% (find-LATEX "2020seelyplc.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020seelyplc.tex" :end))
% (defun d () (interactive) (find-pdf-page      "~/LATEX/2020seelyplc.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020seelyplc.pdf"))
% (defun e () (interactive) (find-LATEX "2020seelyplc.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020seelyplc"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page   "~/LATEX/2020seelyplc.pdf")
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% http://angg.twu.net/LATEX/2020seelyplc.pdf
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% «.title»	(to "title")

\documentclass[oneside,12pt]{article}
\usepackage[colorlinks,citecolor=DarkRed,urlcolor=DarkRed]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
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\usepackage{amssymb}
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%\usepackage{colorweb}                 % (find-es "tex" "colorweb")
%\usepackage{tikz}
%
% (find-dn6 "preamble6.lua" "preamble0")
\usepackage{proof}   % For derivation trees ("%:" lines)
\input diagxy        % For 2D diagrams ("%D" lines)
\xyoption{curve}     % For the ".curve=" feature in 2D diagrams
%
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\addbibresource{catsem-slides.bib}        % (find-LATEX "catsem-slides.bib")
%
% (find-es "tex" "geometry")
\begin{document}

\catcode`\^^J=10
\directlua{dofile "dednat6load.lua"}  % (find-LATEX "dednat6load.lua")

%\printbibliography



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% «title»  (to ".title")

{\setlength{\parindent}{0em}
\footnotesize

Notes on [Seely87], a.k.a.:

``Categorical Semantics for Higher Order Polymorphic Lambda Calculus'', available at:

\url{https://www.jstor.org/stable/2273831} and

\url{http://www.math.mcgill.ca/rags/JSL/PLC.pdf}

\ssk

These notes are at:

\url{http://angg.twu.net/LATEX/2020seelyplc.pdf}

\ssk

See:

\url{http://angg.twu.net/LATEX/2020favorite-conventions.pdf}

\url{http://angg.twu.net/math-b.html\#favorite-conventions}

I wrote these notes mostly to test if the conventions above
are good enough.


}


\bsk
\bsk

% (find-books "__cats/__cats.el" "seely-plc")
% (find-seelyplcpage (+ -967 972) "1.1.2. Operators")
% (find-seelyplctext (+ -967 972) "1.1.2. Operators")

1.1.2: Operators


%:
%:   ---   ----
%:   *∈1   ⊤∈Ω
%:
%:   ^o1   ^oT
%:
%:
%:   σ∈Ω  τ∈Ω    σ∈Ω  τ∈Ω   
%:   --------    --------   
%:    σ∧τ∈Ω       σ⊸τ∈Ω     
%:                          
%:    ^oand       ^oimp     
%:
%:
%:   σ∈Ω         σ∈Ω     
%:   --------    --------
%:   Σα∈A·σ∈Ω    Πα∈A·σ∈Ω
%:                       
%:   ^oSigma     ^oPi
%:
%:
%:   σ∈A  τ∈B        σ∈A×B         σ∈A×B     
%:   ---------(×I)   ------(×E)    ------(×E)
%:   〈σ,τ〉∈A×B       π_1σ∈A        π_2σ∈B    
%:
%:   ^oxI            ^oxE1         ^oxE2
%:
%:
%:   σ∈Ω             τ∈A  σ∈Ω^A
%:   -----------     ----------
%:   [α∈A:σ]∈Ω^A      σ(τ)∈Ω
%:
%:   ^obrI            ^obrE
%:
%:
$$\pu
  \begin{array}{c}
  \ded{o1} \qquad \ded{oT} \\ \\
  \ded{oand} \qquad \ded{oimp} \\ \\
  \ded{oSigma} \qquad \ded{oPi} \\ \\
  \ded{oxI} \qquad \ded{oxE1} \qquad \ded{oxE2} \\ \\
  \ded{obrI} \qquad \ded{obrE} \\
  \end{array}
$$

\newpage

% (find-seelyplcpage (+ -967 972) "1.1.4. Terms")
% (find-seelyplctext (+ -967 972) "1.1.4. Terms")

1.1.4: Terms

%:
%:  ---(⊤I)
%:  *∈⊤
%:
%:  ^tstar
%:
%:
%:  a∈τ                  b∈σ  a∈σ⊸τ
%:  --------------(⊸I)   ----------(⊸E)
%:  (λx∈σ·a)∈(σ⊸τ)         a(b)∈τ
%:
%:  ^tlambda                ^tapp
%:
%:
%:  a∈σ  b∈τ       a∈σ∧τ         a∈σ∧τ
%:  --------(∧I)   ------(∧E)    ------(∧E)
%:  〈a,b〉∈σ∧τ      π_1a∈σ        π_2a∈τ
%:
%:  ^tpair         ^tpi1         ^tpi2
%:
%:
%:  --------------------------(ΣI)
%:  I_{Σα·σ,τ}∈(σ[τ/α]⊸Σα∈A·σ)
%:
%:  ^tSigmaI
%:
%:  a∈(σ⊸ρ)
%:  ----------------------(ΣE)
%:  (𝐛Vα∈A·a)∈((Σα∈A·σ)⊸ρ)
%:
%:  ^tSigmaE
%:
%:  a∈σ
%:  -----------------(ΠI)
%:  (Λα∈A·a)∈(Πα∈A·σ)
%:
%:  ^tPiI
%:
%:  τ∈A  a∈(Πα∈A·σ)
%:  ---------------(ΠE)
%:  a\{τ\}∈σ[τ/α]
%:
%:  ^PiE
%:
%:
$$\pu
  \begin{array}{c}
  \ded{tstar} \\ \\
  \ded{tlambda} \qquad \ded{tapp} \\ \\
  \ded{tpair} \qquad \ded{tpi1} \qquad \ded{tpi2} \\ \\
  \ded{tSigmaI} \\ \\
  \ded{tSigmaE} \\ \\
  \ded{tPiI} \\ \\
  \ded{PiE} \\
  \end{array}
$$

\newpage

% (find-seelyplcpage (+ -967 972) "1.1.4. Terms")
% (find-seelyplctext (+ -967 972) "1.1.4. Terms")

Decyphering $(ΣI)$:

$(ΣI)$ If $α$ is an indeterminate of order $A$, $σ∈Ω$, $τ∈A$, then
$I_{Σα·σ,τ} ∈ σ[τ/α]⊸Σα∈A·σ$. When clear from the context, we shall
denote this term by $I_τ$, or even by $I$; in particular, if
$b∈σ[τ/α]$, then $I(b)∈Σα∈A·σ$.
%:
%:  P(τ)
%:  ---------(∃I)
%:  ∃α∈A.P(α)
%:
%:  ^dei1
%:
%:  
%:  --------------(∃I)
%:  P(τ)⊸∃α∈A.P(α)
%:
%:  ^dei2
%:
%:
%:  --------------(∃I)
%:  P[α:=τ]⊸∃α∈A.P
%:
%:  ^dei3
%:
%:
%:  -------------------(ΣI)
%:  I_τ∈(σ[τ/α]⊸Σα∈A·σ)
%:
%:  ^dei5
%:
%:
%:            -----------------(ΣI)
%:  b∈σ[τ/α]  I∈(σ[τ/α]⊸Σα∈A·σ)
%:  ---------------------------(⊸E)
%:  I(b)∈Σα∈A·σ
%:
%:  ^dei6
%:
%:
%:
$$\pu
  \begin{array}{c}
  \ded{dei1} \\ \\
  \ded{dei2} \\ \\
  \ded{dei3} \\ \\
  \ded{dei5} \\ \\
  \ded{dei6} \\
  \end{array}
$$


% (find-seelyplcpage (+ -967 972) "1.1.4. Terms")
% (find-seelyplctext (+ -967 972) "1.1.4. Terms")


Decyphering $(ΣE)$:

$(ΣE)$ If $a∈σ⊸ρ$, $α$ an indeterminate of order $A$ which is not free
in $p$ nor in the type of any free variable in $a$, then
$𝐛Vα∈A·a∈(Σα∈A·σ)⊸ρ$.


%:
%:
%:          [P]^1
%:           :
%:  ∃α∈A.P   Q
%:  ----------(∃E)
%:      Q
%:
%:      
%:




\end{document}

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% <make>

 (eepitch-shell)
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# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020seelyplc veryclean
make -f 2019.mk STEM=2020seelyplc pdf

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