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% (find-LATEX "2020lambek86.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2020lambek86.tex" :end))
% (defun d () (interactive) (find-pdf-page "~/LATEX/2020lambek86.pdf"))
% (defun d () (interactive) (find-pdftools-page "~/LATEX/2020lambek86.pdf"))
% (defun e () (interactive) (find-LATEX "2020lambek86.tex"))
% (defun u () (interactive) (find-latex-upload-links "2020lambek86"))
% (defun v () (interactive) (find-2a '(e) '(d)) (g))
% (find-pdf-page "~/LATEX/2020lambek86.pdf")
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% file:///tmp/pen/2020lambek86.pdf
% http://angg.twu.net/LATEX/2020lambek86.pdf
% (find-LATEX "2019.mk")
\documentclass[oneside,12pt]{article}
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% (find-dn6 "preamble6.lua" "preamble0")
%\usepackage{proof} % For derivation trees ("%:" lines)
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\begin{document}
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% %L dofile "edrxtikz.lua" -- (find-LATEX "edrxtikz.lua")
% %L dofile "edrxpict.lua" -- (find-LATEX "edrxpict.lua")
% \pu
% (find-books "__cats/__cats.el" "lambek86")
% (find-books "__cats/__cats.el" "lncs0242")
{\setlength{\parindent}{0em}
\footnotesize
Notes on [Lambek86], a.k.a.:
``Cartesian Closed Categories and Typed $λ$-Calculi'', available at:
\url{https://link.springer.com/chapter/10.1007\%2F3-540-17184-3_44}
or as pages 136--175 of LNCS 242:
\url{https://link.springer.com/book/10.1007/3-540-17184-3}
\ssk
These notes are at:
\url{http://angg.twu.net/LATEX/2020lambek86.pdf}
}
\bsk
The introduction says:
{\sl While the material in the first part has been published before [L1974,
1980, LS1984], an attempt is made to look at some of it from a
different point of view and to clarify some difficult points. At any
rate, it is hoped that it will serve as an introduction to the
forthcoming book ``Introduction to higher order categorical logic'',
written in collaboration with Phil Scott.}
\section*{2. Cartesian Categories}
% (find-lambek86page (+ -135 140) "2. CARTESIAN CATEGORIES")
% (find-lambek86text (+ -135 140) "2. CARTESIAN CATEGORIES")
% (find-lambek86page (+ -135 141) "admits the following form of the deduction theorem")
% (find-lambek86text (+ -135 141) "admits the following form of the deduction theorem")
% (find-books "__logic/__logic.el" "hindley-seldin2")
% (find-hindleyseldin2page (+ 14 26) "2C Abstraction in CL")
Page 141:
Although the conjunction calculus contains no symbol for implication,
it admits the following form of the \ColorRed{\sl deduction theorem}:
Proposition 2.1: if $φ(x):B→C$ is a proof from the assumption $x:T→A$,
there is a proof $κ_{x∈A}φ(x):A∧B→C$ in $\calL$ not depending on the
assumption $x$.
(...)
\msk
Proof: there are four cases in the proof of the deduction theorem:
(1) $φ(x) = k:B→C$, a proof in $\calL$;
(2) $φ(x) = x:T→A$, where $B=T$ and $C=A$;
(3) $φ(x) = χ(x)ψ(x)$, where $ψ(x):B→D$ and $χ(x):D→C$;
(4) $φ(x) = 〈ψ(x),χ(x)〉$, where $ψ(x):B→D$, $χ(x):B→E$ and $C=D∧E$.
\msk
We define $κ_{x∈A} φ(x)$ by induction on the ``length'' of $φ(x)$:
(1) $κ_{x∈A} k = kπ'_{A,B}$;
(2) $κ_{x∈A} x = π_{A,T}$;
(3) $κ_{x∈A} (χ(x)ψ(x)) = κ_{x∈A} χ(x) 〈π_{A,B}, κ_{x∈A}ψ(x)〉$;
(4) $κ_{x∈A} 〈ψ(x),χ(x)〉 = 〈κ_{x∈A} ψ(x), κ_{x∈A} χ(x)〉$;
% From: (find-LATEX "2017planar-has-defs.tex" "defub")
%
\def\defub#1#2{\expandafter\def\csname ub-#1\endcsname{#2}}
\def\ifubundefined#1{\expandafter\ifx\csname ub-#1\endcsname\relax}
\def\ub#1{\ifubundefined{#1}
\errmessage{UNDEFINED UB: #1}
\else
\csname ub-#1\endcsname
\fi
}
\def\und#1#2{\underbrace{#1}_{#2}}
\def\ka{κ_{x∈A}}
%UB \ka( k ) &=& k π'_{A,B}
%UB ------ ---- --------
%UB :B→C :B→C :A×B→B
%UB ----------- -------------
%UB :A×B→C :A×B→C
%L
%L defub "cond1"
%L
%UB \ka( x ) &=& π_{A,T}
%UB ------ -------
%UB :T→A :A×T→A
%UB -----------
%UB :A×T→A
%L
%L defub "cond2"
%L
%UB \ka(χ(x) ψ(x)) &=& \kaχ(x) 〈π_{A,B},\kaψ(x)〉
%UB ---- ---- ---- ------- ----
%UB :D→C :B→D :D→C :A×B→A :B→D
%UB --------- ------- -------
%UB :B→C :A×D→C :A×B→D
%UB -------------- -----------------
%UB :A×B→C :A×B→A×D
%UB -------------------------
%UB :A×B→C
%L
%L defub "cond3"
%L
%UB \ka〈ψ(x),χ(x)〉 &=& 〈 \ka ψ(x) , \ka χ(x) 〉
%UB ---- ---- ---- ----
%UB :B→E :B→D :B→E :B→D
%UB ----------- --------- --------
%UB :B→D×E :A×B→E :A×B→D
%UB -------------- ------------------------
%UB :A×B→D×E :A×B→D×E
%L
%L defub "cond4"
%L
%
$$\pu
\begin{array}{rcl}
\ub{cond1} \\ \\
\ub{cond2} \\ \\
\ub{cond3} \\ \\
\ub{cond4} \\
\end{array}
$$
\end{document}
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% <make>
* (eepitch-shell)
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# (find-LATEXfile "2019planar-has-1.mk")
make -f 2019.mk STEM=2020lambek86 veryclean
make -f 2019.mk STEM=2020lambek86 pdf
% Local Variables:
% coding: utf-8-unix
% ee-tla: "l86"
% End: