#benjamin haskell

[auburnlaughter]

Plant Company please and thank you!

Thank you for your patience and the ask. Here are your sentences!

WIP Wednesday Game - Plant Company (original story)

There was no sign of Sorenson in the atrium. The hallway for the docking bay was the closest to medical, but also the busiest, and Haskell was certain he would have received an alert by now if anyone had spotted the captain. On a hunch, he turned toward the hallway that led toward atrium one.

[fundieshaderoom]

Fundie Families and Adjacents I Follow: Burnett

John Haskell Burnett II "Johnny"- July 3, 1958

Cheryl Desnise Clay- July 28, 1959

The couple married on May 2, 1981. They share 8 children, 8 children-in-law, and 21 grandchildren.

John-Clay Byford- August 14, 1986

Hannah Joy- May 10, 1988

Caleb Andrew- August 30, 1990

Abbie Grace- April 16, 1992

Charity Faith- April 13, 1994

Maggie Ruth- August 2, 1996

Carinna Elizabeth- August 2, 1996

Benjamin Josiah- January 30, 2000

John-Clay married Sarah Faith Ferraro (May 2, 1991) on August 22, 2015. They share 4 sons.

Eric Steadfast- June 19, 2016

Samuel John- September 26, 2017

Joseph Byford- October 15, 2019

Nathaniel Wilberforce- July 14, 2022

Hannah married Dustin Gene Dornink (June 20, 1989). They share 5 children.

Evan Trust- June 19, 2014

Ransom James- January 2016

Harrison John- 2017

Brielle- 2019

Boy- 2021

Girl- 2023

Caleb married Kasey Noel Goodwin (October 4, 1995) on July 7, 2018. They share 3 children.

Erin- June 2019

Timothy- 2021

Girl- March/April 2023

Abbie married John-David Duggar (January 12, 1990) on November 3, 2018. She was 26 and he was 28. They now share two children.

Grace Annette "Gracie"- January 7, 2020

Charlie- September 2022

Charity married Samuel Wearden (December 17, 1990) on June 10, 2017. They share 2 children.

Brendan- August 9, 2018

Girl- 2021

Boy- 2023

Maggie married David Andres Becerra Nino on March 11, 2021

Carinna married Daniel S Penn (October 18, 1996) on February 1, 2020. They share 2 daughters.

Girl- November 2020

Girl- October 2022

Ben married Bethany Meckle (September 10, 1999) on November 20, 2021. They share 1 son.

Josiah Thomas- May 23, 2023

[wuggen]

Tagged in by the lovely @gwentrification!

Favorite color: Deep blue and/or purple! (I am mildly colorblind so it's sometimes difficult for me to tell the difference lol)

Currently reading: Mainly Types and Programming Languages by Benjamin C. Pierce and Category Theory by Steve Awodey

Last song listened to: Jesus fuck I do not remember. I think the last song I heard was Mr. Brightside cuz it was playing at the cafe, but like the last thing I listened to? Couldn't tell ya. Possibly something from the new Dwarf Fortress soundtrack?

Last movie watched: Watched The Five Star Stories with @gwentrification, @j-psilas, and @impossiblejellyfishfart the other night!

Last series watched: I think this would be Kaiba? Also with @gwentrification and @j-psilas :3

Currently craving: The abolition of the 40-hour work week

Currently working on: Oh god, many things. I think I've said this before but I perpetually have at least a half dozen personal projects in flight at any given time. Got a tiny little game dev project that I wanna finish, got a larger game concept that I'm still fleshing out a little before I get much implemented, got a haskell library for music composition I've been poking at, I have a steno board that I still need to write the firmware for, there's the Rust parsing combinator library I started making for use while working through Types and Programming Languages... etc etc etc

Tea or coffee: I love them both very dearly, but I think my love for tea is much less situational. I'm pretty much always down to have some tea, whereas I kinda have to be in a particular mood for coffee

@groundlessness @kwarrtz @lady-inkyrius it's chain letter time besties :3

[readyforevolution]

#burningdownblackchurchesbecauseyouhateObama

On November 5, 2008, Thomas A. Gleason, Michael F. Jacques, and Benjamin Haskell set a fire to burn down Macedonia Church of God In Christ, in Springfield, Massachusetts out of protest over the election of then President elect Barack Obama. All the men admitted to varying levels of distaste for minorities, with witnesses telling investigators the trio said they burned the church to denounce Obama's election.

Sources: https://www.masslive.com/news/2011/02/defendant_in_church_arson_case.html

[cynsualc829]

Delicious music on The Smooth Jazz Kitchen Radioshow - Tasty Smooth. Chef Bruce Buege has created another scrumptious musical menu to please your listening appetite. So, tune in with the Chef and be delighted starting at 7 pm EST, 6 pm CST, 4 pm PST, and 11 pm UK time.

Bon Appétit 🍽

🇺🇸 🇬🇧 🇩🇪 🇮🇹 🇬🇷 🇸🇩 🇿🇦 🇨🇦 🇧🇶

𝐖𝐏𝐔𝐑 - 𝐏𝐔𝐋𝐒𝐄 𝐈𝐍𝐓'𝐋 𝐑𝐀𝐃𝐈𝐎

www.pulseintlradio.com

#thesmoothjazzkitchen #SmoothJazz #contemporaryjazz #tasty #tastytreats #delicousmusic #menu #GoodMusicGoodVibes #international #TheListeningExperience

Bunny DeBarge Jeremiah SoundCheck Moore John Spencer Moore Jaee Logan II Renard Gallo Bill Ortiz Jose Rossy DrJohn S MooreJr April Moore Darren Benjamin Mae Short

The Smooth Jazz Kitchen Radioshow - Tasty Smooth

00:00 Greg Chambers feat Lemek Yisrael - Playdate

00:03 Evan Carydakis - Paradise Beach

00:07 3rd Force - Big Shot

00:12 Blair Prince Bryant - Summit

00:15 Ken Navarro - Heaven Sent

00:19 Gota - In The City Life

00:23 The Moore Twins - Is This The Way

00:26 Infinitive Vibes Vibes feat Stabe Wilson - One Love

00:30 Marion Meadows feat Kim Flemming - Be With You

00:35 Rod Best - Into The Light

00:41 Bryan Morris - Tasty Smooth

00:45 Randy Crawford - Give Me The Night

00:50 Stanley Jordan feat Novecento - Easy Love

00:55 Peet Project - Beverly Hills Party

00:58 Tim Cunningham - South Beach

01:03 Deaddream - Eclectic Soul

01:08 Club 1600 - Stay

01:12 Steve Laury - Cuttin Up

01:17 Richard Elliot - Retro Boy

01:21 Eric Evans - Water Sound

01:25 Mark Carter feat Mark Visher & Scottie Haskell - Mi Senorita de San Juan

01:29 Al Nesbitt and The Alchemy - Blowbacks

01:33 PJ Spraggins feat Phil Davis - Metamorphosis

01:37 Brian Culbertson - Feel The Love (Chicago VIP Session-Live)

01:44 David Sanborn - Brother Ray

[ostensiblynone]

White Head December 24, 1818

Sir, I have been informed by friends that there are some envious people seeing to obtain this Light at the expense of my character. I can not find that mariners have any fault with the Light. It appears to be only those who are seeking the office: Neither do I see what grounds they have for engaging it, as the salary is far from being excessive [[?]]. I am informed privately that some have offended others a reward for their assistance in an intrigue against me. Tho. it may or [words with ink smudge] any charge of negligence, or embezzlement. If any thing blamable can be fairly and satisfactorily proved, I will suffer patiently the penalty annexed to the crime. I have, sir, attended punctually & obeyed all instructions, & shall ever pride myself in obeying all your instructions, and to be faithful and careful to keep the States property secure, and the light as bright as the situation will permit. It is a very small light but I have and will keep it as bright as possible.

From Whitehead Light, 1798-1819, National Archives

This file unit consists of correspondence related to Whitehead Light off of Saint George, Maine received by Benjamin Lincoln, Henry Dearborn, and Henry A. S. Dearborn in their capacity as Superintendents of Massachusetts Lighthouses. The correspondence relates to the appointment of lighthouse keepers, the acquisition of land for the lighthouse, and to expenses for supplies. Also included is a 1798 petition to Congress from individuals “employed in boating to and from Penobscot Bay” requesting that a lighthouse be built on Whitehead Island. Included are letters from Treasury Secretary Albert Gallatin, Revenue Commissioner William Miller, contractor and lamp-maker Winslow Lewis, and Lighthouse Keepers Ellis Dowlf, Ebenezer Otis, and Charles Haskell. Of note are several letters from Rebeccah Otis, wife of the late keeper Ebenezer Otis, petitioning to continue operating the lighthouse with her son.

[iamheatherfay]

Directed by 

Drew Harwood

Writing Credits (in alphabetical order)  

Drew Harwood

Cast  

Gareth Koorzen...Jake Drew Harwood...Brandon Sidney Edwards...Sandra Chris R. Taylor...Aj Heather Fay...Cali Aja Nicole...Sofia Mary Charles Miller...Chanler Amber Hutchins...Zoey Katie Garland-Noble...Leanne Johnny Mocker...Ivan Makar Tony Suriano...Vok Chip Heidt...Axl Roger Jerome...Grandpa Jezebel Luxure...Burlesque Dancer Jean Robert De-Cavel...Chef

Other cast:

Bradley Paynter...Party Guest Randy Bruce...Party Goer Richelle Talor...Party Goer Angel Lane...Party Goer Jake Kopronica...Party Goer Brian Lally...Delivery boy Nick Coleman...Kyle (uncredited) Jai Sahai...Priest (uncredited) John Christian Schulte...Spencer (uncredited)

Produced by 

Katie Garland-Noble...producer Drew Harwood...producer Chip Heidt...executive producer Calvin Herbst...line producer Gareth Koorzen...producer Johnny Mocker...executive producer Jared Safier...executive producer

Music by 

Jeanne-Peri Foucault

Cinematography by 

Brandan Haskell

Editing by 

Sam Zarrin

Casting By 

Brittany Kirby

Production Design by 

Rachel Z. Meyer...(as Rachel Meyer) Drew Vecellio

Art Direction by 

Drew Vecellio

Costume Design by 

Robin K Fields...(as Robin Fields)

Makeup Department 

Celeste Blandon...assistant makeup artist Libby Dietrix...assistant makeup artist Patricia Martinez...makeup artist

Production Management 

Calvin Herbst...production manager

Second Unit Director or Assistant Director 

Samantha Sather...second assistant director Jeff Seemann...first assistant director

Art Department 

Rachel Z. Meyer...set dresser (as Rachel Meyer) Tony Suriano...art department assistant Drew Vecellio...set dresser

Sound Department 

Sam Costello...sound designer Chen Shen...ADR & VO recordist Mike Teeters...sound mixer Dean White...re-recording mixer

Stunts 

Bryan Carlisle...boat driver/water safety

Camera and Electrical Department 

Aria Brice...best girl grip Jon Cibella...key grip Jack Miller...second assistant camera Mitchell Sacco...first assistant camera John Christian Schulte...gaffer

Costume and Wardrobe Department 

Elizabeth M Galbraith...costume assistant Julianna Weis-Palacios...assistant set costumer

Editorial Department 

Strack Azar...colorist Jack Kanner...assistant editor

Location Management 

Katie Garland-Noble...location manager Drew Harwood...location scout Gareth Koorzen...location scout

Script and Continuity Department 

Shawn Tira...script supervisor

Additional Crew 

Frank Batten...production assistant Hunter Deno...production assistant Jeremy Jardine...Additional Music Brian Lally...on set paTim Lally...driver double Brodie Mccauley...production assistant Olivia R. Ramage...production assistant Benjamin Willis-Teff...Key PA (as Ben Willis-Teff)

Thanks 

Kimberly Carlisle...special thanks Jean Robert De-Cavel...special thanks (as Jean-Robert de Cavel) Pattie Lally...special thanks Lynn Michele Latta...special thanks Marilou Lind...special thanks D. Lynn Meyers...special thanks Kristen Schlotman...special thanks Tassy Taylor...special thanks

[threeravenspublishing]

Do you love a little Magic to start off your day? That's perfect, because today is #releaseday for our Misfits of Magic anthology, featuring stories by myself, Megan Higgins, Jennifer Brinn, Benjamin Tyler Smith, Jon Michael Kelley, Wayland Smith, Kristina Barnes, Michael Morton, Michael K. Falciani, @wotfcontest quarterly winner N. V. Haskell and an opening by Piers Anthony #Fantasy #anthology #readersofinstagram #readingcommunity #booktok #epicfantasy #swordandsorcery #darkfantasy #Hillbilly #3RP #TRP Grab your copy today https://www.amazon.com/dp/B0BC1KBT65 (at Chickamauga, Georgia) https://www.instagram.com/p/CiSiaGFrUZp/?igshid=NGJjMDIxMWI=

[uksportstravaganza]

The ride home to Oxford. Somebody had a big day at the British Grand Prix: truck track tour, pit lane walk, meet and greet with Drive to Survive celebrity, Haas garage tour, champagne, wine, lamb, beef, cheese plate… I think he got my money’s worth out of the ticket.

Benjamin Lincoln Haskell did not approve the posting of this photo.

[ssunvulcan1981]

Haskel VS Netanyahu In Duel of the Broccoli and Pickles

השכל נגד נתניהו בדו קרב הברוקולי והחמוצים

[luthienne]

Hey, dear! Let me be one [more] follower who asks a few quotes about some things. Could you compile some (just a few, just a bunch) about silence and/or introversion? Thank you dearly. ♡

a compilation v close to my heart ♡

“Solitude: liberation from even the expectation of being seen.”

Kathleen Graber, The Eternal City: Poems; “The Telephone”

“For now she need not think about anybody. She could be herself, by herself. And that was what now she often felt the need of — to think; well not even to think. To be silent, to be alone.”

Virginia Woolf, To the Lighthouse

“I thought of you—wished you were here with me but I get a keen sort of exhilaration from being alone…”

Georgia O’Keeffe, in a letter to Cady Wells, featured in Georgia O’Keeffe: A Life

“But I love such days—rare lonely days. I love above all things, my dear, to be alone.”

Katherine Mansfield, in a letter to J.M. Murry

“I am, oddly, happiest when alone for weeks on end talking to no one there, talking in my mind to the imagined listener who perfectly hears, perfectly understands, and talks back with equal truthfulness.”

Martha Gellhorn, from Selected Letters

Amélie, dir. Jean-Pierre Jeunet (2001)

“For a long time now, every meeting with another human being has been a collision. I feel too much, sense too much, am exhausted by reverberations after even the simplest conversation.”

May Sarton, Journal of a Solitude

“I have packed myself into silence so deeply and for so long that I can never unpack myself using words. When I speak, I only pack myself a little differently.”

Herta Müller, from The Hunger Angel

“I don't see much of anybody these days—I feel rather funny with other people—even those whom I care for. While one's heart is being transformed into a little world, one wants to be alone.”

Kahlil Gibran, in a letter to Mary Haskell, from Beloved Prophet: The Love Letters of Kahlil Gibran and Mary Haskell, and her private journal

Bella Akhmadulina, Fever and Other New Poems; “Longing for Lermontov” (tr. Geoffrey Dutton, Igor Mezhakoff-Koriakin)

“Perhaps I am addicted to solitude and feel safe and easy in it,”

Martha Gellhorn, Selected Letters

“…because there was too much silence within me. In those days I was alone,”

Clarice Lispector, Why This World: A Biography of Clarice Lispector

“My current silence interests only me. It touches too many parts of my personal life for me to explain it to you.”

Albert Camus, Notebooks (1951-1959), Vol 3.

“People love talking, and I have never been a huge talker. I carry on an inner monologue, but the words often don’t reach my lips.”

Gillian Flynn, Gone Girl

“And what were they anyway, sprigs of grass, things of blue? For a long time I wanted to use words, then didn’t.”

Mary Ruefle, Madness, Rack, and Honey

“What words? What words can I trust to convey this fragile heart?”

Jeanette Winterson, Gut Symmetries

Keaton Henson, from “How Could I Have Known”

“So much of what we live goes on inside– The diaries of grief, the tongue-tied aches Of unacknowledged love are no less real For having passed unsaid. What we conceal Is always more than what we dare confide. Think of the letters that we write our dead.”

Dana Gioia, Unsaid

“But I cannot help it. I only want to be alone. I want to be myself and alone and free to breathe, live, look upon the world and find it however it is…”

Martha Gellhorn, from Selected Letters

“… Perhaps love is to give one’s own solitude to others? For it is the very last thing we have to offer.”

Clarice Lispector, Selected Cronicas; “The Gift”

“I’ve never been afraid of loneliness because I’ve never felt the need to justify my feelings to myself. I accept the muteness of feeling too. I have huge respect for my own silence. I let it speak. I allow time to do its trick and lead me back to myself. I don’t want just anyone to share life and myself with me.”

Anaïs Nin, from The Diary of Anaïs Nin, Vol. 5: 1947-1955

“Don’t allow yourself to be imprisoned by any affection. Preserve your solitude. If the day ever comes when a real friendship is bestowed on you there will be no conflict between your inner solitude and this friendship. On the contrary, that is the infallible sign by which you will know it.”

Simone Weil, First and Last Notebooks: Supernatural Knowledge

“…the most precious thing of all: solitude.”

Clarice Lispector, The Hour of the Star (tr. Benjamin Moser)

“She naturally loved solitary places, vast views, and to feel herself for ever and ever and ever alone.”

Virginia Woolf, Orlando

Harold Pinter, Old Times

“Le Notti Bianche” (1957) - Luchino Visconti

“Solitude itself is a way of waiting for the inaudible and the invisible to make itself felt. And that is why solitude is never static and never hopeless.”

May Sarton, Plant Dreaming Deep

“O you—my sacred solitude!”

Anna Akhmatova, The Complete Poems; “Solitude (from Rilke)”

“It is curious for one who has been much alone—this sinking back into silence.”

Katherine Mansfield, in a letter to J.M. Murry

“…I have backed up / into my silence / as inexhaustible as the sun”

Fanny Howe, The Lyrics: Poems; “O’Clock”

“All I want is silence, for myself and for the selves I used to be, a silence like the magical cottage in the forest that lost children find in fairy tales.”

Alejandra Pizarnik, Extracting the Stone of Madness (tr. Yvette Siegert)

Allison Stone, “Persephone��s First Season in Hell”

“…and the heart took shelter behind a parapet of silence;”

Dulce María Loynaz, Absolute Solitude: Selected Poems; “Poema XLV” (tr. James O’Connor)

[Original: “…y el corazón se encastilló en un muro de silencio;”]

“I feel the same way about solitude as some people feel about the blessing of the church. It’s the light of grace for me. I never close my door behind me without the awareness that I am carrying out an act of mercy toward myself.”

Peter Høeg, “Smilla’s Sense of Snow”

“Solitude as necessity, demandable, honorable. Not sinful, indulgent, wasteful, undeserved.”

May Sarton, from a journal entry dated October 18, 1993

“And in that silence, what grace.”

Camille Norton, Corruption: Poems; “Savonarola’s Cape”

“and my chest appears translucent, / heart in its center, / cathedral of dust / and silence”

Milagros Terán, Las luces en la sien (tr. Fiona Griffin)

[Original: “y el pecho lo llevo traslúcido, / corazón en medio / como una catedral de pólvera / y silencio”]

Emily Dickinson, “I felt a Funeral in my Brain”

“I want to meet no one; I want to say nothing; / I want to go down and rest in the black earth of silence.”

Robert Bly, Eating the Honey of Words; “Depression,”

“You would rather have gone on feeling nothing, / emptiness and silence; the stagnant peace / of the deepest sea,”

Margaret Atwood, from “Eurydice,” Selected Poems II: 1976 - 1986

“I don’t know about birds / nor do I know the history of fire. / But I believe that my solitude should have wings.”

Alejandra Pizarnik, Tree of Diana, tr. Joseph Mulligan & Patricia Rossi

“For language to have meaning there must be intervals of silence somewhere, to divide word from word and utterance from utterance.”

Thomas Merton, “Disputed Questions”

“I have a need of silence and of stars. / Too much is said too loudly.”

William Alexander Percy, from “Home”

Camille Norton, Corruption: Poems

“—a space of virgin silence, a place of rest where I wait for myself.”

Alejandra Pizarnik, “A Night Shared in a Memory of Escape” (tr. Yvette Siegert)

[auburnlaughter]

You could say I'm going back to my "roots" for Plant Company ;)

:D Love it! Thank you for the ask and here are your sentences.

WIP Wednesday Game - Plant Company (original story)

"You're just in time to help us plan next week's movie night."

"Movie night?" one of the inspectors asked, making a note on their tablet.

"Technically it's usually more of a TV night," Ben explained as he led the way to the living room. "But yes, we tend to have one evening a week were we watch something together, often an episode of Galaxy Blade or one of the local nature documentaries."

[fundieinfoplace]

The Burnett Family

John (Johny) Haskell Burnett II was born July 3 1958

Cheryl Denise Clay was born July 28 1959

John and Cheryl married May 2 1981. John was 23. Cheryl was 22. They had 8 children.

1. John- Clay Byford was born Aug 14 1986

2. Hannah Joy was born May 10 1988

3. Caleb Andrew was born Aug 30 1990

4. Abbie Grace was born April 16 1992

5. Charity Faith was born April 13 1994

6. Maggie Ruth was born Aug 2 1996

7. Carinna Elizabeth was born Aug 2 1996

8. Benjamin Josiah was born Jan 2000

.

.

John married Sarah Faith Ferraro Aug 22 2015

1. Eric Steadfast June 11 2016

2. Samuel John Oct 5 2017

3. Joseph Byford Oct 14 2019

4. Nathaniel Wilberforce July 14 2022

.

Hannah married Dustin Gene Dornik 2013

1. Evan Trust June 20 2014

2. Randsom James 2016

3. Harrison John 2017

4. Brielle Joy Sept 2019

5. (David Gene) Oct 2021

6. Kelly Marie Aug 2023

.

Caleb married Kasey Noel Goodwin July 7 2018

1. Erin Noel July 14 2019

2. Timothy John Feb 9 2021

3. (Baby Boy) Mar 2023

.

Abbie married John David Duggar Nov 3 2018

1. Grace Annette Jan 7 2020

2. Charles John Sept ? 2022

.

Charity married Samuel Wearden June 10 2017

1. Braedon John Aug 13 2018

2. (Faith Marie) Oct 2021

3. Baby Boy Wearden 2023

.

Maggie married David Andreas Becerra Nino Mar 11 2021

.

Carinna married Daniel Penn Feb 2 2020

1. Alaythia Joy Nov 2022

2. Destiny Elizabeth Oct 2022

.

Benjamin married Bethany Meckle Nov 20 2021

1. Josiah Thomas May 20 2023

The names in () are just Placer names since I'm not too sure on them.

[youzicha]

Generalized Church is the Curry-Howard of Knaster-Tarski

The proofs-as-programs principle (also known as the Curry-Howard isomorphism) sets up a correspondence between logical proof systems and type systems for programming languages. So occasionally a result is independently developed twice, once as a theorem and once as a program. Here’s a short and cute example of this phenomenon. (I read about it in this textbook, but I think it deserves to be better known).

The Program

The program we will consider is the roll and unroll functions. At this point you might like to try yourself to implement them (my own solution is further down this post). It’s an interesting exercise, because the function definitions look completely unintuitive, but they are actually easy to write if you let the types guide you. Using Haskell’s “typed holes” feature, you can start with just an underscore _, the typechecker will tell you “this underscore should be replaced by a term of type …”, and you can gradually refine it by putting in the only thing that has the right type at that point.

This coding of recursive types has been rediscovered many times, but I think it was made widely known by this note by Philip Wadler from 1990. I don’t know if there is a standard name for it, but I’ll call it the generalized Church encoding. (If you instantiate f with option you recover a slight variation of the usual Church encoding of natural numbers.) It’s imporant because what it tells us about the expressiveness of System F—it shows that just System F without any additional features can already encode (positive) recursive datatypes. But it’s not used much in practice because it has the usual drawbacks of Church encodings:

It’s inefficient at runtime, e.g. calculating the predecessor of a natural number takes O(n) time instead of O(1).

You often need impredicative polymorphism to define functions operating on them, which may offend your constructivist sensibilities or confound your type checker.

In standard Type Theory it’s not possible to prove the induction principle, so you can’t prove any interesting theorems about them. (However, as Wadler’s note shows, if you can prove or assume parametricity, that gives you some additional power.)

The Proof

Meanwhile, (part of) the Knaster-Tarski fixpoint theorem states that if f : X → X is a monotone function on a complete lattice, then ⋀{x | f x ≤ x} is the least fixpoint of f.

This theorem is well known to programming language researchers, because it can be used to as a unifying form for different kinds of induction principles. For example, Benjamin Pierce’s textbook Types and Programming Languages appeals to it to explain how induction and coinduction over typing relations work, by taking the lattice X to be a powerset of relation tuples, ordered by subset inclusion. (Note that this is not obviously connected to the problem of encoding recursive types. In Types and Programming Languages, the recursive terms are assumed as already given, and the theorem is used to define sets of those terms.)

The proof is only a few lines, but it’s still kindof tricky, each time I try to remember it I fail and have to look it up again. Here’s how it looks in a random pdf that google turned up:

It’s kindof neat, isn’t it? I particularly like the twist where it goes “in order to prove f(p) ≤ p, we first prove p ≤ f(p)”.

The Correspondence

Here’s the definitions of roll and unroll that I came up with. I particularly like the twist where it goes “in order to define unroll, we first define roll”.

Now, it’s not obvious from just looking at it, but I claim that this is basically a line-by-line translation of the proof I quoted above.

In order help to make them match up, we will allow ourselves to write x≤y as alternative syntax for the function type x→y. So the lattice we will consider is the set of types ordered by function mappings, or (on the other side of proofs-as-programs) logical propositions ordered by implication. Let’s investigate each part of the proof.

Let pre be the set of prefixed points, and p the glb of pre. Existence of p is guaranteed since L is a complete lattice.

What the pdf calls p we will call (RecA f). This part of the proof is the only bit that requires some creative thought to set up a correspondence, because we have to invent the definition of RecA. It turns out that the lattice of types forms a complete lattice, where for any set of types {x | P x}, the greatest lower bound ⋀{x | P x} can be represented as the type ∀x.(P(x) → x). In order see that this really is the greatest lower bound, we prove two theorems, saying that it is a lower bound of {x | P x}, and that it’s greater than any other lower bound.

Then the type (RecA f) is just the type Meet, specialized to take P(x) := (f x ≤ x).

From now on we can forget the definition of ≤, and in the rest of the proof every occurrence of → refers to logical implication. All the rest of the proof is just a line-by-line translation, under the usual proof-as-programs correspondence:

Here’s the program we just constructed:

We can expand out the definitions of isLowerBound, isGreatestLowerBound, and (.) to confirm that this really is exactly the same as the original Haskell program above:

But it’s also worth noting that it will still typecheck for any definition of the type constructor ≤, as long as we provide terms of the right type for the lattice lemmas. So this really is a translation of the proof in its full generality.

I’m not sure exactly what moral to draw from this exercise, except perhaps this: If you feel confused trying to prove the Knaster-Tarski theorem, you shouldn’t! If you pretend it’s a Haskell program, it’s easy to construct the proof by just following the types.

[duggardata]

Info, Please:  Extended Families  (Part 1, Duggars + In–Laws)

So…  I think it would be nuts to actually draw out of a family tree, since it would have to be so, so huge.  But, I’m happy to provide a written Post about this!  I’ll trace each family back to the Great–Grandparents of Josh, etc., usually dubbed the “Second Generation” by Duggar Data.  I’m not going to include any Uncles, Aunts, Great–Uncles, Great–Aunts, or Cousins, except for the Duggars / Bates.  Too much to research, and it’s maybe a bit too invasive for the less high–profile families.

Part 1 will be Duggars + In–Laws.  Part 2 will be Bateses + In–Laws.  Part 3 will be the rest of the Predictor Families.  Onward, to...   

The Duggar Family

Josh, etc.’s Parents are James Robert (“Jim Bob” or “J.B.”) Duggar (b. 1965) + Michelle Annette (Ruark) Duggar (b. 1966).  Their Paternal Grandparents—i.e., Jim Bob’s Parents—are James Lee (“J.L.”) Duggar (1936–2009) + Mary Leona (Lester) Duggar (1941–2019).  Their Maternal Grandparents—i.e., Michelle’s Parents—are Garrett Floyde Ruark (1924–2010) + Ethel Marie (Hardin) Ruark (1927–1991).  Jim Bob has 1 Sibling (Deanna).  Michelle is 1 of 7 Children.

Paternal Grandparents   James Lee + Mary Leona (Lester) Duggar

Maternal Grandparents   Garrett Floyde + Ethel Marie (Hardin) Ruark

Paternal Aunts / Uncles   Deanna (Duggar) Jordan

Maternal Aunts / Uncles   Pamela Ethel Peters, Freda Benderman (d. 2015), Evelyn Alice Ruark, Kathie Ann Arnold (1952–2013), Carolyn Ann Hudgins, Garrett Floyde Ruark, Jr.

Parents   James Robert (“Jim Bob”) + Michelle Annette (Ruark) Duggar

Children   19 Children  (Josh, Jana, etc.)

Grandchildren   17 So Far  (+1 On The Way)        

In–Laws...  After the jump.

The Burnett Family

Abbie (Burnett) Duggar’s Parents are John Haskell Burnett II (b. 1958) + Cheryl Denise (Clay) Burnett (b. 1959).  She has 7 Siblings—namely, John–Clay Byford, Hannah Joy, Caleb Andrew, Charity Faith, Maggie Ruth, Carinna Elizabeth, and Benjamin Josiah.  Neither set of Grandparents is known; however, her Paternal Grandfather is presumably John Haskell Burnett, Sr.  (See Also.)

Paternal Grandparents   John Haskell Burnett, Sr.  +  [ Unknown ]

Maternal Grandparents   [ Unknown ]

Parents   John Haskell Burnett II + Cheryl Denise (Clay) Burnett

Children   John–Clay Byford Burnett, Hannah Joy Dornick, Caleb Andrew Burnett, Abbie Grace Duggar, Charity Faith Wearden, Maggie Ruth Burnett, Carrina Elizabeth Penn, Benjamin Josiah Burnett    

Grandchildren   10 So Far  (inc. Grace Annette Duggar)     

The Caldwell Family

Kendra’s Family consists of her Parents, Pastor Gene Paul (“Paul”) Caldwell (b. 1977) + Christina (Hamrick) Caldwell (b. 1979), and 8 Children—Kendra Renee, Lauren Hope, Micah Joel, Nathan, Timothy, Olivia Grace, Jesiah Mathew, and Isaiah Gabriel.  Her Paternal Grandparents are Marion Lynn + Peggy Ann (Dragland) Caldwell.  Her Maternal Grandparents are Unknown.

Paternal Grandparents   Marion Lynn + Peggy Ann (Dragland) Caldwell

Maternal Grandparents   [ Unknown ]

Parents   Gene Paul (“Paul”) + Christina (Hamrick) Caldwell

Children   Kendra Renee Duggar, Lauren Hope Caldwell, Micah Joel Caldwell, Nathan Caldwell, Timothy Caldwell, Olivia Grace Caldwell, Jesiah Mathew Caldwell, Isaiah Gabriel Caldwell  

Grandchildren   2 So Far  (Garrett David Duggar and Addison Renee Duggar)  

The Dillard Family

As for Jill’s Husband, Derick, his Parents are Cathy (George) Byrum (Previously Dillard) (b. 1957) + Richard (“Rick”) W. Dillard (1957–2008).  His Stepfather is Ronnie Byrum.  He has 1 Sibling—a brother, Daniel David.  His Grandparents are Norman + Sue Dillard (Paternal), and Shuford George + Vergie (Wilkerson) Payne (Previously George / Pipher) (1932–2014) (Maternal).

Paternal Grandparents   Norman + Sue Dillard

Maternal Grandparents   Shuford George + Vergie (Wilkerson) Paine

Parents   Richard “Rick” Dillard + Cathy (George) Byrum

Children   Derick Michael Dillard, Daniel David Dillard

Grandchildren   3 So Far  (Israel David Dillard, Samuel Scott Dillard, and Jaxon Michael Dillard)

The Forsyth Family

Joy’s In–Laws include Austin’s Parents, Terry Lynn Forsyth (b. 1959) + Roxanne (Waters) Forsyth (b. 1963), and his sister, Meagan.  Roxanne is Terry’s 2nd Wife; he was previously married to Julia Ann Rowe, but got divorced in 1982.  Terry + Julia had 2 Children—Brandon Lynn and Rachael Marie—who are Half–Siblings of Austin.  Nothing is known about Austin’s Grandparents.

Paternal Grandparents   [ Unknown ]

Maternal Grandparents   [ Unknown ]

Parents   Terry + Roxanne (Waters) Forsyth; Terry + Julia (Rowe) Forsyth

Children (w/ Roxanne)   Meagan Elizabeth Ballinger, Austin Martyn Forsyth

Children (w/ Julie)   Brandon Lynn Forsyth, Rachael Marie Tunstill  

Grandchildren (w/ Roxanne)   3 So Far  (+2 On The Way)

Grandchildren (w/ Julia)  10 So Far  

The Keller Family

The Keller Family is headed by Parents Michael “Mike” Edward Keller (b. 1959) + Lillie “Suzette” (Stembridge) Keller (b. 1955).  Together, they have 8 Children—Esther Joy, Rebekah Ann, Daniel Michael, Priscilla Lynn, Anna Renee, Nathan Edward, and David Nathaniel.  The Paternal Grandparents are Unknown.  Their Maternal Grandparents are Melvin Rudolph Stembridge, Sr. + Dorothy Mae (Adams) Stembridge (d. 2008).  

Paternal Grandparents   [ Unknown ]

Maternal Grandparents   Melvin Rudolph Stembridge, Sr. + Dorothy Mae (Adams) Stembridge

Parents   Michael Edward + Lillie “Suzette” (Stembridge) Keller

Children   Esther Joy Shrader, Rebekah Ann Hunt, Daniel Michael Keller, Priscilla Lynn Waller, Anna Renee Duggar, Susanna Grace Bridges, Nathan Edward Keller, David Nathaniel Keller

Grandchildren   22 So Far (+1 On The Way)  [Note—This includes non–Quiverful Couples’ children, which I typically don’t discuss.]

The Seewald Family

Looking to the Seewald Clan, Ben’s Parents are Michael Seewald (b. 1974) + Guinnevere (“Guinn”) (Eaton) Seewald (b. 1974).  He has 6 Siblings, namely—Jessica Lane, Danielle, Michelle Alaine, Ethan, Faith, and Tommy Jack.  Ben’s Grandparents include Bob + Arlene Seewald (Paternal), and Robert (“Bob”) Eaton (1950–2019) + Deborah Eaton (Maternal).

Paternal Grandparents   Bob + Arlene Seewald

Maternal Grandparents   Robert (“Bob”) + Deborah Eaton

Parents   Michael + Guinnevere (“Guinn”) (Eaton) Seewald

Children   Benjamin Michael Seewald, Jessica Lane Lester, Danielle Seewald, Michelle Alaine Seewald, Ethan Seewald, Faith Seewald, Tommy Jack Seewald      

Grandchildren   3 So Far  (Spurgeon Elliot Seewald, Henry Wilberforce Seewald, Ivy Jane Seewald)

The Swanson Family

Lauren’s Parents are Dwain Charles Swanson (b. 1975) + Lana M. Swanson (b. 1975).  Nothing is known about the extended family; both sets of Grandparents are Unknown.  Lauren is 1 of 9 Children.  Her siblings include Lily, David, Lydia, Daniel, Dustin, Drew, Luci, and Duke.

Paternal Grandparents   [ Unknown ]

Maternal Grandparents   [ Unknown ]

Parents   Dwain Charles + Lana Swanson

Children   Lauren Milagro Duggar, Lily Swanson, David Swanson, Lydia Swanson, Daniel Swanson, Dustin Swanson, Drew Swanson, Luci Swanson, Duke Matthew Swanson        

Grandchildren   1 So Far  (Bella Milagro Duggar)

The Vuolo Family

Updated 4–22–20 w/ information provided by an Anon...

As for the Vuolo Clan, Jeremy’s Parents are Charles (“Chuck”) Henry Vuolo (b. 1951) + Diana Lynn (Salamon) Vuolo (b. 1955).  Jeremy has 2 Living Siblings—Valerie and Charles—and a 3rd Sibling (Angela; d. 1984) who was stillborn.  (It seems she was Valerie’s twin, since Angela’s death of death is Valerie’s DOB.)  Jeremy’s Paternal Grandparents are Tony + Ann Vuolo, and Jeremy’s Maternal Grandparents are Stanley Peter Salamon (d. 2013) + Mary Elizabeth (Gillespie) Salamon (1926–2009).  

Paternal Grandparents   Tony + Ann Vuolo

Maternal Grandparents   Stanley P. + Mary E. (Gillespie) Salamon

Parents   Charles (“Chuck”) Henry + Diana Lynn (Salamon) Vuolo

Children   Valerie Vuolo, Charles Anthony Vuolo, Jeremy Joseph Vuolo

Grandchildren   1 So Far  (Felicity Nicole Vuolo)

Next up, the Bates Family + In–Laws...  Stay tuned for Part 2.

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