE}_{p}$ for the propositional fragment of  Prawitz' ecumenical logic, where $Restart$ and $Alt$ appear as special cases of the use of {\em stoup}.\footnote{It should be noted that this idea  appears somewhat hidden in~\cite{restall} for the propositional classical case, where Restall uses sequent style
$X\rest A; Y$ (the `score'), where $X$ represent the undischarged assumptions, $A$ the current conclusion, and $Y$ the {\em alternatives}.}

\section{Ecumenical natural deduction system}\label{sec:ec}

The language $\Lscr$ used for ecumenical systems is described as follows. We will use a subscript $c$ for the classical meaning and $i$ for the intuitionistic, dropping such subscripts when formulae/connectives can have either meaning.

Classical and intuitionistic  n-ary predicate symbols ($p_{c}, p_{i},\ldots$) co-exist in $\Lscr$ but have different meanings.
The neutral logical connectives $\{\bot,\neg,\wedge,\forall\}$ are common for classical and intuitionistic fragments, while $\{\iimp,\ivee,\iexists\}$ and $\{\cimp,\cvee,\cexists\}$ are restricted to intuitionistic and classical interpretations, respectively. In order to avoid clashes of variables we make use of a denumerable set $a, b,\ldots$ of special variables called {\em parameters}, which do not appear quantified.

In Fig.~\ref{fig:NE} we present $\mathcal{NE}$ Prawitz' original natural deduction first-order ecumenical system. In the rules for quantifiers, the notation $A(a/x)$  stands for the substitution of $a$ for every (visible) instance of $x$ in $A$. In the rules $\exists_i\mbox{-elim},\forall\mbox{-int}$, $a$ is a fresh parameter, \ie, it does not occur free in any assumption that $A,B$ depends on (apart from the assumption eliminated by $\exists_{i}$-elim).

\begin{figure}[htp]
{\sc Intuitionistic rules}
\[
\begin{array}{lc@{\quad}l}
\infer[{\iimp\mbox{-elim}}]{B}{A\iimp B & A}
&
\infer[{\iimp\mbox{-int}}]{ A\iimp B}
{\deduce{B}{\deduce{}{\deduce{\Pi}{\deduce{}{[A]}}}}}
&
\infer[{\vee_i\mbox{-elim}}]{C}{ A\vee_i B & \deduce{C}{\deduce{}{\deduce{\Pi_1}{\deduce{}{[A]}}}}
 &\deduce{C}{\deduce{}{\deduce{\Pi_2}{\deduce{}{[B]}}}}}
\\[5pt]
\infer[{\vee_i\mbox{-int}_j}]{ A_1\vee_i A_2}{ A_j}
&
\infer[\exists_i\mbox{-elim}]{ B}{ \exists_ix.A &  \deduce{B}{\deduce{}{\deduce{\Pi}{\deduce{}{[A(a/x)]}}}}}
&
\infer[\exists_i\mbox{-int}]{\exists_ix.A }{ A(a/x)}
\end{array}
\]
{\sc Classic rules}
\[
\begin{array}{lc@{\quad}l}
\infer[{\cimp\mbox{-elim}}]{\bot}{ A\cimp B & A & \neg B}
&
\infer[{\cimp\mbox{-int}}]{A\cimp B}
{\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[A,\neg B]}}}}}
&
\infer[{\vee_c\mbox{-elim}}]{\bot}{A\vee_c B & \neg A &  \neg B}
\\[5pt]
\infer[{\vee_c\mbox{-int}}]{A\vee_c B}{\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[\neg A,\neg B]}}}}}
&
\infer[\exists_c\mbox{-elim}]{\bot}{ \exists_cx.A &  \forall x.\neg A}
&
\infer[\exists_c\mbox{-int}]{ \exists_cx.A}{\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[\forall x.\neg A]}}}}}
\\[5pt]
\infer[p_c\mbox{-elim}]{ \bot}
{
p_c &  \neg p_i
}
&
\infer[p_c\mbox{-int}]{ p_c}
{
\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[\neg p_i]}}}}
}
\end{array}
\]
{\sc Neutral rules}
\[
\begin{array}{lc@{\qquad}lc@{\qquad}lc@{\qquad}l}
\infer[{\wedge\mbox{-elim}_j}]{ A_j}{A_1\wedge A_2}
& &
\infer[{\wedge\mbox{-int}}]{A \wedge B}{A \quad  B}
& &
\infer[{\neg\mbox{-elim}}]{\bot}{ A &  \neg A}
& &
\infer[{\neg\mbox{-int}}]{\neg A}{\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[A]}}}}}
\\[5pt]
\infer[\bot\mbox{-elim}]{A}{ \bot}
& &
\infer[\forall\mbox{-elim}]{A(a/x) }{\forall x.A}
& &
\infer[\forall\mbox{-int}]{ \forall x.A}{ A(a/x)}
\end{array}
\]

\caption{Ecumenical natural deduction system $\mathcal{NE}$.  In rules
$\forall\mbox{-int}$ and $\exists_i\mbox{-elim}$, the parameter $a$ is fresh.}\label{fig:NE}
\end{figure}

The propositional fragment of the natural deduction Ecumenical system proposed by Prawitz (here called $\mathcal{NE}_{p}$) has been proved normalizing, sound and complete with respect to  intuitionistic logic's Kripke semantics in~\cite{luiz17}.

The rules for intuitionistic implication are the traditional ones, while the rules for classical implication make sure that $A\to_{c} B$ is treated as $\neg A\lor_c B$, its classical rendering. The surprising facts are that (i) one can have a single constant for absurdity $\bot$ (instead of two, one intuitionistic and one classical, taking that absurd as the unit of disjunction, of which we have two variants) and (ii) that the intuitionistic and classical negations coincide.
If negation was simply implication into false (as it is the case for intuitionistic negation) one might expect two negations, one intuitionistic and one classical.

\section{Sequent Calculus with \textit{stoup}}\label{sec:sc}

$\LC$~\cite{DBLP:journals/mscs/Girard91} is a sequent system for classical logic that separates the rules for {\em positive} and {\em negative} formulas, being a precursor of the notion of {\em focusing} in sequent systems~\cite{andreoli92jlc}. The polarity of a formula in $\LC$ is determined by its outermost connective and the polarity of its subformulas. For example, atoms and unities are always positive and, if $P,Q$ are positive formulas, then  $P\wedge Q$ is also positive. The table of polarities can be checked in~\cite{DBLP:journals/mscs/Girard91}, page 10.

In $\LC$,  sequents have the form
$\seq \Delta ; \Sigma$, where $\Delta , \Sigma$ are multisets of formulas, with $\Sigma$, called the {\em stoup}, containing {\em at most} one formula. The main idea is that the {\em stoup} controls the rule applications, in the sense that a positive active formula in the conclusion of a rule is always placed there, while active negative formulas are handled in the classical context.
Characteristic examples are the rules for the conjunction of positive/negative formulas
\[
\infer[\wedge_p]{\seq \Gamma,\Delta ; P\wedge Q}{\seq \Gamma ; P & \seq \Delta ; Q}\qquad
\infer[\wedge_n]{\seq \Gamma,M\wedge N;\Sigma}{\seq \Gamma, N;\Sigma & \seq \Gamma,M ; \Sigma}
\]
Observe that they also have a multiplicative/additive flavor.

The idea of focusing is also present in $\LC$, with the {\em dereliction} and {\em store} rules
\[
\infer[der]{\seq \Gamma,P;\cdot}{\seq \Gamma;P}\qquad \qquad
\infer[store]{\seq \Gamma;N}{\seq \Gamma,N;\cdot}{}
\]
While in $der$ positive formulas can be chosen to be focused on, in $store$ negative formulas are stored in the classical context, in a bottom-up reading of rules. This enables for a {\em two-phase} proof construction, where the focused formula $P$ is systematically decomposed until reaching a leaf or a negative sub-formula $N$. In this last case, focusing is lost and $N$ is stored, allowing for the beginning of a new focused phase.

Finally, due to polarities, $\LC$ has two admissible cut rules
\[
\infer[p-cut]{\seq \Gamma,\Delta;\Sigma}{\seq \Gamma;P & \seq \neg P,\Delta;\Sigma}\qquad\qquad
\infer[n-cut]{\seq \Gamma,\Delta;\Sigma}{\seq \Gamma,N;\cdot & \seq \neg N,\Delta;\Sigma}
\]

In $\LC$, the sequent $\seq \Delta ; \Sigma$ has
an intuitionistic interpretation: $\neg \Delta\seq \bot$ if $\Sigma$ is empty and
$\neg \Delta\seq A$ if $\Sigma=A$. That is, the context $\Delta$ makes the classical information persistent, via an implicit double negation elimination. This implies that sequents with empty {\em stoup} have also a classical interpretation, using \eg\ G\"{o}del's double negation translation. Sequents with non-empty stoup do not have a classical interpretation, as discussed in~\cite{DBLP:journals/mscs/Girard91}.

Hence one could say that sequents with stoup have a certain ecumenical flavor: formulas with intuitionistic behavior are identified as being {\em positive}, while formulas with classical
behavior are identified as being {\em negative}.

In this work, we will carry out a similar idea under the spectrum of Prawitz' ecumenical natural deduction system.
While it has some similarities with Girard's original proposal, our system will not consider polarities, and all the conclusion formulas of introduction/elimination rules will be placed in the {\em stoup}.

\section{Natural Deduction with \textit{stoup} - the propositional system $\mathcal{LE}_{p}$}

We  will now incorporate the notion of {\em stoups} to natural deduction in the case of propositional logic, showing its natural connection to the ecumenical setting.

Let the expression  $\Delta;\Sigma$ denote an {\em stoup with a context} (abreviated as \textit{stp-c}), an extension of natural deduction formulas, where $\Sigma$ is the {\em stoup} and  $\Delta$ is its accompanying  context (called {\em alternatives} in~\cite{restall}). As for the case of $\LC$, the {\em stoup} will carry the intuitionistic (positive) and neutral information, while the context accumulates the classical information related to it.

In the following, we will construct ecumenical introduction and elimination rules over the {\em stoup} in a step-by-step manner, justifying all our choices.

\subsection{Intuitionistic operators}
\subsubsection*{Implication.}
The following result, proved in~\cite{DBLP:journals/synthese/PimentelPP21} for the sequent calculus ecumenical system, states that logical consequence in $\mathcal{NE}$ is interpreted intuitionistically.
\begin{theorem}\label{prop:ev}
Let $\Gamma$ be a set of ecumenical formulas. Then
$B$ is provable from $\Gamma$ in ecumenical logic iff $\bigwedge\Gamma\iimp B$ is provable in $\mathcal{NE}$.
\end{theorem}
Hence the rule
\[
\infer[{\iimp\mbox{-int}}]{ A\iimp B}
{\deduce{B}{\deduce{}{\deduce{\Pi}{\deduce{}{[A]}}}}}
\]
induces the rule with {\em stoup}
\[
\infer[{\iimp\mbox{-int}}]{\Delta;A\iimp B}{\deduce{\Delta;B}{\deduce{}{\deduce{\Pi}{\deduce{}{[\cdot;A] & \Gamma}}}}}
\]
where the {\em stoup} is preserved in discarded assumptions. Note that all the information $\Gamma$ about contexts should be remembered.

For the implication elimination rule, observe that different {\em stoups} carry different contexts, so we will have the multiplicative version of the rule, combining the classical information
\begin{prooftree}
\AxiomC{$\Delta_{1};A \to_{i} B$}
\AxiomC{$\Delta_{2};A$}
\RightLabel{$\to_{i}$-elim}
\BinaryInfC{$\Delta_{1},\Delta_{2};B$}
\end{prooftree}

\subsubsection*{Disjunction.}
The rule for introduction is
\begin{prooftree}
\AxiomC{$\Delta ;A_i$}
\RightLabel{$\vee_{i}$-int}
\UnaryInfC{$\Delta ;A_1 \vee_{i} A_2$}
\end{prooftree}

\noindent while the elimination rule combines the all the context information

\begin{prooftree}
\AxiomC{$\Gamma_1$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1};A \vee_{i} B$}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_2$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2};C$}
\AxiomC{$[\cdot ;B]$}
\AxiomC{$\Gamma_3$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3};C$}
\RightLabel{$\vee_{i}$-elim}
\TrinaryInfC{$\Delta_{1}, \Delta_{2}, \Delta_{3};C$}
\end{prooftree}

\subsection{Classical operators}
\subsubsection*{Implication.} Observe that the negated assumptions in $\mathcal{NE}$ will correspond to the classical counterpart of the stoup, so the consequent of the implication will be stored in this context. The introduction rule
\[
\infer[{\cimp\mbox{-int}}]{A\cimp B}
{\deduce{\bot}{\deduce{}{\deduce{\Pi}{\deduce{}{[A,\neg B]}}}}}
\]
then becomes
\begin{prooftree}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ,B;\cdot$}
\RightLabel{$\to_{c}$-int}
\UnaryInfC{$\Delta ;A \to_{c} B$}
\end{prooftree}
For the elimination rule, also the negated formula in the premise become classical, this time with empty stoup.

\begin{prooftree}
\AxiomC{$\Delta_1; A \to_{c} B$}
\AxiomC{$\Gamma_2$}
\noLine
\UnaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2};A$}
\AxiomC{$[\cdot;B]$}
\AxiomC{$\Gamma_3$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\Delta_{3} ;\cdot$}
\RightLabel{$\to_{c}$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}

\subsubsection*{Disjunction.}
The same idea applies to disjunction, where negated assumptions become part of the classical context, while positive assumptions are kept as {\em stoups}
\begin{prooftree}
\AxiomC{$\Delta ,A,B; \cdot $}
\RightLabel{$\vee_{c}$-int}
\UnaryInfC{$\Delta ;A \vee_{c} B$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$\Delta_1;A \vee_{c} B$}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2} ;\cdot$}
\AxiomC{$[\cdot ;B]$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\Delta_{3} ;\cdot$}
\RightLabel{$\vee_{c}$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3} ; \cdot$}
\end{prooftree}

\subsection{Neutral operators}
\subsubsection*{Negation.} Since negation can be defined in classical/intuitionistic logic as ``implies bottom'', the rules for $\neg$ can be derived from the ones for implication with an empty {\em stoup}.

\begin{prooftree}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ; \cdot$}
\RightLabel{$\neg$-int}
\UnaryInfC{$\Delta ;\neg A$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$\Delta_{1};A$}
\AxiomC{$\Delta_{2};\neg A$}
\RightLabel{$\neg$-elim}
\BinaryInfC{$\Delta_{1}, \Delta_{2}; \cdot$}
\end{prooftree}

\subsubsection*{Conjunction.} Since our ecumenical system is essentially intuitionistic (in the terms of Proposition~\ref{prop:ev}), all active formulas are placed in the {\em stoup}. Hence we will adopt the multiplicative version of Girard's rule for positive conjuncts.

\begin{prooftree}
\AxiomC{$\Delta_{1};A$}
\AxiomC{$\Delta_{2};B$}
\RightLabel{$\wedge$-int}
\BinaryInfC{$\Delta_{1},\Delta_{2};A \wedge B$}
\end{prooftree}

\begin{prooftree}
\AxiomC{$\Delta ;A_1 \wedge A_2$}
\RightLabel{$\wedge$-elim$_j$}
\UnaryInfC{$\Delta ;A_j$}
\end{prooftree}

\subsection{Hypothesis formation and dereliction}
The hypothesis formation is the usual one and, as in $\LC$, dereliction is needed for guaranteeing  the completeness of the system, since ecumenical active formulas are always placed in the {\em stoup}.

\noindent
\textit{Hypothesis formation}
\begin{prooftree}
\AxiomC{$\cdot;A$}
\end{prooftree}

\textit{Dereliction}
\begin{prooftree}
\AxiomC{$\Delta ;A$}
\RightLabel{der}
\UnaryInfC{$\Delta ,A;\cdot $}
\end{prooftree}

\subsection{Structural rules}
Finally, on choosing the multiplicative version of rules, we need structural rules acting in the classical context, so to transform multisets into sets. As usual in intuitionistic systems, weakening is also allowed in the {\em stoup}.

\textit{Weakening}
\begin{prooftree}
\AxiomC{$\Delta ; \cdot$}
\RightLabel{$W_i$}
\UnaryInfC{$\Delta  ; A$}
\DisplayProof
\qquad \qquad
\AxiomC{$\Delta ; C$}
\RightLabel{$W_c$}
\UnaryInfC{$\Delta , A ; C$}
\end{prooftree}

\textit{Contraction}
\begin{prooftree}
\AxiomC{$\Delta , A,A ; C$}
\RightLabel{$C_c$}
\UnaryInfC{$\Delta , A ; C$}
\end{prooftree}

Derivations are then  inductively defined in the usual way.
\begin{definition}\label{def:der}
We say that the \textit{stp-c} $\Delta ;\Sigma$ is {\em derivable} from a set $\Gamma$ of \textit{stp-cs}  in $\mathcal{LE}_{p}$ (denoted by $\Gamma \vdash_{\mathcal{LE}_p} \Delta ;\Sigma $) if and only if there is a derivation of $\Delta ;\Sigma$ from $\Gamma$. A formula $A$ is a {\em theorem} of $\mathcal{LE}_{p}$ if and only if $\vdash_{\mathcal{LE}_p} \cdot ; A$.
\end{definition}
As usual, we also may add indices in derivations, for relating a discharged assumption with a specific rule application.

\section{Examples}
We present below the proofs of some classical tautologies in $\mathcal{LE}_{p}$.
\begin{enumerate}
\item \textit{Peirce's Law}
\begin{prooftree}
\AxiomC{$[\cdot ; ((A \to_{c} B) \to_{c} A)]^{3}$}
\AxiomC{$[\cdot ;A]^{1}$}
\RightLabel{der}
\UnaryInfC{$A;\cdot$}
\RightLabel{$W_{c}$}
\UnaryInfC{$A,B;\cdot$}
\RightLabel{$\to_{c}$-int}
\LeftLabel{1}
\UnaryInfC{$A;(A \to_{c} B)$}
\AxiomC{$[\cdot ;A]^{2}$}
\RightLabel{der}
\UnaryInfC{$A;\cdot$}
\RightLabel{$\to_{c}$-elim}
\LeftLabel{2}
\TrinaryInfC{$A,A; \cdot $}
\RightLabel{$C_{c}$}
\UnaryInfC{$A; \cdot $}
\LeftLabel{3}
\RightLabel{$\to_{c}$-int}
\UnaryInfC{$\cdot ; (((A \to_{c} B) \to_{c} A) \to_{c} A)$}
\end{prooftree}
Observe that the only difference of the proof above w.r.t. $Alt$ or $Restart$ systems lies in the use of structural rules in the classical context.

More interestingly, note that any sequent of the form  $(((A \to_{j} B) \to_{k} A) \to_{c} A)$ with $j,k\in\{i,c\}$ is provable in  $\mathcal{LE}_{p}$. That is, provability is maintained if the outermost implication is classical.
\item \textit{Excluded-middle}
\begin{prooftree}
\AxiomC{$[\cdot ; A]^{1}$}
\RightLabel{der}
\UnaryInfC{$A; \cdot $}
\RightLabel{$\neg$-int}
\LeftLabel{1}
\UnaryInfC{$ A ; \neg A$}
\RightLabel{der}
\UnaryInfC{$ A , \neg A ; \cdot$}
\RightLabel{$\vee_{c}$-int}
\UnaryInfC{$ \cdot ; (A \vee_{c} \neg A) $}
\end{prooftree}
Of course, $ \cdot ;A \vee_{i} \neg A$ is not a theorem in $\mathcal{LE}_{p}$.
\item \textit{Dummett's linearity axiom}
\begin{prooftree}
\AxiomC{$[\cdot ;A]^{1}$}
\RightLabel{der}
\UnaryInfC{$A; \cdot$}
\RightLabel{$W_{c}$}
\UnaryInfC{$A, B;\cdot $}
\RightLabel{$\to_{c}$-int}
\LeftLabel{$1$}
\UnaryInfC{$A;(A \to_{c} B)$}
\RightLabel{der}
\UnaryInfC{$A, (A \to_{c} B);\cdot $}
\RightLabel{$\to_{c}$-int}
\UnaryInfC{$(A \to_{c} B); (B \to_{c} A)$}
\RightLabel{der}
\UnaryInfC{$(A \to_{c} B), (B \to_{c} A)); \cdot$}
\RightLabel{$\vee_{c}$-int}
\UnaryInfC{$\cdot ; ((A \to_{c} B) \vee_{c}  (B \to_{c} A))$}
\end{prooftree}
This is also an interesting case, where any sequent of the form  $ ((A \to_{j} B) \vee_{c}  (B \to_{k} A))$ with $j,k\in\{i,c\}$ is provable in  $\mathcal{LE}_{p}$. That is, provability is maintained if the outermost conjunction is classical.
\end{enumerate}

\section{Systems equivalence}
In the following, we will show that $\mathcal{LE}_{p}$ is correct and complete w.r.t. $\mathcal{NE}_p$. We will use the following extra notation:
\begin{itemize}
\item[-] Given a multiset $\Delta$ of
formulas,  we denote by $\neg \Delta $ the multiset formed by the negation of each formula in $\Delta$.
\item[-] If $\Gamma$ is a set of formulas, we denote by $\Gamma \vdash_{\mathcal{NE}_p} A$ the fact that the formula $A$ depends on the set $\Gamma$ of assumptions  in $\mathcal{NE}_{p}$.
\item[-] If $\Gamma=\bigcup\{\cdot;A_i\}_{0\leq i\leq n}$ is a multiset of {\em stoup} with empty contexts  in $\mathcal{LE}_{p}$,  we will abuse the notation and also represent by $\Gamma$ the underlying set of formulas in these {\em stp-c}, that is, $\Gamma=\bigcup\{A_i\}_{0\leq i\leq n}$ in $\mathcal{NE}_{p}$.
\end{itemize}
\begin{theorem}
Let $\Gamma=\bigcup\{\cdot;A_i\}_{0\leq i\leq n}$ be a set of hypothesis. Then
$\Gamma  \vdash_{\mathcal{LE}_p} \Delta;\Sigma$ iff $\Gamma , \neg \Delta\vdash_{\mathcal{NE}_p} \Sigma$. In case $\Sigma$ is empty, we have $\Gamma , \neg \Delta\vdash_{\mathcal{NE}_p} \bot$.
\end{theorem}
\begin{proof}
By induction on the length of derivations in $\mathcal{LE}_{p}$ and $\mathcal{NE}_{p}$. The only interesting cases are the ones involving classical connectives and structural rules.
\begin{itemize}
\item Case $\cvee$-int. Suppose that we have the following derivation  in $\mathcal{LE}_p$:
\begin{prooftree}
\AxiomC{$\Gamma$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta,A,B;\cdot$}
\RightLabel{$\cvee$-int}
\UnaryInfC{$\Delta;A\cvee B $}
\end{prooftree}
By the inductive hypothesis, $\Gamma , \neg A,\neg B,\neg \Delta  \vdash_{\mathcal{NE}_p}\bot$. We can then take the desired derivation to be:
\begin{prooftree}
\AxiomC{$\Gamma\quad \neg \Delta\quad  [\neg A]^{n}\quad  [\neg B]^{m}$}
\noLine
\UnaryInfC{$\Pi'$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$n,m$}
\RightLabel{$\cvee$-int}
\UnaryInfC{$A\cvee B $}
\end{prooftree}

On the other hand, suppose that $ \Gamma , \neg \Delta \vdash_{\mathcal{NE}_p} A\cvee B$  with proof
\begin{prooftree}
\AxiomC{$\Gamma\quad \neg \Delta\quad  [\neg A] \quad [\neg B]$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$\cvee$-int}
\UnaryInfC{$A\cvee B $}
\end{prooftree}
By the inductive hypothesis, we have $\Gamma \vdash_{\mathcal{LE}_p} \Delta,A,B;\cdot$. We can then take the desired derivation to be:
\begin{prooftree}
\AxiomC{$\Gamma$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta,A,B;\cdot$}
\RightLabel{$\cvee$-int}
\UnaryInfC{$\Delta;A\cvee B $}
\end{prooftree}

\item Case $\cvee$-elim.
Suppose that $\Gamma_{1}, \Gamma_{2}, \Gamma_{3} \vdash_{\mathcal{LE}_p}\Delta_{1},\Delta_{2},\Delta_{3};\cdot$  with proof
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_1$}
\noLine
\UnaryInfC{$\Delta_1; A \cvee B$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$[\cdot;A]$}
\noLine
\BinaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2};\cdot$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$[\cdot;B]$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\Delta_{3} ;\cdot$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}
By the inductive hypothesis we have: $\Gamma_{1}, \neg \Delta_1 \vdash_{\mathcal{NE}_p} A \cvee B$, $\Gamma_{2}, \neg \Delta_2, A\vdash_{\mathcal{NE}_p} \bot$ and
$\Gamma_{3}, \neg \Delta_3,B\vdash_{\mathcal{NE}_p} \bot$. We can the obtain the desired derivation as follows:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\neg\Delta_1$}
\noLine
\BinaryInfC{$\Pi_1'$}
\noLine
\UnaryInfC{$A \cvee B$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$\neg \Delta_2$}
\AxiomC{$[A]^{n}$}
\noLine
\TrinaryInfC{$\Pi_2'$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$n$}
\RightLabel{$\neg$-int}
\UnaryInfC{$\neg A$}
\AxiomC{$\Gamma_ {3}$}
\AxiomC{$\neg\Delta_3$}
\AxiomC{$[B]^{m}$}
\noLine
\TrinaryInfC{$\Pi_3'$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$m$}
\RightLabel{$\neg$-int}
\UnaryInfC{$\neg B$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\bot$}
\end{prooftree}
On the other hand, suppose that $\Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \neg [\Delta_1],\neg [\Delta_2],\neg [\Delta_3]\vdash_{\mathcal{NE}_p} \bot$  with the following derivation:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\neg \Delta_1$}
\noLine
\BinaryInfC{$\Pi_1$}
\noLine
\UnaryInfC{$A \cvee B$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$\neg \Delta_2$}
\noLine
\BinaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\neg A$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$\neg \Delta_3$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\neg B$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\bot$}
\end{prooftree}
By the  inductive hypothesis: $\Gamma_{1}, \vdash_{\mathcal{LE}_p} \Delta_1;A\cvee B$, $\Gamma_{2}, \vdash_{\mathcal{LE}_p} \Delta_2;\neg A$ and $\Gamma_{3} \vdash_{\mathcal{LE}_p} \Delta_3;\neg B$. We can then construct our desired derivation as:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_1'$}
\noLine
\UnaryInfC{$\Delta_1; A \cvee B$}
\AxiomC{$\Gamma_{2}$}
\noLine
\UnaryInfC{$\Pi_2'$}
\noLine
\UnaryInfC{$\Delta_{2};\neg A$}
\AxiomC{$[\cdot; A]$}
\RightLabel{$\neg$-int}
\BinaryInfC{$\Delta_{2};\cdot$}
\AxiomC{$\Gamma_3$}
\noLine
\UnaryInfC{$\Pi_3'$}
\noLine
\UnaryInfC{$\Delta_{3};\neg B$}
\AxiomC{$[\cdot; B]$}
\RightLabel{$\neg$-int}
\BinaryInfC{$\Delta_{2};\cdot$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}

\item Case $\cimp$-int.
Suppose that $\Gamma \vdash_{\mathcal{LE}_p}\Delta;A\cimp B$  with a derivation as:
\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$[\cdot ;A]$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ,B;\cdot$}
\RightLabel{$\to_{c}$-int}
\UnaryInfC{$\Delta ;A \to_{c} B$}
\end{prooftree}
By the inductive hypothesis, we have $ \Gamma , A,\neg B,\neg \Delta \vdash_{\mathcal{NE}_p}\bot$. We can then construct our desired derivation in $\mathcal{NE}_{p}$ as:

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$[A]$}
\AxiomC{$[\neg B]$}
\AxiomC{$\neg\Delta$}
\noLine
\QuaternaryInfC{$\Pi'$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$\cimp$-int}
\UnaryInfC{$A\cimp B $}
\end{prooftree}
In the other direction, suppose that $ \Gamma ,\neg \Delta  \vdash_{\mathcal{NE}_p} A\cimp B$ with a derivation as:
\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$\neg \Delta$}
\AxiomC{$[A]$}
\AxiomC{$[\neg B]$}
\noLine
\QuaternaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$\cimp$-int}
\UnaryInfC{$A\cimp B $}
\end{prooftree}
By the inductive hypothesis, we have $ \Gamma\cup\{  \cdot ; A \} \vdash_{\mathcal{LE}_p} \Delta ,B;\cdot$. Thus, we can obtain the desired derivation in $\mathcal{LE}_{p}$ as:
\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$[\cdot;A]$}
\noLine
\BinaryInfC{$\Pi'$}
\noLine
\UnaryInfC{$\Delta,B;\cdot$}
\RightLabel{$\cvee$-elim}
\UnaryInfC{$\Delta;A\cimp B $}
\end{prooftree}
\item Case $\cimp$-elim.
Suppose that $ \Gamma_{1}, \Gamma_{2}, \Gamma_{3}  \vdash_{\mathcal{LE}_p}\Delta_{1},\Delta_{2},\Delta_{3};\cdot$   with a derivation as:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_1$}
\noLine
\UnaryInfC{$\Delta_1; A \to_{c} B$}
\noLine
\AxiomC{$\Gamma_{2}$}
\noLine
\UnaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2};A$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$[\cdot;B]$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\Delta_{3} ;\cdot$}
\RightLabel{$\to_{c}$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}
By inductive hypothesis, we have $ \Gamma_{1} ,\neg \Delta_1  \vdash_{\mathcal{NE}_p} A \to_{c} B$, $ \Gamma_{2}, \neg \Delta_2 \vdash_{\mathcal{NE}_p} A$ and
$ \Gamma_{3}, \neg\Delta_3 ,B \vdash_{\mathcal{NE}_p} \bot$. Then, we can obtain the desired derivation in $\mathcal{NE}_{p}$ as:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\neg \Delta_1$}
\noLine
\BinaryInfC{$\Pi_1'$}
\noLine
\UnaryInfC{$A \to_{c} B$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$\neg \Delta_2$}
\noLine
\BinaryInfC{$\Pi_2'$}
\noLine
\UnaryInfC{$A$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$\neg \Delta_3 \quad[B]$}
\noLine
\BinaryInfC{$\Pi_3'$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$\neg$-int}
\UnaryInfC{$\neg B$}
\RightLabel{$\to_{c}$-elim}
\TrinaryInfC{$\bot$}
\end{prooftree}
In the other direction, suppose that $ \Gamma_{1}, \Gamma_{2}, \Gamma_{3}, \neg \Delta_1,\neg \Delta_2,\neg \Delta_3  \vdash_{\mathcal{NE}_p} \bot$   with a derivation as:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\neg \Delta_1$}
\noLine
\BinaryInfC{$\Pi_1$}
\noLine
\UnaryInfC{$A \cimp B$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$\neg \Delta_2$}
\noLine
\BinaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$A$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$\neg \Delta_3$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\neg B$}
\RightLabel{$\cimp$-elim}
\TrinaryInfC{$\bot$}
\end{prooftree}
By the inductive hypothesis, we have that $\Gamma_{1} \vdash_{\mathcal{LE}_p} \Delta_1;A\cimp B$, $\Gamma_{2} \vdash_{\mathcal{LE}_p} \Delta_2; A$ and $\Gamma_{3} \vdash_{\mathcal{LE}_p} \Delta_3;\neg B$. We can then  obtain the desired derivation in $\mathcal{LE}_{p}$ as:

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_1'$}
\noLine
\UnaryInfC{$\Delta_1; A \cimp B$}
\AxiomC{$\Gamma_{2}$}
\noLine
\UnaryInfC{$\Pi_2'$}
\noLine
\UnaryInfC{$\Delta_{2};A$}
\AxiomC{$\Gamma_{3}$}
\noLine
\UnaryInfC{$\Pi_3'$}
\noLine
\UnaryInfC{$\Delta_{3};\neg B$}
\AxiomC{$[\cdot; B]$}
\RightLabel{$\neg$-int}
\BinaryInfC{$\Delta_{2};\cdot$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}

\item Case der. Consider the derivation
\begin{prooftree}
\AxiomC{$\Gamma$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ;A$}
\RightLabel{der}
\UnaryInfC{$\Delta ,A;\cdot $}
\end{prooftree}
By the  inductive hypothesis, we have $ \Gamma ,\neg \Delta \vdash_{\mathcal{NE}_p} A$. We can then obtain the desired derivation in $\mathcal{NE}_{p}$ as:

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$\neg\Delta$}
\noLine
\BinaryInfC{$\Pi'$}
\noLine
\UnaryInfC{$A$}
\AxiomC{$[\neg A]$}
\RightLabel{$\neg$-int}
\BinaryInfC{$\bot $}
\end{prooftree}
The cases of the other structural rules are trivial.
\end{itemize}
\end{proof}

\section{Normalization}
We will now describe how normalization works in the natural deduction with {\em stoup} setting.
The idea follows the usual one for natural deduction systems: show how to {\em compose} derivations, so to eliminate detours. The presence of {\em stoups}, however, adds an extra case analysis, since the composition may occur in the {\em stoup} or in the classical context. Both processes will be carefully described in what follows.
\subsection{Composition}
Before we define reductions and prove the normalization theorem for $\mathcal{LE}_{p}$, we must guarantee that the process of \textit{composition} of derivations is preserved in $\mathcal{LE}_{p}$. As quickly mentioned in Sec.~\ref{sec:sc}, sequent systems with {\em stoup} usually allow for two types of cut: a cut where the left cut-formula is in the \textit{stoup} and a cut where the left cut-formula is in the classical region. These two types of cut will correspond to two modes of composition: a composition that occurs in the \textit{stoup} and a composition that occurs in the classical context. We will detail these two forms of compositions below.
\begin{enumerate}
\item Composition in the \textit{stoup}.
\begin{theorem}\label{thm:comp-stoup}
Let $\Pi_{1}$ be a derivation of $\Gamma_{1} \vdash_{\mathcal{LE}_p} \Delta_{1} ; A$ and $\Pi_{2}$ be a derivation of $\Gamma_{2} \cup \{\cdot ;A\} \vdash_{\mathcal{LE}_p} \Delta_{2} ;B$. Then, the result of replacing the assumption $\cdot ;A$ in $\Pi_{2}$ by the derivation $\Pi_{1}$ is a derivation $\Pi$ of $\Gamma_{1} , \Gamma_{2} \vdash_{\mathcal{LE}_p} \Delta_{1}  , \Delta_{2}; B$.
\end{theorem}
\begin{proof}
By induction on the length of $\Pi_{2}$. We will examine two cases, the other cases being treated in a similar way.\\
\begin{enumerate}
\item The last rule applied in $\Pi_{2}$ is $\vee_{c}$-elim:
\begin{prooftree}
\AxiomC{$\Gamma_{2}^{1}$}
\AxiomC{$\cdot ;A$}
\noLine
\BinaryInfC{$\Pi_{2}^1$}
\noLine
\UnaryInfC{$\Delta_{2}^{1};C \vee_{c} D$}
\AxiomC{$[\cdot ;C]$}
\AxiomC{$\cdot ;A$}
\AxiomC{$\Gamma_{2}^{2}$}
\noLine
\TrinaryInfC{$\Pi_{2}^2$}
\noLine
\UnaryInfC{$\Delta_{2}^{2} ;\cdot$}
\AxiomC{$[\cdot ;D]$}
\AxiomC{$\cdot ;A$}
\AxiomC{$\Gamma_{2}^{3}$}
\noLine
\TrinaryInfC{$\Pi_{2}^3$}
\noLine
\UnaryInfC{$\Delta_{2}^{3} ;\cdot$}
\RightLabel{$\vee_{c}$-elim}
\TrinaryInfC{$\Delta_{2} ; \cdot$}
\end{prooftree}
where $\Delta_2^{i}$ indicates a partition of $\Delta_2$, the same with $\Gamma_2$.
By the induction hypothesis we obtain the following derivations:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\Gamma_{2}^{1}$}
\noLine
\BinaryInfC{$\Pi_{1}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{1} ; (C \vee_{c} D)$}
\end{prooftree}

and

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$[\cdot ; C]$}
\AxiomC{$\Gamma_{2}^{2}$}
\noLine
\TrinaryInfC{$\Pi_{2}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{2} ; \cdot$}
\end{prooftree}

and

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$[\cdot ; D]$}
\AxiomC{$\Gamma_{2}^{3}$}
\noLine
\TrinaryInfC{$\Pi_{3}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{3} ; \cdot$}
\end{prooftree}

The resulting derivation $\Pi$ is:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\Gamma_{2}^{1}$}
\noLine
\BinaryInfC{$\Pi_{1}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{1} ; (C \vee_{c} D)$}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$[\cdot ; C]$}
\AxiomC{$\Gamma_{2}^{2}$}
\noLine
\TrinaryInfC{$\Pi_{2}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{2} ; \cdot$}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$[\cdot ; D]$}
\AxiomC{$\Gamma_{2}^{3}$}
\noLine
\TrinaryInfC{$\Pi_{3}^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{3} ; \cdot$}
\RightLabel{$\vee_{c}$-elim}
\TrinaryInfC{$\Delta_{1} , \Delta_{1} , \Delta_{1} , \Delta_{2} ; \cdot$}
\doubleLine
\RightLabel{$C_c$}
\UnaryInfC{$\Delta_{1} , \Delta_{2} ; \cdot$}
\end{prooftree}

\noindent Where the \textit{double line} indicates several applications of $C_c$.

\item The last rule applied in $\Pi_{2}$ is dereliction:

\begin{prooftree}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$\cdot ;A$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ;B$}
\RightLabel{der}
\UnaryInfC{$\Delta_{2} ,B;\cdot $}
\end{prooftree}

By the induction hypothesis we obtain the following derivation:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ; B$}
\end{prooftree}

The resulting derivation $\Pi$ is then:

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi^{*}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ; B$}
\RightLabel{der}
\UnaryInfC{$\Delta_{1} , \Delta_{2} , B; \cdot $}
\end{prooftree}

\end{enumerate}
\end{proof}
\item Composition in the \textit{context}.
\begin{theorem}
Let $\Pi_{1}$ be a derivation of $\Gamma_{1} \vdash_{\mathcal{LE}_p} \Delta_{1};C$, where $A \in \Delta_{1}$, and $\Pi_{2}$ be a derivation of $ \Gamma_{2}\cup\{\cdot; A\} \vdash_{\mathcal{LE}_p} \Delta_{2} ;B$. Then, the result of replacing the assumption $\cdot ;A$ in $\Pi_{2}$ by the derivation $\Pi_{1}$ is a derivation $\Pi$ of $\Gamma_{1} , \Gamma_{2} \vdash_{\mathcal{LE}_p} \Delta_{1}^{*}  , \Delta_{2}, B;C$, where  $\Delta_{1}^{*}$ is obtained from $\Delta_{1}$ by means of the elimination of the occurrences of $A$.
\end{theorem}
\begin{proof}
The proof is by straightforward induction on the length of $\Pi_{1}$. The only exception is dereliction, which uses Theorem~\ref{thm:comp-stoup}.\\ \\
Let the last rule applied in $\Pi_{1}$ be \textit{dereliction}:

\begin{prooftree}
\AxiomC{$\Gamma_1$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; A$}
\RightLabel{der}
\UnaryInfC{$\Delta_{1} , A; \cdot $}
\end{prooftree}
\end{proof}

By the induction hypothesis we can obtain a derivation $\Pi'$  of  $\Gamma_{1}, \Gamma_{2} \vdash_{\mathcal{LE}_p} \Delta_{1}^{*} , \Delta_{2}, B;A$. By Theorem 2 we can obtain a derivation $\Pi''$ of
$\Gamma_{1}, \Gamma_{2} \vdash_{\mathcal{LE}_p} \Delta_{1}^{*} , \Delta_{2}, \Delta_{2}, B;B$. We can now take the derivation $\Pi$ to be:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi''$}
\noLine
\UnaryInfC{$\Delta_{1}^{*} , \Delta_{2}, \Delta_{2}, B;B$}
\RightLabel{$der$}
\UnaryInfC{$\Delta_{1}^{*} , \Delta_{2}, \Delta_{2}, B, B; \cdot$}
\doubleLine
\RightLabel{$C_{c}$}
\UnaryInfC{$\Delta_{1}^{*} , \Delta_{2}, B; \cdot$}
\end{prooftree}

\end{enumerate}

In what follows, we shall use the following notation to indicate composition in the \textit{stoup}  and composition  in the \textit{classical context}.
\begin{itemize}
\item Composition in the \textit{stoup}:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; [\frac{A}{\cdot ;A}]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ;B$}
\end{prooftree}

\item Composition in the \textit{classical context}:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} , [\frac{A}{\cdot ;A}]; B$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ;B$}
\end{prooftree}

\item A more concise notation for composition in general: $\Pi_{1}/[\frac{A}{\cdot ;A}]/\Pi_{2}$.

\end{itemize}

\subsection{Reductions}
Derivations in $\mathcal{LE}_{p}$ may contain \textit{detours}. These detours are of two types: we may introduce a formula by an application of an introduction rule to immediately use it as major premiss of an application of an elimination rule; or we may introduce a formula by an application of an introduction rule and use it as major premiss of an application of an elimination rule after several applications of $\vee_{i}$-elim. The \textit{reductions} defined in this section are intended, as usual, to eliminate \textit{detours} that may occur in a derivation.
\begin{definition}
A {\em segment} in a derivation $\Pi$ is a sequence $A_{1},. . . , A_{n}$  of consecutive formulas  in a thread in $\Pi$ such that:
\begin{itemize}
\item $A_{1}$ is not in the stoup of the consequence of an application of $\vee_{i}$-elim or of an application of $C_{c}$;
\item $A_{j}$, for $j < n$, is in the stoup of the minor premiss of an application of $\vee_{i}$-elim or of an application of $C_{c}$; and
\item $A_{n}$ is not in the stoup of  the minor premiss of an application of $\vee_{i}$-elim or of an application of $C_{c}$.
\end{itemize}
\end{definition}
We note  the presence of contraction in the last definition. The idea is that contractions move down on reductions, just like in a sequent calculus' cut-elimination process.

\begin{definition}
A segment that begins with with the consequence of an application of an introduction rule or $W_{i}$ and ends with an application of an elimination rule is called a {\em maximal segment}. A maximal segment of length $1$ is called a {\em maximum formula}.
\end{definition}

\begin{definition}
Let $\Pi$ be a derivation in $\mathcal{LE}_{p}$. The {\em degree of $\Pi$}, $d[\Pi ]$, is defined as $\max\{d[A]:$ A is the end-formula of maximal segment in $\Pi \}$, where $d[A]$ is the {\em weight} of the formula $A$, defined inductively by
$$
\begin{array}{lcl}
d[\bot]=d[p] &=& 0 \quad p\mbox{ atomic.}\\
d[A \circ B] &=& d[A] + d[B] + 1 \mbox{ for } \circ\in\{\to_{i,c},\vee_{i,c},\wedge\}\\
d[\neg A] &=& d[A] +1.
\end{array}
$$
\end{definition}

\begin{definition}
A derivation $\Pi$ is called {\em normal} if and only if $d[\Pi ] = 0$.
\end{definition}
We will present next all the reduction steps in $\mathcal{LE}_{p}$ that will be used in the elimination of maximal segments.
\begin{enumerate}
\item $\wedge$-reduction:\\

The derivation
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ;A_{1}$}
\AxiomC{$\Gamma_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ;A_{2}$}
\RightLabel{$\wedge$-int}
\BinaryInfC{$\Delta_{1} , \Delta_{2} ; (A_{1} \wedge A_{2})$}
\RightLabel{$\wedge_j$-elim}
\UnaryInfC{$\Delta_{1} , \Delta_{2} ;A_{j}$}
\end{prooftree}

Reduces to
\begin{prooftree}
\AxiomC{$\Gamma_{j}$}
\noLine
\UnaryInfC{$\Delta_{j} ;A_{j}$}
\RightLabel{$W_c$}
\doubleLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ; A_{j}$}
\end{prooftree}

\item $\to_{i}$-reduction:\\

The derivation
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; A$}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; B$}
\RightLabel{$\iimp$-int}
\UnaryInfC{$\Delta_{2} ; (A \to_{i} B)$}
\RightLabel{$\iimp$-elim}
\BinaryInfC{$\Delta_{1} , \Delta_{2} ;B$}
\end{prooftree}

Reduces to:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; [\frac{A}{\cdot ;A}]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} ; B$}
\end{prooftree}
Observe that the case for negation is analogous.

\item $\vee_{i}$-reduction:\\

The derivation

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ;A_{j}$}
\RightLabel{$\ivee$-int}
\UnaryInfC{$\Delta_{1} ; (A_{1} \vee_{i} A_{2})$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$[\cdot ;A_{1}]$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; B$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$[\cdot ;A_{2}]$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3} ; B$}
\RightLabel{$\ivee$-elim}
\TrinaryInfC{$\Delta_{1} , \Delta_{2} ,\Delta_{3} ; B$}

\end{prooftree}

Reduces to:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; [\frac{A_j}{\cdot ;A_{j}}]$}
\AxiomC{$\Gamma_{j}$}
\noLine
\BinaryInfC{$\Pi_{j}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{j} ; B$}
\doubleLine
\RightLabel{$W_c$}
\UnaryInfC{$\Delta_{1} , \Delta_{2}, \Delta_{3} ; B$}
\end{prooftree}

\item $\to_{c}$-reduction:\\

The derivation
\begin{prooftree}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{1}$}
\noLine
\BinaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{2} , B; \cdot$}
\RightLabel{$\cimp$-int}
\UnaryInfC{$\Delta_{2} ; (A \to_{c} B)$}
\AxiomC{$\Gamma_2$}
\noLine
\UnaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2};A$}
\AxiomC{$[\cdot;B]$}
\AxiomC{$\Gamma_3$}
\noLine
\BinaryInfC{$\Pi_3$}
\noLine
\UnaryInfC{$\Delta_{3} ;\cdot$}
\RightLabel{$\to_{c}$-elim}
\TrinaryInfC{$\Delta_{1},\Delta_{2},\Delta_{3};\cdot$}
\end{prooftree}

Reduces to:

\begin{prooftree}
\AxiomC{$\Gamma_2$}
\noLine
\UnaryInfC{$\Pi_2$}
\noLine
\UnaryInfC{$\Delta_{2};[\frac{A}{\cdot ;A}]$}
\AxiomC{$\Gamma_{1}$}
\noLine
\BinaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2} , [\frac{B}{\cdot ;B}]; \cdot$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{1} , \Delta_{2}^{*} , \Delta_{3} ; \cdot$}
\doubleLine
\RightLabel{$W_{c}$}
\UnaryInfC{$\Delta_{1} , \Delta_{2}, \Delta_{3} ; \cdot$}
\end{prooftree}

\item $\vee_{c}$-reduction:\\

The derivation

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} , A,B;\cdot$}
\RightLabel{$\cvee$-int}
\UnaryInfC{$\Delta_{1} ; (A \vee_{c} B)$}
\AxiomC{$\Gamma_{2}$}
\AxiomC{$[\cdot ;A]$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; \cdot$}
\AxiomC{$\Gamma_{3}$}
\AxiomC{$[\cdot ;B]$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3} ; \cdot$}
\RightLabel{$\cvee$-elim}
\TrinaryInfC{$\Delta_{1} , \Delta_{2} ,\Delta_{3} ; \cdot$}

\end{prooftree}

Reduces to:

\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} , [\frac{A}{\cdot ;A}],B;\cdot$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{1}^{*} , \Delta_{2} , [\frac{B}{\cdot ;B}];\cdot$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{1}^{*} , \Delta_{2}^{*} , \Delta_{3};\cdot$}
\doubleLine
\RightLabel{$W_{c}$}
\UnaryInfC{$\Delta_{1}, \Delta_{2}, \Delta_{3};\cdot$}
\end{prooftree}

\item Permutative reductions:\\
\begin{enumerate}
\item The derivation

{\small
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; (A \vee_{i} B)$}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; C$}
\AxiomC{$[\cdot ;B]$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3} ; C$}
\TrinaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} ; C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
}
where $C$ is the major premiss of an elimination rule with minor premisses $\Theta_{1};\Lambda_{1} \ldots \Theta_{m};\Lambda_{m}$ (if any), reduces to \\

\resizebox{\textwidth}{!}{
$
\infer={\Delta_1,\Delta_2,\Delta_3,\Theta_1,\ldots,\Theta_m;\Lambda}
{\infer{\Delta_1,\Delta_2,\Delta_3,\Theta_1,\ldots,\Theta_m,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_1;A\ivee B}
{\deduce{\vspace{0.1cm}\Pi_1}{\vspace{0.1cm}\Gamma_1}}&
\infer{\Delta_2,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_2;C}{\deduce{\vspace{0.1cm}\Pi_2}
{\vspace{0.1cm}[\cdot;A]&\vspace{0.1cm}\Gamma_2}}&
\deduce{\Theta_1;\Lambda_1}{\vspace{0.1cm}\Sigma_1}&
\ldots&
\deduce{\Theta_m;\Lambda_m}{\vspace{0.1cm}\Sigma_m}
}&
\infer{\Delta_3,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_3;C}{\deduce{\vspace{0.1cm}\Pi_3}
{\vspace{0.1cm}[\cdot;B]&\vspace{0.1cm}\Gamma_3}}&
\deduce{\Theta_1;\Lambda_1}{\vspace{0.1cm}\Sigma_1}&
\ldots&
\deduce{\Theta_m;\Lambda_m}{\vspace{0.1cm}\Sigma_m}
}}}
$
}

\item The derivation
{\small
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1}, A, A;C$}
\RightLabel{$C_{c}$}
\UnaryInfC{$\Delta_{1}, A;C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
}
where $C$ is the major premiss of an elimination rule with minor premisses $\Theta_{1};\Lambda_{1} \ldots \Theta_{m};\Lambda_{m}$ (if any), reduces to \\

{\small
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta, A, A:C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta, A, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$}
\RightLabel{$C_{c}$}
\UnaryInfC{$\Delta , A,  \Theta_{1} ,\ldots, \Theta_{m}, A; \Lambda$}
\end{prooftree}
}

\end{enumerate}
\end{enumerate}

\subsection{Normalization}
We shall use Pottinger's \textit{critical derivation} strategy ~\cite{Pottinger76} to prove the normalization theorem for $\mathcal{LE}_{p}$. But before the proof of normalization, we need some definitions and preparatory lemmas that relate reductions and composition to the degree of derivations. The proof of the next lemma is obvious.

\begin{lemma}\label{lemma:junction}
Let $\Pi$ be $\Pi_{1} /[\frac{A}{\cdot ;A}]/\Pi_{2}$, the composition of derivations $\Pi_{1}$ with $\Pi_ {2}$ at junction point $A$. Then, $d[\Pi ] = max\{d[\Pi_{1} ] , d[\Pi_{2} ], d[A]\}$
\end{lemma}

\begin {lemma} \label{lemma:reduction}
If $\Pi$ reduces to $\Pi'$, then $d[\Pi ] \leq d[\Pi']$.
\end{lemma}
\begin{proof}
Directly from the form of the reductions and Lemma~\ref{lemma:junction}.
\end{proof}
\begin{definition}
A derivation $\Pi$ is {\em critical} iff:
\begin{itemize}
\item $\Pi$ ends with an elimination rule $\alpha$;
\item The major premiss $A$ of $\alpha$ is the end of maximal segment;
\item $d[\Pi ]$ = $d[A]$; and
\item For every proper subderivation $\Pi'$ of $\Pi$, $d[\Pi' ]$ $<$ $d[\Pi ]$.
\end{itemize}
\end{definition}

\begin{lemma}\label{lemma:critical}
\textit{(\textbf{Critical Lemma}}): Let $\Pi$ be a critical derivation of $\Gamma\vdash_{\mathcal{LE}_p} \Delta ; \Sigma$  in $\mathcal{LE}_{p}$. Then, $\Pi$ reduces to a derivation $\Pi'$ of $\Gamma'\vdash_{\mathcal{LE}_p}\Delta ; \Sigma$ with $\Gamma ' \subseteq \Gamma$, such that $d[\Pi' ]$ $<$ $d[\Pi ]$.
\end{lemma}
\begin{proof}
By induction on the length of $\Pi$.
\begin{itemize}
\item Case 1: The major premiss of the last rule applied in $\Pi$ is a maximum formula. The result follows directly from the form of the reductions and Lemma~\ref{lemma:reduction}.
\item Case 2: The major premiss of the last rule applied in $\Pi$ is the end formula of maximum segment of length $>$1. There are two sub-cases to be examined:
\begin{enumerate}
\item $\Pi$ is:
{\small
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; (A \vee_{i} B)$}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; C$}
\AxiomC{$[\cdot ;B]$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3} ; C$}
\TrinaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} ; C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
}
By a permutative reduction, $\Pi$ reduces to the following derivation $\Pi^{*}$:\\

\resizebox{\textwidth}{!}{
$
\infer={\Delta_1,\Delta_2,\Delta_3,\Theta_1,\ldots,\Theta_m;\Lambda}
{\infer{\Delta_1,\Delta_2,\Delta_3,\Theta_1,\ldots,\Theta_m,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_1;A\ivee B}
{\deduce{\vspace{0.1cm}\Pi_1}{\vspace{0.1cm}\Gamma_1}}&
\infer{\Delta_2,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_2;C}{\deduce{\vspace{0.1cm}\Pi_2}
{\vspace{0.1cm}[\cdot;A]&\vspace{0.1cm}\Gamma_2}}&
\deduce{\Theta_1;\Lambda_1}{\vspace{0.1cm}\Sigma_1}&
\ldots&
\deduce{\Theta_m;\Lambda_m}{\vspace{0.1cm}\Sigma_m}
}&
\infer{\Delta_3,\Theta_1,\ldots,\Theta_m;\Lambda}
{\deduce{\Delta_3;C}{\deduce{\vspace{0.1cm}\Pi_3}
{\vspace{0.1cm}[\cdot;B]&\vspace{0.1cm}\Gamma_3}}&
\deduce{\Theta_1;\Lambda_1}{\vspace{0.1cm}\Sigma_1}&
\ldots&
\deduce{\Theta_m;\Lambda_m}{\vspace{0.1cm}\Sigma_m}
}}}
$
}
\\ \\

By Lemma 2, $d[\Pi^{*}] \leq d[\Pi ]$. If $d[\Pi^{*}] < d[\Pi ]$, then we take $\Pi' = \Pi^{*}$. If $d[\Pi^{*}] = d[\Pi ]$, then at least one of the derivations of the minor premisses has degree = $d[\Pi ]$. For the sake of the argument, let's assume that both have degree = $d[\Pi ]$. By the induction hypothesis, the derivation
\small{

\begin{prooftree}
\AxiomC{$[\cdot ;A]$}
\AxiomC{$\Gamma_{2}$}
\noLine
\BinaryInfC{$\Pi_{2}$}
\noLine
\UnaryInfC{$\Delta_{2} ; C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{2} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
}

reduces to a derivation $\Pi_{2}'$ of $\Delta_{2} , \Theta_{1} , \ldots, \Theta_{m} ; \Lambda$ such that $d[\Pi_{2}']$ $<$ $d[\Pi_{2}]$, and the derivation

\begin{prooftree}
\AxiomC{$[\cdot ;B]$}
\AxiomC{$\Gamma_{3}$}
\noLine
\BinaryInfC{$\Pi_{3}$}
\noLine
\UnaryInfC{$\Delta_{3} ; C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}

Reduces to a derivation $\Pi_{3}'$ of $\Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$ such that $d[\Pi_{3}']$ $<$ $d[\Pi_{3}]$. \\

Let $\Pi'$ be:
\begin{prooftree}
\AxiomC{$\Gamma_{1}$}
\noLine
\UnaryInfC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1} ; (A \vee_{i} B)$}
\AxiomC{$\Pi_{2}'$}
\noLine
\UnaryInfC{$\Delta_{2} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\AxiomC{$\Pi_{3}'$}
\noLine
\UnaryInfC{$\Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\TrinaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
We can easily see that $\Pi$ reduces to $\Pi'$ and that $d[\Pi' ]$ $<$ $d[\Pi ]$.

\item $\Pi$ is
{\small
\begin{prooftree}
\AxiomC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta_{1}, A, A:C$}
\RightLabel{$C_{c}$}
\UnaryInfC{$\Delta_{1}, A:C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta_{1} , \Delta_{2} , \Delta_{3} , \Theta_{1} ,\ldots, \Theta_{m} ; \Lambda$}
\end{prooftree}
}
By a permutative reduction, $\Pi$ reduces to the following derivation $\Pi^{*}$:\\
{\small
\begin{prooftree}
\AxiomC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta, A, A:C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta, A, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$}
\RightLabel{$C_{c}$}
\UnaryInfC{$\Delta , A,  \Theta_{1} ,\ldots, \Theta_{m}, A; \Lambda$}
\end{prooftree}
}
As in the previous case, by Lemma 2, $d[\Pi^{*}] \leq d[\Pi ]$. If $d[\Pi^{*}] < d[\Pi ]$, then we take $\Pi' = \Pi^{*}$. If $d[\Pi^{*}] = d[\Pi ]$, then by the induction hypothesis, the derivation
{\small
\begin{prooftree}
\AxiomC{$\Pi_{1}$}
\noLine
\UnaryInfC{$\Delta, A, A:C$}
\AxiomC{$\Sigma_{1}$}
\noLine
\UnaryInfC{$\Theta_{1};\Lambda_{1}$}
\AxiomC{$\ldots$}
\AxiomC{$\Sigma_{m}$}
\noLine
\UnaryInfC{$\Theta_{m};\Lambda_{m}$}
\QuaternaryInfC{$\Delta, A, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$}
\end{prooftree}
}
reduces to a derivation $\Pi^{**}$ of $\Gamma^{*} \subseteq \Gamma \vdash_{\mathcal{LE}_p} \Delta, A, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$ such that $d[\Pi^{**}] < d[\Pi ]$. We can then take the desired derivation $\Pi'$ as
\begin{prooftree}
\AxiomC{$\Pi^{**}$}
\noLine
\UnaryInfC{$\Delta, A, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$}
\UnaryInfC{$\Delta, A, \Theta_{1} ,\ldots, \Theta_{m}; \Lambda$}
\end{prooftree}
\end{enumerate}
\end{itemize}
\end{proof}

\begin{lemma}\label{lemma:main}
Let $\Pi$ be a derivation of $\Gamma\vdash_{\mathcal{LE}_p} \Delta ; \Sigma$ with $d[\Pi]>0$.
Then, $\Pi$ reduces to a derivation $\Pi'$ of $\Gamma'\vdash_{\mathcal{LE}_p} \Delta ; \Sigma$ with $\Gamma ' \subseteq \Gamma$, such that $d[\Pi' ]$ $<$ $d[\Pi ]$.
\end{lemma}
\begin{proof}
By induction on the length of $\Pi$.
\begin{itemize}
\item Case 1: $\Pi$ ends with an application of an introduction rule. \\
This case follows directly from the induction hypothesis.
\item Case 2: $\Pi$ ends with an application of an elimination rule. The general form of $\Pi$ is:
\begin{prooftree}
\AxiomC{$\Pi_{1}$}
\AxiomC{$\Pi_{2}$}
\AxiomC{$\Pi_{3}$}
\TrinaryInfC{$\Delta ; \Lambda$}
\end{prooftree}
By the induction hypothesis, $\Pi_{k}$ reduces to a derivation $\Pi_{k}'$ such that $d[\Pi_{k}' ]$ $<$ $d[\Pi_{k} ]$ ($1 \leq k \leq 3$).\\
Let $\Pi^{*}$ be:
\begin{prooftree}
\AxiomC{$\Pi_{1}'$}
\AxiomC{$\Pi_{2}'$}
\AxiomC{$\Pi_{3}'$}
\TrinaryInfC{$\Delta ; \Lambda$}
\end{prooftree}
By Lemma~\ref{lemma:reduction}, $d[\Pi^{*}]$ $\leq$ $d[\Pi ]$. If $d[\Pi^{*}]$ $<$ $d[\Pi ]$, we take $\Pi^{*} = \Pi'$. If $d[\Pi^{*}] = d[\Pi ]$, the $\Pi^{*}$ is a critical derivation and the result follows from Lemma~\ref{lemma:critical}.

\end{itemize}
\end{proof}

\begin{theorem} (\textbf{Normalization Theorem} for $\mathcal{LE}_{p}$)
Let $\Pi$ be a derivation of $\Gamma\vdash_{\mathcal{LE}_p} \Delta ; \Sigma$ in $\mathcal{LE}_p$. Then, $\Pi$ reduces to a normal derivation $\Pi'$ of $\Gamma'\vdash_{\mathcal{LE}_p} \Delta ; \Sigma$ where $\Gamma ' \subseteq \Gamma$.
\end{theorem}
\begin{proof}
 Directly from Lemma~\ref{lemma:main} by induction on $d[\Pi ]$.
\end{proof}

\section{Concluding remarks and future work}
There are lots of things to be done in the domain of ecumenical systems and more specifically in connection with pure ecumenical systems. We conclude this paper by mentioning  two possible lines of work we are pursuing.

\subsection{Pure First-order ecumenical systems}
We can easily show that, in Prawitz' ecumenical system, if the main operator of a formula $A$ is classical, if $\Gamma , \neg A \vdash \bot$, then $\Gamma \vdash A$. For example, assume a derivation $\Pi$ as follows:

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$\neg (B \vee_{c} C)$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\end{prooftree}

Then, we can construct the following derivation of $\Gamma \vdash (B \vee_{c} C)$:

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$[(B \vee_{c} C)]^{1}$}
\AxiomC{$[\neg B]^{2}$}
\AxiomC{$[\neg C]^{3}$}
\TrinaryInfC{$\bot$}
\RightLabel{1}
\UnaryInfC{$[\neg (B \vee_{c} C)]$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\RightLabel{$2,3$}
\UnaryInfC{$(B \vee_{c} C)$}
\end{prooftree}

The same result holds for the classical existential quantifier. From the derivation

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$\neg  \exists_{c} xB(x)$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\end{prooftree}

We can easily obtain the derivation:

\begin{prooftree}
\AxiomC{$\Gamma$}
\AxiomC{$[\exists_{c} x(Bx)]^{1}$}
\AxiomC{$[\forall x\neg B(x)]^{2}$}
\BinaryInfC{$\bot$}
\LeftLabel{1}
\UnaryInfC{$[\neg \exists_{c} c(B(x)]$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{2}
\UnaryInfC{$ \exists_{c} xB(x)$}
\end{prooftree}

\noindent From these results it follows that the classical implication $\to_{c}$ satisfies \textit{modus ponens} for classical succedents:

\begin{theorem}
Let the main operator of the formula $B$ be classical. Then we can prove in Prawitz' system that $\{ (A \to_{c} B), A \} \vdash B$.
\end{theorem}

\noindent In the propositional system with the \textit{stoup}, we can  obtain the same results for the classical propositional operators. For example, given a derivation of

\begin{prooftree}
\AxiomC{$\cdot ; \neg (B \vee_{c} C)$}
\AxiomC{$\Gamma$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ; \cdot$}
\end{prooftree}

We can obtain a derivation of

\begin{prooftree}
\AxiomC{$\Gamma$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta ; (B \vee_{c} C)$}
\end{prooftree}

\noindent  But in the first-order case, a \textit{pure rule} for the classical existential quantifier in the {\em stoup} format requires an extra attention. Consider the following pure rules with {\em stoup} for the classical existential quantifier:

\begin{prooftree}
\AxiomC{$\Delta_{1} , A(t); \cdot$}
\UnaryInfC{$\Delta ; \exists_{c} xA(x)$}
\DisplayProof
\qquad \qquad
\AxiomC{$\Delta_{1} ; \exists_{c} xA(x)$}
\AxiomC{$\cdot ;A(a)$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\Delta_{2} ;\cdot$}
\BinaryInfC{$\Delta_{1} ; \Delta_{2} ;\cdot$}
\end{prooftree}

\noindent It is easy to show that the first order system with {\em stoup} obtained by means of the addition of the intuitionistic rules for $\exists_{i}$, $\forall$ and these rules for $\exists_{c}$ is not complete with respect to Prawitz' first-order ecumenical natural deduction. The important relation $\neg  \forall x\neg A(x) \vdash \exists_{c} xA(x)$ between $\forall$ and $\exists_{c}$ is not derivable in this first-order system with {\em stoup}. As a future work, we propose to investigate the first-order system obtained by the addition of the rules mentioned above plus a new structural rule, the \textit{store} rule:

\begin{center}
\begin{prooftree}
\AxiomC{$\Delta , A;\cdot$}
\RightLabel{{\em store}}
\UnaryInfC{$\Delta ;A$}
\end{prooftree}
\end{center}
with the side condition that the main operator of $A$ is classical.

\subsection{A different approach to \textit{purity}}
A different and interesting approach to pure systems worth exploring is based on some ideas proposed by Julien Murzi in~\cite{Murzi2018}. Murzi proposes a \textit{pure} single-conclusion Natural Deduction system that satisfies the basic inferentialist requirements of harmony and separability. Murzi's proposal combines (in a very interesting way!) Peter Schroeder-Heister's  idea of \textit{higher-level rules} with Neil Tennant's idea that the sign $\bot$ for the \textit{absurd} should be conceived as a \textit{punctuation mark}. Using Murzi's idea we can formulate a new \textit{pure} ecumenical natural deduction system for classical and intuitionistic logic.\\

(1) The \textit{impure} rule for $\vee_{c}$-Int becomes

\begin{prooftree}
\AxiomC{$[\frac{A}{\bot}]^{j}$}
\AxiomC{$[\frac{B}{\bot}]^{k}$}
\noLine
\BinaryInfC{$\underbrace{}$}
\noLine
\UnaryInfC{$\vdots$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$j,k$}
\UnaryInfC{$(A \vee_{c} B)$}
\end{prooftree}

(2) $\vee_{c}$-Elim.

\begin{prooftree}
\AxiomC{$(A \vee_{c} B)$}
\AxiomC{$[A]^{j}$}
\noLine
\UnaryInfC{$\vdots$}
\noLine
\UnaryInfC{$\bot$}
\AxiomC{$[B]^{k}$}
\noLine
\UnaryInfC{$\vdots$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$j,k$}
\TrinaryInfC{$\bot$}
\end{prooftree}

(3) The \textit{impure} rule for $\to_{c}$-Int becomes

\begin{prooftree}
\AxiomC{$[A]^{j}$}
\AxiomC{$[\frac{B}{\bot}]^{k}$}
\noLine
\BinaryInfC{$\underbrace{}$}
\noLine
\UnaryInfC{$\vdots$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$j,k$}
\UnaryInfC{$(A \to_{c} B)$}
\end{prooftree}

(4) $\to_{c}$-Elim.

\begin{prooftree}
\AxiomC{$(A \to_{c} B)$}
\AxiomC{$A$}
\AxiomC{$[B]^{k}$}
\noLine
\UnaryInfC{$\vdots$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$k$}
\TrinaryInfC{$\bot$}
\end{prooftree}

It is easy to show that the impure rules can be obtained from the new pure rules. In the case of $\vee_{c}$, for example, given a derivation $\Pi$ of $\bot$ from $\neg A$ and $\neg B$, we can construct the following derivation:
\begin{prooftree}
\AxiomC{$[\frac{A}{\bot}]^{j}$}
\UnaryInfC{$[\neg A]$}
\AxiomC{$[\frac{B}{\bot}]^{k}$}
\UnaryInfC{$[\neg B]$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$j,k$}
\UnaryInfC{$(A \vee_{c} B)$}
\end{prooftree}

In order to prove the other direction, it is convenient to add a new general rule that allows us to conclude rules. In the formulation of the rule we will use (as Murzi does) the expression $\Delta/A$ as an alternative to the rule $\frac{\Delta }{A}$.

\begin{prooftree}
\AxiomC{$[A]^{j}$}
\noLine
\UnaryInfC{$\Pi$}
\noLine
\UnaryInfC{$B$}
\LeftLabel{$j$}
\UnaryInfC{$(A /B)$}
\end{prooftree}

Suppose now that we have a derivation

\begin{prooftree}
\AxiomC{$\frac{A}{\bot}$}
\AxiomC{$\frac{B}{\bot}$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\end{prooftree}

We can then construct the following derivation:
\begin{prooftree}
\AxiomC{$[A]^{1}$}
\AxiomC{$[\neg A]^{3}$}
\BinaryInfC{$\bot$}
\LeftLabel{1}
\UnaryInfC{$[(A /\bot )]$}
\AxiomC{$[B]^{2}$}
\AxiomC{$[\neg B]^{4}$}
\BinaryInfC{$\bot$}
\LeftLabel{2}
\UnaryInfC{$[(B /\bot )]$}
\noLine
\BinaryInfC{$\Pi$}
\noLine
\UnaryInfC{$\bot$}
\LeftLabel{$3,4$}
\UnaryInfC{$(A \vee_{c} B)$}
\end{prooftree}

An example: \textit{Peirce's law}:

\begin{prooftree}
\AxiomC{$((A \to_{c} B) \to_{c} A)^{2}$}
\AxiomC{$[A]^{1}$}
\AxiomC{$[\frac{A}{\bot}]^{3}$}
\BinaryInfC{$\bot$}
\LeftLabel{1}
\UnaryInfC{$(A \to_{c} B)$}
\AxiomC{$[\frac{A}{\bot}]^{3}$}
\TrinaryInfC{$\bot$}
\LeftLabel{$2,3$}
\UnaryInfC{$(((A \to_{c} B) \to_{c} A) \to_{c} A)$}
\end{prooftree}

redpajama_set_name: RedPajamaArXiv


This entry was posted in Atlantic City Surf, Camden Riversharks, Newark Bears by Neil deMause. Bookmark the permalink.
There was also a Ballpark in Sussex Nj called Skylands park not sure if it was Atlantic league but the team folded there too. I think it was Stl Cards Affiliate.
I have often written in support of modest subsidies for minor league facilities as an "enabling" factor (and not in the bad way). A minor league franchise worth maybe $500k (if that) cannot possibly float a construction mortgage on a $10-15m stadium (unlike their professional counterparts). They can and should, of course, pay rent once that facility is built.
If the Gov. committed tens of millions to these minor league facilities, what was the percentage of public funding on average? 100?
Were there also private financial commitments to the facilities or were these another "pay us what you can if you can, we'll build and operate it for you" deals?
When I speak in favour of minor league stadium subsidies, I tend to think of smallish cities kicking in $4-5m for infrastructure and so forth. Not $40m up front costs with no prospect of earning any annual revenue from either ticket taxes or team rent payments.
250,000 fans annually in a short season league isn't great, but it's not terrible either (4-5k/gm if the above figure represents the home attendance for just one team).
If your gate is $2-3m (at a modest $10/head avg) with potentially another $1-2m in concessions and merch and you can't fund a minor league team with a payroll of maybe $250-300k and operating costs of maybe double that…. I guess I would wonder where the rest of the cash is going. If the team is responsible for stadium operations (not unreasonable), this could eat up another million or so… but it still doesn't take the total budget anywhere near $4-5m.
The Atlantic League went around telling smallish northeastern cities, "Hey, build us a stadium and you'll get an independent league team! It's the wave of the future, better than affiliated ball and sliced bread combined!" It wasn't.
From the little I know of it, this seems a very optimistic average for such a small team.
It may be, but if we are to believe that 250,000 people attended games in an average season, and that (short) season is approx 100-120 games… then 50-60 home games would produce an average attendance of 4-5k.
Some of the articles also suggested that the (roughly 10k seater) stadia were 'around half full'.
So unless the average ticket price was $2 rather than $10, I'm really not seeing how the total revenue is so low that a baseball team with maybe 35 employees (most of whom make less than $2k/month and are seasonal employees) and does not pay anything beyond stadium operating costs struggles to break even.
In fact, if you cut the paid attendance to an average of 2600 at an avg ticket price of $10, the club is still generating close to $1.5m in gate alone (say $2.5m including concessions and merch).
It's not staggering revenue, certainly. But for a team with total expenses that should be less than $1.5m… the numbers really don't add up to me.
1) Don't believe that 250,000 people attended games in an average season. Minor-league teams tend to be, uh, "creative" with their attendance counts.
2) Their stadium sat 6,400, not 10,000.
3) Where are you getting "less than $1.5m" from? Atlantic League team budgets are over $2 million easily, and some are over $3m.
Actually, that 250k number came from Prof. Matheson. So, presumably, at the top end they do attract that many paying fans. He would know.
This stadium may have had a 6,400 cap, but some of those linked in the articles were clearly larger venues.
Finally, the heart of the question originally is what are the total legitimate expenses of a run of the mill Atlantic or other Independent league team (be it Rookie, A, AA or other level)?
Most minor league players (even in the affiliated system) make very little, it is reasonable to assume that most independent league players make a similar amount. One MLB team just announced they were increasing all their MiLB player salaries by 50%. When asked, the GM of that club said he expected the total cost of that decision to be in the hundreds of thousands, but well under a million dollars. And that's for an MLB team with 4-6 real farm teams in organized ball.
Non MLB players make garbage money. So, even if these independent teams have to pay players more than the $1100/mo some MiLB players are working for, the player payroll for a five month season is still likely under $250k. Non playing staff could equal that easily. Travel and accommodations are expensive even for fairly narrow geographic leagues, so lets say $500k for that and another $200k for equipment and promotion. Add 20-25% for contingencies if you like… we are still around $1.5m. If the team pays for stadium ops during the season that could bump it up to $2.5 or $3m, but most don't as I understand it.
Unless you are paying a hefty owner's salary and travel expenses on top of the actual team costs, of course.
Consider the New. Indoor Arena which was built in Newark. At taxpayer expense to lure the pro teams from Hackensack. Nj. .. Plus the new Meadowland Fb. Stadium Built at taxpayer. Then The expense three small baseball std.. Are small potatoes..


text:
Home News Remedy Situation Forthwith, Ensure 700MT of Oxygen on Daily Basis to Delhi,...
Remedy Situation Forthwith, Ensure 700MT of Oxygen on Daily Basis to Delhi, SC Tells Centre
The Supreme Court has directed the Centre to remedy the situation forthwith and to ensure the availability of 700 MT of oxygen to Delhi on a daily basis, pending further orders. It said that the Centre is under an obligation to ensure a daily supply of 700 MT of oxygen to meet the existing requirements of the national capital and directed that there shall be no reduction in the allocation and availability of medical oxygen to Delhi.
A bench of justices D Y Chandrachud and M R Shah said that the Centre has on record stated on April 21 that there is no dearth of oxygen and a quantity of 16,000 MT of liquid medical oxygen is available in the country. "We direct the Union of India to remedy the situation forthwith and to ensure that the direction issued by this court for the availability of 700 MT is strictly observed on a daily basis, pending further orders," the bench said in its order passed on May 6 and uploaded on Saturday.
It noted that on May 6, the total quantity of oxygen delivered to Delhi was 577 MT, resulting in a shortfall of 123 MT and as of 9 am on May 7, the total quantity which has been received at Delhi border was 87.97 MT, while 9.64 MT was under transit. The bench said that the plan of the Centre for complying with its direction to ensure supply of 700MT of oxygen to Delhi, which has been placed before this Court "is subject to caveats and conditions which cannot be accepted".
"What is sought to be assured in the first part of the plan is diluted with the next segment. 700 MT was not intended to be a requirement to be fulfilled for one day or sporadically, but on a daily basis. Daily basis means for every day. We accordingly direct, that there shall be no reduction in the allocation and availability of medical oxygen to NCTD and the direction in regard to the provision of 700 MT per day shall continue to be observed," the bench said.
It said that the top court's direction has been based on the assurance of the Centre to the court and on May 5, it has reiterated the direction for maintaining the supplies to NCTD at 700 MT per day. The bench said that the Delhi government has submitted that it has computed the requirement of oxygen on the basis of the formula which has been adopted by the Centre.
"The Union Government has not disputed the correctness of the computation on the basis of the formula. At this stage, no contrary material has been placed on the record by the Union Government. The attention of the Court has not been drawn to any error in the methodology of computation which has been adopted by GNCTD. GNCTD has drawn the attention of the Court to the serious deficiency in the availability of oxygen," it said.
The top court said that in its April 30 order in which it was said that though the projected daily demand of Delhi as of April 20 had increased from 300 MT to 700 MT, the existing allocation of oxygen had remained at 490 MT per day. It noted the submission of Solicitor General Tushar Mehta, appearing for the Centre, that many of the demands made by the states and union territories, including Delhi, for the provision of medical oxygen were "unrealistic" and the problem of shortage in the national capital is due to a systemic failure to ensure proper distribution of oxygen.
The bench said, "except for a bare assertion that an increase of 210 MT to NCTD would result in a corresponding reduction to other states, no material has been produced on the record by the Union of India." On Friday, the top court had made it clear to the Centre that it will have to keep supplying 700 MT of oxygen to Delhi every day till its order is modified, saying it meant "business" and non-compliance may force it to go "firm".
Earlier, the top court had stayed the contempt proceedings initiated by Delhi High Court against central government officials for non-compliance of direction to supply 700 MT of oxygen for COVID patients to Delhi, saying "putting officers in jail" would not bring oxygen and efforts should be made to save lives. The top court had said that this is an all-India pandemic situation and it will have to find ways to ensure oxygen supply to the national capital.
It had said that this court cannot review its own order of April 30 and the Centre will have to inch towards supply of 700 MT of oxygen daily for Delhi and asked the Union government to place before it a plan as to how it will revamp the supply to 700MT. On May 6, the Centre had apprised the top court that it had complied with the order and instead of 700 MT liquid medical oxygen it ensured supply of 730 MT to Delhi.
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Elections were held for all the seats of the Mazhilis provided for in the new Constitution of August 1995.
In March 1994, the first post-independence general elections for the former unicameral Parliament took place. A year later, Kazakstan's Constitutional Court invalidated this poll and declared the legislature illegal. President of the Republic Narsultan Nazarbayev thereupon announced the dissolution of the Parliament and plans to rule by decree pending new elections. A majority of the 177 Deputies challenged the dissolution.
On 30 August 1995, a new Constitution providing, inter alia, for a smaller bicameral legislature was approved by popular referendum. On 2 October, the President announced the December election dates. The overall conduct of the polling was overseen by the Central Election Commission (CEC). According to it, 285 candidates (128 self-nominated, 157 registered to parties or public associations) ran for the 67 Mazhilis seats. As for the Senate, 49 candidates vied for the 40 elective seats, with 14 of the 20 constituencies being uncontested. There were a series of limitations on pre-election campaigning.
The 9 December polling day for the Mazhilis was monitored by domestic and international observers and saw a reported turnout of over 79%. Observers from the OSCE questioned this figure, citing a number of irregularities, including multiple voting by the same individual. At the end of the day, only 41 candidates were declared elected with the required thresholds. Two weeks later, on 23 December, another 13 Deputies were chosen. Finally, on 4 February 1996, the remaining 13 seats were filled. Of the opposition parties, many boycotted the poll.
On 30 January, the newly-constituted Parliament met in joint session for the first time. The Council of Ministers is headed by Prime Minister Akezhan Kazhegeldin.