……重新翻开 Thompson 的《Type Theory and Functional Programming》，显然我已经能看懂很多了。如果你像我一样喜欢老书的那种怀旧感，那就去搞一本 Per Martin-Löf 主编的《Intuitionistic Type Theory》原版吧。
…re-start Thompson's "Type Theory and Functional Programming" and it was clear that I started to understand a lot of it. If you enjoy the nostalgia of older books like me, then pick up an original copy of "Intuitionistic Type Theory" edited by Per Martin-Löf.
……Thompson（随便搜一下就能找到 PDF）Rob Nederpelt / Herman Geuvers 的《Type Theory and Formal Proof》前 120 页。关于 dependent types 的一些早期内容（上面那些书里也有涉及），可以看看 Per Martin-Lof 的《Intuitionistic Type Theory》、罗朝晖（Zhaohui Luo）的《Computation and Reasoning: A Type Theory for Computer Science》、Edwin Brady 的《Type-Driven Development with Idris》、《Gentle Introduction to Dependent Types with Idris》……
…Thompson (PDF available via quick search) First 120 pages of "Type Theory and Formal Proof" by Rob Nederpelt / Herman Geuvers For some early stuff on dependent types (also covered in the above), check out "Intuitionistic Type Theory" by Per Martin-Lof "Computation and Reasoning: A Type Theory for Computer Science" by Zhaouhui Luo "Type-Driven Development with Idris" by Edwin Brady "Gentle Introduction to Dependent Types with Idris"…
……瞥了一眼 Ada，不过我很肯定：类型论学者说的 dependently typed languages 根本不是这个意思。这类语言源自 Per Martin-Löf 的形式化理论（也称为直觉主义类型论 Intuitionistic Type Theory），它们包含 dependent sum 和 dependent product 类型，并允许类型检查器（type checker）证明关于代码的复杂陈述。（dependent types 的历史与形式化……的历史交织在一起……
…glance at Ada, I am nevertheless sure: this is not what type theorists mean when they talk about dependently typed languages. Such languages derive from the formulation of Per Martin-Löf (also called Intuitionistic Type Theory), they include dependent sum and dependent product types, and they allow type checkers to prove complex statements about code. (The history of dependent types is intertwined with the history of formalizing…
……罗素悖论 Russell's paradox）。解决方案是让每个类型都处于一个“universe”中，使得 `Type l : Type (l + 1)`，其中 `l` 是 universe level。去看看 Martin-Lof type theory：https://en.wikipedia.org/wiki/Intuitionistic_type_theory
…Russell's paradox). The solution is for each type to be in a "universe" so that `Type l : Type (l + 1)`, where `l` is the universe level. Read about Martin-Lof type theory: https://en.wikipedia.org/wiki/Intuitionistic_type_theory
> After 1970 with the introduction of Martin-Löf's Intuitionistic Type Theory, an inbreed of abstract math and computer science, a period of intense research on new type languages as Agda and Epigram started. This ended up forming the basic support layer for functional programming…
…dependency issues and just some general PITA's. [1] https://www.youtube.com/watch?v=YR5WdGrpoug&t=1504s [2] https://math.stackexchange.com/questions/4486995/what-is-an-... [3] https://en.wikipedia.org/wiki/Intuitionistic_type_theory#Ext... [4] https://youtu.be/YR5WdGrpoug?t=2668 [5] https://www.quora.com/In-type-theory-what-is-an-eliminator-a... [6] https://youtu.be/YR5WdGrpoug?t=2823
……重新翻开 Thompson 的《Type Theory and Functional Programming》，显然我已经能看懂很多了。如果你像我一样喜欢老书的那种怀旧感，那就去搞一本 Per Martin-Löf 主编的《Intuitionistic Type Theory》原版吧。
…re-start Thompson's "Type Theory and Functional Programming" and it was clear that I started to understand a lot of it. If you enjoy the nostalgia of older books like me, then pick up an original copy of "Intuitionistic Type Theory" edited by Per Martin-Löf.
……Thompson（随便搜一下就能找到 PDF）Rob Nederpelt / Herman Geuvers 的《Type Theory and Formal Proof》前 120 页。关于 dependent types 的一些早期内容（上面那些书里也有涉及），可以看看 Per Martin-Lof 的《Intuitionistic Type Theory》、罗朝晖（Zhaohui Luo）的《Computation and Reasoning: A Type Theory for Computer Science》、Edwin Brady 的《Type-Driven Development with Idris》、《Gentle Introduction to Dependent Types with Idris》……
…Thompson (PDF available via quick search) First 120 pages of "Type Theory and Formal Proof" by Rob Nederpelt / Herman Geuvers For some early stuff on dependent types (also covered in the above), check out "Intuitionistic Type Theory" by Per Martin-Lof "Computation and Reasoning: A Type Theory for Computer Science" by Zhaouhui Luo "Type-Driven Development with Idris" by Edwin Brady "Gentle Introduction to Dependent Types with Idris"…
……瞥了一眼 Ada，不过我很肯定：类型论学者说的 dependently typed languages 根本不是这个意思。这类语言源自 Per Martin-Löf 的形式化理论（也称为直觉主义类型论 Intuitionistic Type Theory），它们包含 dependent sum 和 dependent product 类型，并允许类型检查器（type checker）证明关于代码的复杂陈述。（dependent types 的历史与形式化……的历史交织在一起……
…glance at Ada, I am nevertheless sure: this is not what type theorists mean when they talk about dependently typed languages. Such languages derive from the formulation of Per Martin-Löf (also called Intuitionistic Type Theory), they include dependent sum and dependent product types, and they allow type checkers to prove complex statements about code. (The history of dependent types is intertwined with the history of formalizing…
……罗素悖论 Russell's paradox）。解决方案是让每个类型都处于一个“universe”中，使得 `Type l : Type (l + 1)`，其中 `l` 是 universe level。去看看 Martin-Lof type theory：https://en.wikipedia.org/wiki/Intuitionistic_type_theory
…Russell's paradox). The solution is for each type to be in a "universe" so that `Type l : Type (l + 1)`, where `l` is the universe level. Read about Martin-Lof type theory: https://en.wikipedia.org/wiki/Intuitionistic_type_theory
> 1970 年之后，随着 Martin-Löf 的直觉主义类型论（Intuitionistic Type Theory）——一种抽象数学与计算机科学的杂交产物——的提出，针对 Agda 和 Epigram 等新型类型语言的密集研究阶段拉开帷幕。这最终构成了函数式编程的基础支撑层……
> After 1970 with the introduction of Martin-Löf's Intuitionistic Type Theory, an inbreed of abstract math and computer science, a period of intense research on new type languages as Agda and Epigram started. This ended up forming the basic support layer for functional programming…
……依赖问题，以及一些普遍让人头疼的破事（PITA）。[1] https://www.youtube.com/watch?v=YR5WdGrpoug&t=1504s [2] https://math.stackexchange.com/questions/4486995/what-is-an-... [3] https://en.wikipedia.org/wiki/Intuitionistic_type_theory#Ext... [4] https://youtu.be/YR5WdGrpoug?t=2668 [5] https://www.quora.com/In-type-theory-what-is-an-eliminator-a... [6] https://youtu.be/YR5WdGrpoug?t=2823
…dependency issues and just some general PITA's. [1] https://www.youtube.com/watch?v=YR5WdGrpoug&t=1504s [2] https://math.stackexchange.com/questions/4486995/what-is-an-... [3] https://en.wikipedia.org/wiki/Intuitionistic_type_theory#Ext... [4] https://youtu.be/YR5WdGrpoug?t=2668 [5] https://www.quora.com/In-type-theory-what-is-an-eliminator-a... [6] https://youtu.be/YR5WdGrpoug?t=2823