Notes on textual substitution.
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@ -289,6 +289,13 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
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%%ANKI
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Basic
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How is tautology $e$ written equivalently with a quantifier?
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Back: For free identifiers $i_1, \ldots, i_n$ in $e$, as $\forall (i_1, \ldots, i_n), e$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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The term "equivalent" refers to a comparison between what two objects?
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@ -619,7 +626,8 @@ END%%
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## Textual Substitution
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**Textual substitution** refers to the simultaneous replacement of a free identifier with an expression, introducing parentheses as necessary. This concept is just the [[#Equivalence Rules|Substitution Rule]] with different notation. For example, let $E$ and $e$ be expressions and $x$ an identifer. Then $$E_e^x$$ denotes the simultaneous replacement of all free occurrences of $x$ with $e$.
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**Textual substitution** refers to the simultaneous replacement of a free identifier with an expression, introducing parentheses as necessary. This concept is just the [[#Equivalence Rules|Substitution Rule]] with different notation. Let $\bar{x}$ denote a list of distinct identifiers. If $\bar{e}$ is a list of expressions of the same length as $\bar{x}$, then simultaneous substitution of $\bar{x}$ by $\bar{e}$ in expression $E$ is denoted as $$E_{\bar{e}}^{\bar{x}}$$
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Note that simultaneous substitution is different than sequential substitution.
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%%ANKI
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Basic
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@ -631,24 +639,45 @@ END%%
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%%ANKI
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Basic
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What is $E$'s role in textual substitution $E_e^x$?
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Back: It is the expression that free occurrences of $x$ are replaced with $e$ in.
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What identifier is guaranteed to not occur in $E_e^x$?
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Back: None.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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What identifier is guaranteed to not occur in $E_{s(e)}^x$?
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Back: $x$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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*Why* does $x$ not occur in $E_{s(e)}^x$?
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Back: Because $s(e)$ evaluates to a constant proposition.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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What is the role of $E$ in textual substitution $E_e^x$?
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Back: It is the expression in which free occurrences of $x$ are replaced.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304126-->
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END%%
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%%ANKI
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Basic
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What is $e$'s role in textual substitution $E_e^x$?
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Back: It is the expression that free occurrences of $x$ in $E$ are substituted with.
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What is the role of $e$ role in textual substitution $E_e^x$?
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Back: It is the expression that is evaluated and substituted into $E$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304127-->
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END%%
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%%ANKI
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Basic
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What is $x$'s role in textual substitution $E_e^x$?
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Back: It is the identifier matching free occurrences in $E$ that are replaced with $e$.
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What is the role of $x$ in textual substitution $E_e^x$?
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Back: It is the identifier matching free occurrences in $E$ that are replaced.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304129-->
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END%%
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@ -717,10 +746,41 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
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%%ANKI
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Basic
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In textual substitution, what does e.g. $\bar{x}$ denote?
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Back: A list of *distinct* identifiers.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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What is the role of $E$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
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Back: It is the expression in which free occurrences of $\bar{x}$ are replaced.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304126-->
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END%%
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%%ANKI
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Basic
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What is the role of $\bar{e}$ role in textual substitution $E_{\bar{e}}^{\bar{x}}$?
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Back: It is the expressions that are evaluated and substituted into $E$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304127-->
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END%%
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%%ANKI
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Basic
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What is the role of $\bar{x}$ in textual substitution $E_{\bar{e}}^{\bar{x}}$?
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Back: It is the distinct identifiers matching free occurrences in $E$ that are replaced.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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<!--ID: 1707762304129-->
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END%%
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### Theorems
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* $(E_u^x)_v^x = E_{u_v^x}^x$
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* The only possible free occurrences of $x$ that may appear after the first of the sequential substitutions occur in $u$.
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* If $y$ is not free in $E$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
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* $y$ may not be free in $E$ but substituting $x$ with $u$ can introduce a free occurrence. It doesn't matter if we perform the substitution first or second though.
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%%ANKI
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Basic
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@ -746,6 +806,9 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
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* If $y$ is not free in $E$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
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* $y$ may not be free in $E$ but substituting $x$ with $u$ can introduce a free occurrence. It doesn't matter if we perform the substitution first or second though.
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%%ANKI
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Basic
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In what two scenarios is $(E_u^x)_v^y = E_{u_v^y}^x$ always an equivalence?
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@ -770,6 +833,76 @@ Reference: Gries, David. *The Science of Programming*. Texts and Monographs in
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END%%
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%%ANKI
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Basic
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How does Gries denote state $s$ with identifer $x$ set to value $v$?
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Back: $(s; x{:}v)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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How is $(s; x{:}v)$ written instead using set-theoretical notation?
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Back: $(s - \{(x, s(x))\}) \cup \{(x, v)\}$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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Execution of `x := e` in state $s$ terminates in what new state?
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Back: $(s; x{:}s(e))$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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Given state $s$, what statement does $(s; x{:}s(e))$ derive from?
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Back: `x := e`
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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What missing value guarantees state $s$ satisfies equivalence $s = (s; x{:}\_)$?
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Back: $s(x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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Given state $s$, why is it that $s = (s; x{:}s(x))$?
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Back: Evaluating $x$ in state $s$ yields $s(x)$.
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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* $s(E_e^x) = s(E_{s(e)}^x)$
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* Substituting $x$ with $e$ and then evaluating is the same as substituting $x$ with the evaluation of $e$.
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* This follows directly from the recursive definition of state evaluation.
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%%ANKI
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Basic
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How can we simplify $s(E_{s(e)}^x)$?
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Back: As $s(E_e^x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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%%ANKI
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Basic
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What equivalence relates $E_e^x$ with $E_{s(e)}^x$?
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Back: $s(E_e^x) = s(E_{s(e)}^x)$
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Reference: Gries, David. *The Science of Programming*. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.
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END%%
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TODO
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* Let $s$ be a state and $s' = (s; x{:}s(e))$. Then $s'(E) = s(E_e^x)$.
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TODO
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* Given identifiers $\bar{x}$ and fresh identifiers $\bar{u}$, $(E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E$.
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TODO
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## References
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* Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
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