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

I am making my first steps in lambda calculus, so please bear with me. I want to create a lambda function, that given two boolean expressions (either $F$ or $T$ – defined below), simulates the formula $\neg p\vee q$ where the $p$ is its first argument and $q$ is the second. Here is my attempt […]

The substitution lemma in lambda-calculus is proved by the following way, but I just did not understand the application of induction hypothesis in it. The lemma as shown below, where x and y are distinct and x is not among the free variables of L: M[x:=N][y:=L] equals M[y:=L][x:=N[y:=L]] to prove that in the case where […]

The wikipedia definition of lambda terms is: The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: a variable, $x$, is itself a valid lambda term if $t$ is a lambda term, and $x$ is a variable, then $(\lambda x.t)$ is a lambda term (called a […]

In Hindley (Lambda-Calculus and Combinators, an Introduction), Corollary 3.3.1 to the fixed-point theorem states: In $\lambda$ and CL: for every $Z$ and $n \ge 0$ the equation $$xy_1..y_n = Z$$ can be solved for $x$. That is, there is a term $X$ such that $$Xy_1..y_n =_{\beta,w} [X/x]Z$$ I dont understand how to even think about […]

I think I’m not understanding it, but eta-conversion looks to me as a beta-conversion that does nothing, a special case of beta-conversion where the result is just the term in the lambda abstraction because there is nothing to do, kind of a pointless beta-conversion. So maybe eta-conversion is something really deep and different from this, […]

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 Simply-typed lambda calculus Other typed lambda calculi Church’s Theory of Types (I’m not sure where this fits in). (As I understand, this should provide a solid basis for […]

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