From cab03d63ec7aa5c56ab5a94573a3ad58781e8091 Mon Sep 17 00:00:00 2001 From: Tan Kian-ting Date: Wed, 25 Dec 2024 00:38:19 +0800 Subject: [PATCH] initial --- .gitignore | 1 + TypeTheoryAndFormalProof第3章.pdf | Bin 0 -> 1429858 bytes TypeTheoryAndFormalProof第3章.typ | 50 ++++++ dated/TypeTheory2第2章part2.pdf | Bin 0 -> 1510323 bytes dated/TypeTheory2第2章part2.typ | 223 ++++++++++++++++++++++++ dated/TypeTheoryAndFormalProof第2章.pdf | Bin 0 -> 2034933 bytes dated/TypeTheoryAndFormalProof第2章.typ | 113 ++++++++++++ 7 files changed, 387 insertions(+) create mode 100644 .gitignore create mode 100644 TypeTheoryAndFormalProof第3章.pdf create mode 100644 TypeTheoryAndFormalProof第3章.typ create mode 100644 dated/TypeTheory2第2章part2.pdf create mode 100644 dated/TypeTheory2第2章part2.typ create mode 100644 dated/TypeTheoryAndFormalProof第2章.pdf create mode 100644 dated/TypeTheoryAndFormalProof第2章.typ diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..0c6117d --- /dev/null +++ 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M$,$M$中所有裡面的$x$變成bound variable(約束變數)。 + +$lambda x : sigma : M$ 取決於(depends on)term $x$ => $lambda_arrow.r$可以terms depends on the terms + +$M N$是型別的應用 + +如此可說是一階(first order)抽象 + +本章討論terms depend on types (second order operation/second order dependency) 可以理解成依型別值(?) + +本章討論$lambda 2$:二階具型別lambda演算 + +範例:$id(x)=x$函數 + +若是自然數,則有$lambda x : \n\a\t . space x : \n\a\t -> \n\a\t $ + +若是布林值,則有$lambda x : italic("bool") . space x : \b\o\o\l -> \b\o\o\l $ + +還有函數$lambda x : (\n\a\t->\b\o\o\l) . space x : (\n\a\t->\b\o\o\l) -> (\n\a\t->\b\o\o\l) $ + +可是這樣函數要每個型別就重複定義一次,有無更好的方法? + +我們需要定義一個任意型別$alpha$,變成:$f equiv lambda x : alpha space . x$ + +但是對於$M in \n\a\t$,$f space M$是invalid,因為$alpha equiv.not \n\a\t$。所以要進行抽象化: + +$lambda alpha : ast . lambda x : alpha . x$ + +其中$alpha$ 是type,$ast$是kind。 + +所以 + +$(lambda alpha : ast . lambda x : alpha . x ) \n\a\t $ + +$arrow.twohead.r_beta lambda x : \n\a\t . space x$ + +且 +$(lambda alpha : ast . lambda x : alpha . x ) (\b\o\o\l->\n\a\t) $ + 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