Image operations and b-nf class characterization.
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},
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"fields_dict": {
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"Basic": [
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---
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title: "2024-07-10"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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* Notes on an alternative classification for $\beta\text{-nf}$.
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* Small notes on the difference of images.
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---
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title: "2024-07-09"
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---
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- [x] Anki Flashcards
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- [x] KoL
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- [x] OGS
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- [ ] Sheet Music (10 min.)
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- [ ] Korean (Read 1 Story)
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@ -210,7 +210,7 @@ END%%
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## Normal Form
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A term $Q$ which contains no $\beta$-redexes is called a **$\beta$-normal form** (or a **term in $\beta$-normal form** or just a **$\beta$-nf**). The class of all $\beta$-normal forms is called $\beta$-nf or $\lambda\beta$-nf. If a term $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf, then $Q$ is called a **$\beta$-normal form of $P$**.
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A term $Q$ which contains no $\beta$-redexes is called a **$\beta$-normal form** (or a **term in $\beta$-normal form** or just a **$\beta\text{-nf}$**). The class of all $\beta$-normal forms is called $\beta\text{-nf}$ or $\lambda\beta\text{-nf}$. If a term $P$ $\beta$-reduces to a term $Q$ in $\beta\text{-nf}$, then $Q$ is called a **$\beta$-normal form of $P$**.
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%%ANKI
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Basic
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@ -246,14 +246,14 @@ END%%
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%%ANKI
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Cloze
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The class of {all $\beta$-normal forms} is called {$\beta$-nf/$\lambda\beta$-nf}.
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The class of {all $\beta$-normal forms} is called {$\beta\text{-nf}$/$\lambda\beta\text{-nf}$}.
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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: 1719065185812-->
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END%%
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%%ANKI
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Basic
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What ambiguity does term "$\beta$-nf" introduce?
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What ambiguity does term "$\beta\text{-nf}$" introduce?
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Back: It refers to a specific $\beta$-normal form or the class of $\beta$-normal forms.
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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: 1719065185815-->
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@ -262,23 +262,23 @@ END%%
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%%ANKI
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Basic
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What does it mean for term $Q$ to be a $\beta$-normal form of term $P$?
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Back: $P$ $\beta$-reduces to a term $Q$ in $\beta$-nf.
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Back: $P$ $\beta$-reduces to a term $Q$ in $\beta\text{-nf}$.
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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: 1719065185819-->
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END%%
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%%ANKI
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Basic
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How is the class $\beta$-nf alternatively denoted?
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Back: As $\lambda\beta$-nf.
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How is the class $\beta\text{-nf}$ alternatively denoted?
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Back: As $\lambda\beta\text{-nf}$.
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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: 1719065185823-->
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END%%
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%%ANKI
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Basic
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How is the class $\lambda\beta$-nf alternatively denoted?
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Back: As $\beta$-nf.
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How is the class $\lambda\beta\text{-nf}$ alternatively denoted?
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Back: As $\beta\text{-nf}$.
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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: 1719065185799-->
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END%%
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@ -325,20 +325,36 @@ END%%
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%%ANKI
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Basic
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Why isn't $x(\lambda u. uv)$ in $\beta$-normal form?
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Back: N/A. It is.
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Is $a(\lambda u. uv)x$ in $\beta$-normal form?
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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: 1720645031207-->
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END%%
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%%ANKI
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Basic
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*Why* is $x(\lambda u. uv)$ in $\beta$-normal form?
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Back: It contains no $\beta$-redex.
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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: 1719103644324-->
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END%%
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%%ANKI
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Basic
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Why isn't $(\lambda u. uv)x$ in $\beta$-normal form?
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*Why* isn't $(\lambda u. uv)x$ in $\beta$-normal form?
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Back: Because $(\lambda u. uv)x$ is a $\beta$-redex.
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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: 1719103644325-->
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END%%
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%%ANKI
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Basic
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*Why* is $a(\lambda u. uv)x$ in $\beta$-normal form?
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Back: With parentheses, $(a(\lambda u. uv))x$ clearly contains no $\beta$-redex.
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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: 1720645031212-->
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END%%
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%%ANKI
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Basic
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Let $P \,\triangleright_\beta\, Q$. How do $FV(P)$ and $FV(Q)$ relate to one another?
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@ -363,11 +379,127 @@ Reference: Hindley, J Roger, and Jonathan P Seldin. “Lambda-Calculus and Combi
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<!--ID: 1719406791436-->
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END%%
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As an alternative characterization, the class $\beta\text{-nf}$ is the smallest class such that
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* all atoms are in $\beta\text{-nf}$;
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* $M, N \in \beta\text{-nf} \Rightarrow aMN \in \beta\text{-nf}$ for all atoms $a$;
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* $M \in \beta\text{-nf} \Rightarrow \lambda x. M \in \beta\text{-nf}$
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%%ANKI
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Basic
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What proposition explains how atoms relate to the definition of $\beta\text{-nf}$?
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Back: All atoms are in $\beta\text{-nf}$.
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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: 1720645978919-->
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END%%
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%%ANKI
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Basic
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What proposition explains how applications relate to the definition of $\beta\text{-nf}$?
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Back: For all atoms $a$, if $M, N \in \beta\text{-nf}$, then $aMN \in \beta\text{-nf}$.
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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: 1720645978924-->
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END%%
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%%ANKI
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Basic
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Given atom $a$ and $M \in \beta\text{-nf}$, what application is in $\beta\text{-nf}$?
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Back: $aM$
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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: 1720646122613-->
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END%%
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%%ANKI
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Basic
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$M, N \in \beta\text{-nf}$ implies what application is in $\beta\text{-nf}$?
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Back: $aMN$ for any atom $a$.
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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: 1720645978929-->
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END%%
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%%ANKI
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Basic
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Given $M, N \in \beta\text{-nf}$, when is $MN \in \beta\text{-nf}$?
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Back: When $M$ is not an abstraction.
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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: 1720645978933-->
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END%%
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%%ANKI
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Basic
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Given $M, N \in \beta\text{-nf}$, when is $MN \not\in \beta\text{-nf}$?
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Back: When $M$ is an abstraction.
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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: 1720646122620-->
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END%%
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%%ANKI
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Basic
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What proposition explains how abstractions relate to the definition of $\beta\text{-nf}$?
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Back: If $M \in \beta\text{-nf}$, then $\lambda x. M \in \beta\text{-nf}$.
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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: 1720645978936-->
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END%%
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%%ANKI
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Basic
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$M \in \beta\text{-nf}$ implies what abstraction is in $\beta\text{-nf}$?
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Back: $\lambda x. M$
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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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<!--ID: 1720645978940-->
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END%%
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%%ANKI
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Basic
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Given atom $a$, if $M \equiv aM_1\ldots M_n$ and $M \,\triangleright_\beta\, N$, what form does $N$ have?
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Back: $aN_1\ldots N_n$ where $M_i \,\triangleright_\beta\, N_i$ for $i = 1, \ldots, n$.
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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: 1720649942775-->
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END%%
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%%ANKI
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Basic
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Given atom $a$, if $M \equiv aM_1\ldots M_n$ and $M \,\triangleright_\beta\, N$, *why* does $N$ have form $aN_1\ldots N_n$?
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Back: Since $M \equiv ((\cdots((aM_1)M_2)\cdots)M_n)$, every $\beta$-redex must be in an $M_i$.
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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: 1720649942780-->
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END%%
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%%ANKI
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Basic
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What does it mean for a $\lambda$-term to *be* a $\beta\text{-nf}$?
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Back: The $\lambda$-term contains no $\beta$-redex.
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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: 1720649942783-->
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END%%
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%%ANKI
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Basic
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What does it mean for a $\lambda$-term to *have* a $\beta\text{-nf}$?
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Back: The $\lambda$-term can be $\beta$-reduced into a term in $\beta\text{-nf}$.
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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: 1720649942787-->
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END%%
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%%ANKI
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Basic
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Suppose $[N/x]M$ is a $\beta\text{-nf}$. Is $M$ a $\beta\text{-nf}$?
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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: 1720665224642-->
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END%%
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%%ANKI
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Basic
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Suppose $[N/x]M$ has a $\beta\text{-nf}$. Does $M$ have a $\beta\text{-nf}$?
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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: 1720665224645-->
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END%%
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## Church-Rosser Theorem
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|
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If $P \,\triangleright_\beta\, M$ and $P \,\triangleright_\beta\, N$, then there exists a term $T$ such that $M \,\triangleright_\beta\, T$ and $N \,\triangleright_\beta\, T$.
|
||||
|
||||
As an immediate corollary, if $P$ has a $\beta$-normal form then it it is unique modulo $\equiv_\alpha$.
|
||||
If $P \,\triangleright_\beta\, M$ and $P \,\triangleright_\beta\, N$, then there exists a term $T$ such that $M \,\triangleright_\beta\, T$ and $N \,\triangleright_\beta\, T$. As an immediate corollary, if $P$ has a $\beta$-normal form then it it is unique modulo $\equiv_\alpha$.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
|
|
|
|||
|
|
@ -232,7 +232,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
Which identifiers in the following are bound? $$\exists x, P(x) \land P(y)$$
|
||||
Which identifiers in the following are bound? $\exists x, P(x) \land P(y)$
|
||||
Back: Just $x$.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1707674796777-->
|
||||
|
|
@ -240,7 +240,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
Which identifiers in the following are free? $$\exists x, P(x) \land P(y)$$
|
||||
Which identifiers in the following are free? $\exists x, P(x) \land P(y)$
|
||||
Back: Just $y$.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1707674796779-->
|
||||
|
|
@ -248,12 +248,20 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is the following rewritten in PNF? $$(\exists x, P(x)) \land (\exists y, P(y))$$
|
||||
Back: $\exists x \;y, P(x) \land P(y)$
|
||||
How is the following rewritten in PNF? $(\exists x, P(x)) \land (\exists y, Q(y))$
|
||||
Back: $\exists x \;y, P(x) \land Q(y)$
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1707675399517-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How is the following rewritten in PNF? $(\exists x, P(x)) \land (\forall y, Q(y))$
|
||||
Back: N/A.
|
||||
Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
<!--ID: 1720665224639-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
|
||||
|
|
|
|||
|
|
@ -1271,12 +1271,15 @@ The following holds for any sets $F$, $A$, $B$, and $\mathscr{A}$:
|
|||
* The image of unions is the union of the images:
|
||||
* $F[\![\bigcup\mathscr{A}]\!] = \bigcup\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
* The image of intersections is a subset of the intersection of images:
|
||||
* $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$
|
||||
* $F[\![\bigcap \mathscr{A}]\!] \subseteq \bigcap\,\{F[\![A]\!] \mid A \in \mathscr{A}\}$ for $\mathscr{A} \neq \varnothing$
|
||||
* Equality holds if $F$ is single-rooted.
|
||||
* The image of a difference includes the difference of the images:
|
||||
* $F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$
|
||||
* Equality holds if $F$ is single-rooted.
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the image of unions relate to the union of images?
|
||||
How does the image of unions relate to the union of the images?
|
||||
Back: They are equal.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720382880557-->
|
||||
|
|
@ -1284,7 +1287,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the union of images relate to the images of unions?
|
||||
How does the union of images relate to the images of the unions?
|
||||
Back: They are equal.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023254-->
|
||||
|
|
@ -1324,7 +1327,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the image of intersections relate to the intersection of images?
|
||||
How does the image of intersections relate to the intersection of the images?
|
||||
Back: The former is a subset of the latter.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023257-->
|
||||
|
|
@ -1332,7 +1335,7 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the intersection of images relate to the image of intersections?
|
||||
How does the intersection of images relate to the image of the intersections?
|
||||
Back: The latter is a subset of the former.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023261-->
|
||||
|
|
@ -1380,8 +1383,8 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
*Why* is the following identity intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$
|
||||
Back: $A \cap B$ could be empty but $F[\![A]\!] \cap F[\![B]\!]$ could be nonempty.
|
||||
What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A \cap B]\!] \subseteq F[\![A]\!] \cap F[\![B]\!]$$
|
||||
Back: $A$ and $B$ might be disjoint even if $F[\![A]\!]$ and $F[\![B]\!]$ are not.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720386023280-->
|
||||
END%%
|
||||
|
|
@ -1402,6 +1405,54 @@ Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Pre
|
|||
<!--ID: 1720386023288-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the image of differences relate to the difference of the images?
|
||||
Back: The latter is a subset of the former.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665224629-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
How does the difference of images relate to the image of the differences?
|
||||
Back: The former is a subset of the latter.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665351075-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What $\varnothing$-based example is used to show the following is intuitively true? $$F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$$
|
||||
Back: $F[\![A]\!]$ and $F[\![B]\!]$ might be the same sets even if $A \neq B$.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665224636-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A - B]\!] \subseteq F[\![A]\!] - F[\![B]\!]$
|
||||
Back: $F$ is single-rooted.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665351101-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A - B]\!] = F[\![A]\!] - F[\![B]\!]$
|
||||
Back: $F$ is single-rooted.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665351105-->
|
||||
END%%
|
||||
|
||||
%%ANKI
|
||||
Basic
|
||||
What condition on set $F$ makes the following true? $F[\![A]\!] - F[\![B]\!] \subseteq F[\![A - B]\!]$
|
||||
Back: N/A. This is always true.
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720665351109-->
|
||||
END%%
|
||||
|
||||
## Bibliography
|
||||
|
||||
* “Bijection, Injection and Surjection,” in _Wikipedia_, May 2, 2024, [https://en.wikipedia.org/w/index.php?title=Bijection_injection_and_surjection](https://en.wikipedia.org/w/index.php?title=Bijection,_injection_and_surjection&oldid=1221800163).
|
||||
|
|
|
|||
|
|
@ -251,8 +251,8 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How many members are in set $\{P(y) \mid y \in B\}$?
|
||||
Back: As many as the number of unique $P(y)$ for each $y \in B$.
|
||||
Given function $P$, how is set $\{P(y) \mid y \in B\}$ more compactly denoted?
|
||||
Back: $P[\![B]\!]$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720369624737-->
|
||||
END%%
|
||||
|
|
@ -307,8 +307,8 @@ END%%
|
|||
|
||||
%%ANKI
|
||||
Basic
|
||||
How do you rewrite $\{A \mid A \in B\}$ with an existential in the entrance requirement?
|
||||
Back: $\{v \mid A \in B \land v = A\}$
|
||||
How is $\{A \mid A \in B\}$ rewritten with an existential in the entrance requirement?
|
||||
Back: $\{v \mid \exists A \in B \land v = A\}$
|
||||
Reference: Herbert B. Enderton, *Elements of Set Theory* (New York: Academic Press, 1977).
|
||||
<!--ID: 1720381621849-->
|
||||
END%%
|
||||
|
|
|
|||
Loading…
Reference in New Issue