r
/
notebook
1
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity
notebook/notes/lambda-calculus/alpha-conversion.md

479 lines
25 KiB
Markdown
Raw Blame History

This file contains ambiguous Unicode characters!

This file contains ambiguous Unicode characters that may be confused with others in your current locale. If your use case is intentional and legitimate, you can safely ignore this warning. Use the Escape button to highlight these characters.

---
title: α-conversion
TARGET DECK: Obsidian::STEM
FILE TAGS: λ-calculus
tags:
- λ-calculus
---
## Overview
Let $\lambda$-term $P$ contain an occurrence of $\lambda x. M$, and let $y \not\in FV(M)$. The act of replacing this occurrence of $\lambda x. M$ with $\lambda y. [y/x]M$ is called a **change of bound variable** or an $\alpha$-conversion in $P$.
If $P$ can be changed to $\lambda$-term $Q$ by a finite series of changes of bound variables, we shall say **$P$ is congruent to $Q$**, or **$P$ $\alpha$-converts to $Q$**, or $P \equiv_\alpha Q$.
%%ANKI
Basic
If $P \equiv Q$, does $P \equiv_\alpha Q$?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744134-->
END%%
%%ANKI
Basic
What two ways can we pronounce $P \equiv_\alpha Q$?
Back: "$P$ is congruent to $Q$" and "$P$ $\alpha$-converts to $Q$".
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475477173-->
END%%
%%ANKI
Basic
If $P \equiv_\alpha Q$, does $P \equiv Q$?
Back: Not necessarily.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744141-->
END%%
%%ANKI
Basic
What does an $\alpha$-conversion refer to?
Back: The act of replacing an occurrence of $(\lambda x. M)$ with $\lambda y. [y/x]M$.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475424870-->
END%%
%%ANKI
Basic
What distinguishes terms "$\alpha$-conversion" and "$\alpha$-converts"?
Back: The latter refers to 0 or more applications of the former.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475424871-->
END%%
%%ANKI
Basic
$\alpha$-conversion is most related to what kind of $\lambda$-term?
Back: Abstractions.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744144-->
END%%
%%ANKI
Basic
What property must $y$ satisfy for $\lambda x. M \equiv_\alpha \lambda y. M$?
Back: $y \not\in FV(M)$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744147-->
END%%
%%ANKI
Cloze
"$\alpha$-{conversion}" refers to exactly one change of bound variable.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475424873-->
END%%
%%ANKI
Cloze
"$\alpha$-{converts}" refers to zero or more change of bound variables.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475424874-->
END%%
%%ANKI
Basic
What *kind* of conversion is a change of bound variable?
Back: An $\alpha$-conversion.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744150-->
END%%
%%ANKI
Basic
Given $\lambda$-terms $P$ and $Q$, what does it mean for $P$ to be congruent to $Q$?
Back: $P \equiv_\alpha Q$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744154-->
END%%
%%ANKI
Basic
Given $\lambda$-terms $P$ and $Q$, $P \equiv_\alpha Q$ if and only if what?
Back: $P$ can be changed to $Q$ with a finite number of changes of bound variables.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744157-->
END%%
%%ANKI
Basic
Is the following identity true? $$\lambda x y. x(xy) \equiv \lambda x. (\lambda y. x(xy))$$
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744162-->
END%%
%%ANKI
Basic
Is the following identity true? $$\lambda x y. x(xy) \equiv_\alpha \lambda x. (\lambda y. x(xy))$$
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744166-->
END%%
%%ANKI
Basic
Is the following identity true? $$\lambda x y. x(xy) \equiv \lambda u v. u(uv))$$
Back: No.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744169-->
END%%
%%ANKI
Basic
Is the following identity true? $$\lambda x y. x(xy) \equiv_\alpha \lambda u v. u(uv))$$
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744173-->
END%%
%%ANKI
Cloze
$\alpha$-conversion is known as a change of {bound variable}.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717687744176-->
END%%
%%ANKI
Basic
What greek-prefixed term is a change of bound variable known as?
Back: An $\alpha$-conversion.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718475424876-->
END%%
%%ANKI
Basic
If $P \equiv_\alpha Q$, what can be said about the free variables of $P$ and $Q$?
Back: $FV(P) = FV(Q)$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717848027572-->
END%%
%%ANKI
Basic
What does it mean for $\equiv_\alpha$ to be reflexive?
Back: $P \equiv_\alpha P$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717848027575-->
END%%
%%ANKI
Basic
What does it mean for $\equiv_\alpha$ to be symmetric?
Back: $P \equiv_\alpha Q \Rightarrow Q \equiv_\alpha P$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717848027579-->
END%%
%%ANKI
Basic
What does it mean for $\equiv_\alpha$ to be transitive?
Back: $P \equiv_\alpha Q \land Q \equiv_\alpha R \Rightarrow P \equiv_\alpha R$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717848027582-->
END%%
%%ANKI
Basic
What three properties make $\equiv_\alpha$ an equivalence relation?
Back: $\equiv_\alpha$ is reflexive, symmetric, and transitive.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717848027586-->
END%%
Let $x$, $y$, and $v$ be distinct variables. Then
* $v \not\in FV(M) \Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$
* $v \not\in FV(M) \Rightarrow [x/v][v/x]M \equiv_\alpha M$
* $y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
* $x \not\in FV(Q) \land y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$
* $[P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
%%ANKI
Basic
$[N/x]M$ corresponds to which equivalence-transformation inference rule?
Back: Substitution.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717449830572-->
END%%
%%ANKI
Basic
$[P/v][v/x]M \equiv [P/x]M$ corresponds to which equivalence-transformation inference rule?
Back: Transitivity.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717449830601-->
END%%
%%ANKI
Basic
Rewrite $(E_u^x)_v^x$ using $\lambda$-calculus syntax.
Back: $[v/x][u/x]E$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717449830608-->
END%%
%%ANKI
Basic
Rewrite $[x/v][v/x]M$ using equivalence-transformation syntax.
Back: $(M^x_v)^v_x$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717449830614-->
END%%
%%ANKI
Cloze
{$v \not\in FV(M)$} $\Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810777-->
END%%
%%ANKI
Basic
What happens if the antecedent is false in $v \not\in FV(M) \Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$?
Back: The LHS of the identity has more occurrences of $P$ than the right.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810781-->
END%%
%%ANKI
Basic
If $v \in FV(M)$ and $x \not\in FV(M)$, does $[P/v][v/x]M \equiv_\alpha [P/x]M$?
Back: No.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495760-->
END%%
%%ANKI
Basic
If $v \not\in FV(M)$ and $x \in FV(M)$, does $[P/v][v/x]M \equiv_\alpha [P/x]M$?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495763-->
END%%
%%ANKI
Basic
If $v \not\in FV(M)$, what simpler term is $[P/v][v/x]M$ congruent to?
Back: $[P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495766-->
END%%
%%ANKI
Basic
If $v \not\in FV(M)$ and $x \in FV(M)$, does $[x/v][v/x]M \equiv_\alpha M$?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495775-->
END%%
%%ANKI
Basic
If $v \not\in FV(M)$ and $x \in FV(M)$, does $[v/x][x/v]M \equiv_\alpha M$?
Back: No.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495778-->
END%%
%%ANKI
Basic
If $v \in FV(M)$ and $x \not\in FV(M)$, does $[v/x][x/v]M \equiv_\alpha M$?
Back: Yes.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495782-->
END%%
%%ANKI
Basic
If $v \in FV(M)$ and $x \not\in FV(M)$, does $[x/v][v/x]M \equiv_\alpha M$?
Back: No.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717850495785-->
END%%
%%ANKI
Basic
If $y \not\in FV(P)$, "commuting" substitution in $[P/x][Q/y]M$ yields what congruent term?
Back: $[([P/x]Q)/y][P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717853380814-->
END%%
%%ANKI
Cloze
{$y \not\in FV(P)$} $\Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810784-->
END%%
%%ANKI
Cloze
{$x \not\in FV(Q) \land y \not\in FV(P)$} $\Rightarrow [P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422235903-->
END%%
%%ANKI
Basic
$[P/x][Q/y]M \equiv_\alpha [Q/y][P/x]M$ is a specialization of what more general congruence?
Back: $[P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422235909-->
END%%
%%ANKI
Cloze
{$F$} $\Rightarrow [P/x][Q/x]M \equiv_\alpha [([P/x]Q)/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422235912-->
END%%
%%ANKI
Basic
What expression containing nested substitutions is congruent to $[P/x][Q/x]M$?
Back: $[([P/x]Q)/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422235916-->
END%%
%%ANKI
Basic
What expression containing adjacent substitutions is congruent to $[([P/x]Q)/x]M$?
Back: $[P/x][Q/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422279995-->
END%%
%%ANKI
Basic
What happens if the antecedent of the following lemma is false? $$y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$$
Back: $y$ is subbed in $M$ on the LHS but subbed in both $P$ and $M$ on the RHS.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810787-->
END%%
%%ANKI
Basic
Free occurrences of $x$ are substituted in which $\lambda$-terms of $[P/x][Q/y]M$?
Back: $Q$ and $M$.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810790-->
END%%
%%ANKI
Basic
Free occurrences of $y$ are substituted in which $\lambda$-terms of $[P/x][Q/y]M$?
Back: $M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810794-->
END%%
%%ANKI
Basic
Free occurrences of $x$ are substituted in which $\lambda$-terms of $[([P/x]Q)/y][P/x]M$?
Back: $Q$ and $M$.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810798-->
END%%
%%ANKI
Basic
Free occurrences of $y$ are substituted in which $\lambda$-terms of $[([P/x]Q)/y][P/x]M$?
Back: $P$ and $M$.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1717855810802-->
END%%
For $\lambda$-terms $M$, $M'$, $N$, and $N'$, and variable $x$, $$M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$$
%%ANKI
Basic
The proof of which implication shows "substitution is well-behaved w.r.t. $\alpha$-conversion"?
Back: $M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422973129-->
END%%
%%ANKI
Basic
What does Hindley et al. mean by "substitution is well-behaved w.r.t. $\alpha$-conversion"?
Back: $\alpha$-converting substitution inputs yields congruent outputs.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422973135-->
END%%
%%ANKI
Cloze
{$M \equiv_\alpha M' \land N \equiv_\alpha N'$} $\Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422973141-->
END%%
%%ANKI
Basic
What does Hindley et al. say the following implication says about substitution? $$M \equiv_\alpha M' \land N \equiv_\alpha N' \Rightarrow [N/x]M \equiv_\alpha [N'/x]M'$$
Back: It is well-defined with respect to $\alpha$-conversion.
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718422981125-->
END%%
## Simultaneous Substitution
Substitution can be generalized in the natural way to define simultaneous substitution $$[N_1/x_1, N_2/x_2, \ldots, N_n/x_n]M$$ for $n \geq 2$. As in [[equiv-trans#Substitution|equivalence-transformation]], simultaneous substitution is different from sequential substitution.
%%ANKI
Basic
How is simultaneous substitution of $N_1$ for $x_1$ and $N_2$ for $x_2$ in $M$ denoted?
Back: $[N_1/x_1, N_2/x]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718473252304-->
END%%
%%ANKI
Basic
How is $[N_1/x_1, N_2/x_2]M$ denoted in the equivalence-transformation system?
Back: $M_{N_1, N_2}^{x_1, x_2}$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718473252307-->
END%%
%%ANKI
Basic
How is $M_{N_1, N_2}^{x_1, x_2}$ denoted in $\lambda$-calculus?
Back: $[N_1/x_1, N_2/x_2]M$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718473252312-->
END%%
%%ANKI
Basic
Suppose $M \equiv x_1x_2$. What is the result of $[u/x_1]([x_1/x_2]M)$?
Back: $uu$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718473252309-->
END%%
%%ANKI
Basic
Suppose $M \equiv x_1x_2$. What is the result of $[u/x_1, x_1/x_2]M$?
Back: $ux_1$
Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
<!--ID: 1718473252311-->
END%%
## Bibliography
* Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combinators, an Introduction,” n.d. [https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf](https://www.cin.ufpe.br/~djo/files/Lambda-Calculus%20and%20Combinators.pdf).
Reference in New Issue View Git Blame Copy Permalink