318 lines
16 KiB
Markdown
318 lines
16 KiB
Markdown
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---
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title: α-conversion
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TARGET DECK: Obsidian::STEM
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FILE TAGS: λ-calculus
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tags:
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- λ-calculus
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---
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## Overview
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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 $P$.
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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 $P \equiv_\alpha Q$.
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%%ANKI
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Basic
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If $P \equiv Q$, does $P \equiv_\alpha Q$?
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Back: Yes.
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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).
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END%%
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%%ANKI
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Basic
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If $P \equiv_\alpha Q$, does $P \equiv Q$?
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Back: Not necessarily.
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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).
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<!--ID: 1717687744141-->
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END%%
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%%ANKI
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Basic
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$\alpha$-conversion is most related to what kind of $\lambda$-term?
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Back: Abstractions.
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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).
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END%%
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%%ANKI
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Basic
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What property must $y$ satisfy for $\lambda x. M \equiv_\alpha \lambda y. M$?
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Back: $y \not\in FV(M)$
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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).
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<!--ID: 1717687744147-->
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END%%
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%%ANKI
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Basic
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What *kind* of conversion is a change of bound variable?
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Back: An $\alpha$-conversion.
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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).
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<!--ID: 1717687744150-->
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END%%
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%%ANKI
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Basic
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Given $\lambda$-terms $P$ and $Q$, what does it mean for $P$ to be congruent to $Q$?
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Back: $P \equiv_\alpha Q$
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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).
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END%%
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%%ANKI
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Basic
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Given $\lambda$-terms $P$ and $Q$, $P \equiv_\alpha Q$ if and only if what?
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Back: $P$ can be changed to $Q$ with a finite number of changes of bound variables.
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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).
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<!--ID: 1717687744157-->
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END%%
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%%ANKI
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Basic
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Is the following identity true? $$\lambda x y. x(xy) \equiv \lambda x. (\lambda y. x(xy))$$
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Back: Yes.
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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).
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<!--ID: 1717687744162-->
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END%%
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%%ANKI
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Basic
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Is the following identity true? $$\lambda x y. x(xy) \equiv_\alpha \lambda x. (\lambda y. x(xy))$$
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Back: Yes.
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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).
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<!--ID: 1717687744166-->
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END%%
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%%ANKI
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Basic
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Is the following identity true? $$\lambda x y. x(xy) \equiv \lambda u v. u(uv))$$
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Back: No.
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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).
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<!--ID: 1717687744169-->
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END%%
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%%ANKI
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Basic
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Is the following identity true? $$\lambda x y. x(xy) \equiv_\alpha \lambda u v. u(uv))$$
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Back: Yes.
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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).
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END%%
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%%ANKI
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Cloze
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$\alpha$-conversion is known as a change of {bound variables}.
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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).
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<!--ID: 1717687744176-->
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END%%
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%%ANKI
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Basic
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If $P \equiv_\alpha Q$, what can be said about the free variables of $P$ and $Q$?
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Back: $FV(P) = FV(Q)$
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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).
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END%%
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%%ANKI
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Basic
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What does it mean for $\equiv_\alpha$ to be reflexive?
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Back: $P \equiv_\alpha P$
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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).
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END%%
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%%ANKI
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Basic
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What does it mean for $\equiv_\alpha$ to be symmetric?
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Back: $P \equiv_\alpha Q \Rightarrow Q \equiv_\alpha P$
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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).
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<!--ID: 1717848027579-->
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END%%
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%%ANKI
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Basic
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What does it mean for $\equiv_\alpha$ to be transitive?
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Back: $P \equiv_\alpha Q \land Q \equiv_\alpha R \Rightarrow P \equiv_\alpha R$
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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).
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END%%
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%%ANKI
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Basic
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What three properties make $\equiv_\alpha$ an equivalence relation?
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Back: $\equiv_\alpha$ is reflexive, symmetric, and transitive.
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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).
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END%%
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Let $x$, $y$, and $v$ be distinct variables. Then
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* $v \not\in FV(M) \Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$
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* $v \not\in FV(M) \Rightarrow [x/v][v/x]M \equiv_\alpha M$
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* $y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$
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%%ANKI
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Basic
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$[N/x]M$ corresponds to which equivalence-transformation inference rule?
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Back: Substitution.
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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).
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END%%
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%%ANKI
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Basic
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$[P/v][v/x]M \equiv [P/x]M$ corresponds to which equivalence-transformation inference rule?
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Back: Transitivity.
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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).
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END%%
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%%ANKI
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Basic
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Rewrite $(E_u^x)_v^x$ using $\lambda$-calculus syntax.
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Back: $[v/x][u/x]E$
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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).
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END%%
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%%ANKI
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Basic
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Rewrite $[x/v][v/x]M$ using equivalence-transformation syntax.
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Back: $(M^x_v)^v_x$
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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).
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END%%
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%%ANKI
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Basic
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$[P/v][v/x]M \equiv_\alpha [P/x]M$ is necessary for what condition?
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Back: $v \not\in FV(M)$
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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).
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END%%
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%%ANKI
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Basic
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What happens if the antecedent is false in $v \not\in FV(M) \Rightarrow [P/v][v/x]M \equiv_\alpha [P/x]M$?
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Back: The LHS of the identity has more occurrences of $P$ than the right.
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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).
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END%%
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%%ANKI
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Basic
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If $v \in FV(M)$ and $x \not\in FV(M)$, does $[P/v][v/x]M \equiv_\alpha [P/x]M$?
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Back: No.
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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).
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END%%
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%%ANKI
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Basic
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If $v \not\in FV(M)$ and $x \in FV(M)$, does $[P/v][v/x]M \equiv_\alpha [P/x]M$?
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Back: Yes.
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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).
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END%%
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%%ANKI
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Basic
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If $v \not\in FV(M)$, what simpler term is $[P/v][v/x]M$ congruent to?
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Back: $[P/x]M$
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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).
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END%%
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%%ANKI
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Basic
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If $v \not\in FV(M)$ and $x \in FV(M)$, does $[x/v][v/x]M \equiv_\alpha M$?
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Back: Yes.
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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).
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END%%
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%%ANKI
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Basic
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If $v \not\in FV(M)$ and $x \in FV(M)$, does $[v/x][x/v]M \equiv_\alpha M$?
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Back: No.
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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).
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END%%
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%%ANKI
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Basic
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If $v \in FV(M)$ and $x \not\in FV(M)$, does $[v/x][x/v]M \equiv_\alpha M$?
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Back: Yes.
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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).
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END%%
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%%ANKI
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Basic
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If $v \in FV(M)$ and $x \not\in FV(M)$, does $[x/v][v/x]M \equiv_\alpha M$?
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Back: No.
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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).
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END%%
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%%ANKI
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Basic
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If $y \not\in FV(P)$, "commuting" substitution in $[P/x][Q/y]M$ yields what congruent term?
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Back: $[([P/x]Q)/y][P/x]M$
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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).
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END%%
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%%ANKI
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Basic
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$[P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$ is necessary for what condition?
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Back: $y \not\in FV(P)$
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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).
|
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|
<!--ID: 1717855810784-->
|
|||
|
END%%
|
|||
|
|
|||
|
%%ANKI
|
|||
|
Basic
|
|||
|
What happens if the antecedent is false in $y \not\in FV(P) \Rightarrow [P/x][Q/y]M \equiv_\alpha [([P/x]Q)/y][P/x]M$?
|
|||
|
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%%
|
|||
|
|
|||
|
## 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).
|