0
votes
0answers
33 views

Proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in $\lambda\pi $ calculus $\equiv$

What is the right representation of the proof of $(\forall x. \varepsilon(x)) \Rightarrow \bot $ in simple type theory as a term of $\lambda\pi $ calculus $\equiv$? Note on notation: The epsilon ...
2
votes
1answer
55 views

Simple type theory: Proof inexistance of closed term

In simple type theory, how can I prove that there is no closed term of type? $$((P \Rightarrow Q) \Rightarrow Q) \Rightarrow P$$
1
vote
0answers
20 views

Reductions under lambda in dependently typed lambda calculus

I am currently reading a Simon Thompson's Type Theory book. In chapter 5 he introduces a system TT(0,C), which limits a notion of reduction . In this notion of reduction, reductions under lambda are ...
4
votes
3answers
82 views

What's a good resource to learn about the simply typed lambda calculus?

I've read An Introduction to Functional Programming Through Lambda Calculus by Greg Michaelson, and found it to be a very good resource to learn about the untyped lambda calculus. However, I want to ...
7
votes
3answers
144 views

How or why does intutionistic logic proof negations from within the theory, constructively?

I'm having a little of a cognitive dissonance why, in intuitionistic logic, it's possible to show stentences like $(\neg A \land \neg B) \implies \neg(A\lor B).$ In plain text: If 'A isn't true' as ...
3
votes
1answer
99 views

Give an example of a program in Simply Typed Lambda that produces Bottom.

I'm not sure how bottom applies to simply typed lambda calculus. not A is a common abbreviation for A -> ⊥ But I see no way to construct a function of that signature within the theory. Edit: A more ...
0
votes
1answer
67 views

What can be proven within Simply Typed lambda calculus?

I was reading http://en.wikipedia.org/wiki/Simply_typed_lambda_calculus and I'm having a hard time thinking of anything remotely interesting that can be proven within Simply Typed lambda calculus. Am ...
3
votes
3answers
184 views

Book on lambda calculus logic and type theory

Can someone recommend me a book for self study which will cover topics of logic, lambda calculus and type theory. I know about "Computability and Logic" written by Bolos but it describe recursive ...
11
votes
1answer
606 views

If $f(x)=g(x)$ for all $x:A$, why is it not true that $\lambda x{.}f(x)=\lambda x{.}g(x)$?

There's something about lambda calculus that keeps me puzzled. Suppose we have $x:A\vdash f(x):P(x)$ and $x:A\vdash g(x):P(x)$ for some dependent type $P$ over a type $A$. Then it is not necessarily ...
38
votes
6answers
4k views

Learning Lambda Calculus

What are some good online/free resources (tutorials, guides, exercises, and the like) for learning Lambda Calculus? Specifically, I am interested in the following areas: Untyped lambda calculus ...