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---
title: Equivalence Transformation
TARGET DECK: Obsidian::STEM
FILE TAGS: logic::equiv-trans
tags:
- logic
- equiv-trans
---
## Overview
**Equivalence-transformation** refers to a class of calculi for manipulating propositions derived from negation ($\neg$), conjunction ($\land$), disjunction ($\lor$), implication ($\Rightarrow$), and equality ($=$). Gries covers two in "The Science of Programming": a system of evaluation and a formal system. The system of evaluation mirrors how a computer processes instructions, at least in an abstract sense. The formal system serves as a theoretical framework for reasoning about propositions and their transformations without resorting to "lower-level" operations like substitution.
%%ANKI
Basic
Who is the author of "The Science of Programming"?
Back: David Gries
Reference: Gries, David.
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END%%
%%ANKI
Basic
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What are constant propositions?
Back: Propositions that contain only constants as operands.
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Reference: Gries, David.
2024-02-09 15:08:41 +00:00
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END%%
%%ANKI
Basic
What are the basic propositional logical operators?
Back: $\neg$, $\land$, $\lor$, $\Rightarrow$, and $\Leftrightarrow$/$=$
Reference: Gries, David.
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END%%
%%ANKI
Cloze
Gries replaces logical operator {$\Leftrightarrow$} in favor of {$=$}.
Reference: Gries, David.
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END%%
%%ANKI
Basic
How does Lean define propositional equality?
Back: Expressions `a` and `b` are propositionally equal iff `a = b` is true.
Reference: Avigad, Jeremy. ‘ ’
Tags: lean
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END%%
%%ANKI
Basic
How does Lean define `propext` ?
Back:
```lean
axiom propext {a b : Prop} : (a ↔ b) → (a = b)
```
Reference: Avigad, Jeremy. ‘ ’
Tags: lean
<!-- ID: 1706994861300 -->
END%%
%%ANKI
Basic
What Lean theorem justifies Gries' choice of $=$ over $\Leftrightarrow$?
Back: `propext`
Reference: Gries, David.
Tags: lean
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END%%
%%ANKI
Basic
What name is given to $\land$ operands?
Back: Conjuncts
Reference: Gries, David.
<!-- ID: 1706994861304 -->
END%%
%%ANKI
Basic
What name is given to $\lor$ operands?
Back: Disjuncts
Reference: Gries, David.
<!-- ID: 1706994861306 -->
END%%
%%ANKI
Basic
What name is given to operand $a$ in $a \Rightarrow b$?
Back: The antecedent
Reference: Gries, David.
<!-- ID: 1706994861308 -->
END%%
%%ANKI
Basic
What name is given to operand $b$ in $a \Rightarrow b$?
Back: The consequent
Reference: Gries, David.
<!-- ID: 1706994861310 -->
END%%
%%ANKI
Basic
Is $(b \land c)$ well-defined in $\{(b, T), (c, F)\}$?
Back: Yes
Reference: Gries, David.
<!-- ID: 1706994861318 -->
END%%
%%ANKI
Basic
Is $(b \lor d)$ well-defined in $\{(b, T), (c, F)\}$?
Back: No
Reference: Gries, David.
<!-- ID: 1706994861320 -->
END%%
%%ANKI
Basic
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What C logical operator corresponds to $\neg$?
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Back: `!`
Reference: Gries, David.
Tags: c
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END%%
%%ANKI
Basic
2024-02-12 21:42:11 +00:00
What C logical operator corresponds to $\land$?
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Back: There isn't one.
Reference: Gries, David.
Tags: c
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END%%
%%ANKI
Basic
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What C logical operator corresponds to $\lor$?
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Back: There isn't one.
Reference: Gries, David.
Tags: c
<!-- ID: 1706994861329 -->
END%%
%%ANKI
Basic
2024-02-12 21:42:11 +00:00
What C logical operator corresponds to $\Rightarrow$?
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Back: There isn't one.
Reference: Gries, David.
Tags: c
<!-- ID: 1706994861331 -->
END%%
%%ANKI
Basic
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What C logical operator corresponds to $\Leftrightarrow$?
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Back: `==`
Reference: Gries, David.
Tags: c
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END%%
%%ANKI
Basic
What proposition represents states $\{(b, T)\}$ and $\{(c, F)\}$?
Back: $b \lor \neg c$
Reference: Gries, David.
<!-- ID: 1706994861337 -->
END%%
%%ANKI
Basic
What set of states does $a \land b$ represent?
Back: The set containing just state $\{(a, T), (b, T)\}$.
Reference: Gries, David.
<!-- ID: 1706994861339 -->
END%%
%%ANKI
Basic
What is sloppy about phrase "the states in $b \lor \neg c$"?
Back: $b \lor \neg c$ is not a set but a representation of a set (of states).
Reference: Gries, David.
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END%%
%%ANKI
Basic
What is the weakest proposition?
Back: $T$
Reference: Gries, David.
<!-- ID: 1706994861348 -->
END%%
%%ANKI
Basic
What set of states does $T$ represent?
Back: The set of all states.
Reference: Gries, David.
<!-- ID: 1706994861350 -->
END%%
%%ANKI
Basic
What is the strongest proposition?
Back: $F$
Reference: Gries, David.
<!-- ID: 1706994861352 -->
END%%
%%ANKI
Basic
What set of states does $F$ represent?
Back: The set of no states.
Reference: Gries, David.
<!-- ID: 1706994861354 -->
END%%
%%ANKI
Basic
What does a proposition *represent* ?
Back: The set of states in which it is true.
Reference: Gries, David.
<!-- ID: 1706994861335 -->
END%%
%%ANKI
Basic
When is $p$ stronger than $q$?
Back: When $p \Rightarrow q$.
Reference: Gries, David.
<!-- ID: 1706994861343 -->
END%%
%%ANKI
Basic
When is $p$ weaker than $q$?
Back: When $q \Rightarrow p$.
Reference: Gries, David.
<!-- ID: 1706994861346 -->
END%%
%%ANKI
Basic
A proposition is well-defined with respect to what?
Back: A state to evaluate against.
Reference: Gries, David.
<!-- ID: 1706994861316 -->
END%%
%%ANKI
Basic
Why is $b \land c$ stronger than $b \lor c$?
Back: The former represents a subset of the states the latter represents.
Reference: Gries, David.
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END%%
%%ANKI
Basic
What is a state?
Back: A function mapping identifiers to values.
Reference: Gries, David.
<!-- ID: 1706994861314 -->
END%%
%%ANKI
Basic
What are the two calculi Gries describes equivalence-transformation with?
Back: A formal system and a system of evaluation.
Reference: Gries, David.
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END%%
## Equivalence Schemas
A proposition is said to be a **tautology** if it evaluates to $T$ in every state it is well-defined in. We say propositions $E1$ and $E2$ are **equivalent** if $E1 = E2$ is a tautology. In this case, we say $E1 = E2$ is an **equivalence** .
%%ANKI
Basic
What does it mean for a proposition to be a tautology?
Back: That the proposition is true in every state it is well-defined in.
Reference: Gries, David.
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END%%
%%ANKI
Basic
The term "equivalent" refers to a comparison between what two objects?
Back: Propositions.
Reference: Gries, David.
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END%%
%%ANKI
Basic
What does it mean for two propositions to be equivalent?
Back: Given propositions $E1$ and $E2$, it means $E1 = E2$ is a tautology.
Reference: Gries, David.
<!-- ID: 1707251673347 -->
END%%
%%ANKI
Basic
What is an equivalence?
Back: Given propositions $E1$ and $E2$, tautology $E1 = E2$.
Reference: Gries, David.
<!-- ID: 1707251673348 -->
END%%
* Commutative Laws
* $(E1 \land E2) = (E2 \land E1)$
* $(E1 \lor E2) = (E2 \lor E1)$
* $(E1 = E2) = (E2 = E1)$
%%ANKI
Basic
Which of the basic logical operators do the commutative laws apply to?
Back: $\land$, $\lor$, and $=$
Reference: Gries, David.
<!-- ID: 1707251673350 -->
END%%
%%ANKI
Basic
What do the commutative laws allow us to do?
Back: Reorder operands.
Reference: Gries, David.
<!-- ID: 1707251673351 -->
END%%
%%ANKI
Basic
What is the commutative law of e.g. $\land$?
Back: $E1 \land E2 = E2 \land E1$
<!-- ID: 1707251673353 -->
END%%
* Associative Laws
* $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
* $E1 \lor (E2 \lor E3) = (E1 \lor E2) \lor E3$
%%ANKI
Basic
Which of the basic logical operators do the associative laws apply to?
Back: $\land$ and $\lor$
Reference: Gries, David.
<!-- ID: 1707251673354 -->
END%%
%%ANKI
Basic
What do the associative laws allow us to do?
Back: Remove parentheses.
Reference: Gries, David.
<!-- ID: 1707251673355 -->
END%%
%%ANKI
Basic
What is the associative law of e.g. $\land$?
Back: $E1 \land (E2 \land E3) = (E1 \land E2) \land E3$
Reference: Gries, David.
<!-- ID: 1707251673357 -->
END%%
* Distributive Laws
* $E1 \lor (E2 \land E3) = (E1 \lor E2) \land (E1 \lor E3)$
* $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
%%ANKI
Basic
Which of the basic logical operators do the distributive laws apply to?
Back: $\land$ and $\lor$
Reference: Gries, David.
<!-- ID: 1707251673358 -->
END%%
%%ANKI
Basic
What do the distributive laws allow us to do?
Back: "Factor" propositions.
Reference: Reference: Gries, David.
<!-- ID: 1707251673360 -->
END%%
%%ANKI
Basic
What is the distributive law of e.g. $\land$ over $\lor$?
Back: $E1 \land (E2 \lor E3) = (E1 \land E2) \lor (E1 \land E3)$
Reference: Gries, David.
<!-- ID: 1707251673361 -->
END%%
* De Morgan's Laws
* $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
* $\neg (E1 \lor E2) = \neg E1 \land \neg E2$
%%ANKI
Basic
Which of the basic logical operators do De Morgan's Laws apply to?
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Back: $\neg$, $\land$, and $\lor$
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Reference: Gries, David.
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END%%
%%ANKI
Basic
What is De Morgan's Law of e.g. $\land$?
Back: $\neg (E1 \land E2) = \neg E1 \lor \neg E2$
Reference: Gries, David.
<!-- ID: 1707251673364 -->
END%%
* Law of Negation
* $\neg (\neg E1) = E1$
%%ANKI
Basic
What does the Law of Negation say?
Back: $\neg (\neg E1) = E1$
Reference: Gries, David.
<!-- ID: 1707251673365 -->
END%%
* Law of the Excluded Middle
* $E1 \lor \neg E1 = T$
%%ANKI
Basic
Which of the basic logical operators does the Law of the Excluded Middle apply to?
Back: $\lor$
Reference: Gries, David.
<!-- ID: 1707251673367 -->
END%%
%%ANKI
Basic
What does the Law of the Excluded Middle say?
Back: $E1 \lor \neg E1 = T$
Reference: Gries, David.
<!-- ID: 1707251673368 -->
END%%
%%ANKI
Basic
Which equivalence schema is "refuted" by sentence, "This sentence is false."
Back: Law of the Excluded Middle
Reference: Gries, David.
<!-- ID: 1707251779153 -->
END%%
* Law of Contradiction
* $E1 \land \neg E1 = F$
%%ANKI
Basic
Which of the basic logical operators does the Law of Contradiction apply to?
Back: $\land$
Reference: Gries, David.
<!-- ID: 1707251673370 -->
END%%
%%ANKI
Basic
What does the Law of Contradiction say?
Back: $E1 \land \neg E1 = F$
Reference: Gries, David.
<!-- ID: 1707251673371 -->
END%%
%%ANKI
Cloze
The Law of {1:the Excluded Middle} is to {2:$\lor$} whereas the Law of {2:Contradiction} is to {1:$\land$}.
Reference: Gries, David.
<!-- ID: 1707251673373 -->
END%%
Gries lists other "Laws" but they don't seem as important to note here.
%%ANKI
Basic
How is $\Rightarrow$ written in terms of other logical operators?
Back: $p \Rightarrow q$ is equivalent to $\neg p \lor q$.
Reference: Gries, David.
<!-- ID: 1706994861358 -->
END%%
%%ANKI
Basic
How is $\Leftrightarrow$/$=$ written in terms of other logical operators?
Back: $p \Leftrightarrow q$ is equivalent to $(p \Rightarrow q) \land (q \Rightarrow p)$.
Reference: Gries, David.
<!-- ID: 1706994861360 -->
END%%
2024-02-08 14:02:59 +00:00
%%ANKI
Basic
What distinguishes an equality from an equivalence?
Back: An equivalence is an equality that is also a tautology.
Reference: Gries, David.
<!-- ID: 1707316178709 -->
END%%
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## Equivalence Rules
* Rule of Substitution
* Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
* Rule of Transitivity
* Let $E1 = E2$ and $E2 = E3$ be equivalences. Then $E1 = E3$ is an equivalence.
%%ANKI
Basic
What two inference rules make up the equivalence-transformation formal system?
Back: Substitution and transitivity.
Reference: Gries, David.
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END%%
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%%ANKI
Basic
Which of the two inference rules that make up the equivalence-transformation formal system is redundant?
Back: Transitivity.
Reference: Gries, David.
<!-- ID: 1707432641598 -->
END%%
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%%ANKI
Basic
What does the rule of substitution say in the system of evaluation?
Back: Let $P(r)$ be a predicate and $E1 = E2$ be an equivalence. Then $P(E1) = P(E2)$ is an equivalence.
Reference: Gries, David.
<!-- ID: 1707253246452 -->
END%%
%%ANKI
Basic
How is the rule of substitution written as an inference rule (in standard form)?
Back:
$$
\begin{matrix}
E1 = E2 \\
\hline P(E1) = P(E2)
\end{matrix}
$$
<!-- ID: 1707253246454 -->
END%%
%%ANKI
Basic
What does the rule of transitivity state in the system of evaluation?
Back: Let $E1 = E2$ and $E2 = E3$. Then $E1 = E3$.
Reference: Gries, David.
<!-- ID: 1707253246455 -->
END%%
%%ANKI
Basic
How is the rule of transitivity written as an inference rule (in standard form)?
Back:
$$
\begin{matrix}
E1 = E2, E2 = E3 \\
\hline E1 = E3
\end{matrix}
$$
<!-- ID: 1707253246457 -->
END%%
%%ANKI
Cloze
The system of evaluation has {equivalences} whereas the formal system has {theorems}.
Reference: Gries, David.
<!-- ID: 1707253246458 -->
END%%
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%%ANKI
Basic
What is a "theorem" in the equivalence-transformation formal system?
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Back: An equivalence derived from the axioms and inference rules.
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Reference: Gries, David.
<!-- ID: 1707316178712 -->
END%%
%%ANKI
Basic
How is e.g. the Law of Implication proven in the system of evaluation?
Back: With truth tables
Reference: Gries, David.
<!-- ID: 1707316178714 -->
END%%
%%ANKI
Basic
How is e.g. the Law of Implication proven in the formal system?
Back: It isn't. It is an axiom.
Reference: Gries, David.
<!-- ID: 1707316178715 -->
END%%
%%ANKI
Cloze
The system of evaluation and formal system are connected by the following biconditional: {$e$ is a tautology} iff {$e = T$ is a theorem}.
Reference: Gries, David.
<!-- ID: 1707316178717 -->
END%%
%%ANKI
Cloze
The {1:system of evaluation} is to {2:"$e$ is a tautology"} whereas the {2:formal system} is to {1:"$e = T$ is a theorem"}.
Reference: Gries, David.
<!-- ID: 1707316276203 -->
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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%%ANKI
Basic
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Textual substitution is derived from what equivalence rule?
Back: The substitution rule.
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Reference: Gries, David.
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END%%
%%ANKI
Basic
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What is $E$'s role in textual substitution $E_e^x$?
Back: It is the expression that free occurrences of $x$ are replaced with $e$ in.
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Reference: Gries, David.
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END%%
%%ANKI
Basic
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What is $e$'s role in textual substitution $E_e^x$?
Back: It is the expression that free occurrences of $x$ in $E$ are substituted with.
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Reference: Gries, David.
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<!-- ID: 1707762304127 -->
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END%%
%%ANKI
Basic
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What is $x$'s role in textual substitution $E_e^x$?
Back: It is the identifier matching free occurrences in $E$ that are replaced with $e$.
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Reference: Gries, David.
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<!-- ID: 1707762304129 -->
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END%%
%%ANKI
Basic
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How is textual substitution $E_e^x$ interpreted as a function?
Back: As $E(e)$, where $E$ is a function of $x$.
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Reference: Gries, David.
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END%%
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%%ANKI
Basic
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Why does Gries prefer notation $E_e^x$ over e.g. $E(e)$?
Back: The former indicates the identifier to replace.
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Reference: Gries, David.
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<!-- ID: 1707762304132 -->
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END%%
%%ANKI
Basic
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What two scenarios ensure $E_e^x = E$ is an equivalence?
Back: $x = e$ or no free occurrences of $x$ exist in $E$.
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Reference: Gries, David.
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<!-- ID: 1707762304133 -->
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END%%
%%ANKI
Basic
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Why might $E_e^x = E$ be an equivalence despite identifier $x$ existing in $E$?
Back: If the only occurrences of $x$ in $E$ are bound.
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Reference: Gries, David.
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<!-- ID: 1707762304135 -->
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END%%
%%ANKI
Basic
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What is required for $E_e^x$ to be valid?
Back: Substitution must result in a syntactically valid expression.
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Reference: Gries, David.
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<!-- ID: 1707762304137 -->
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END%%
%%ANKI
Basic
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What is the result of the following? $$(x < y \land ( \forall i : 0 \leq i < n : b [ i ] < y )) _z ^ x $$
Back: $$(z < y \land ( \forall i : 0 \leq i < n : b [ i ] < y ))$$
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Reference: Gries, David.
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<!-- ID: 1707762304139 -->
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END%%
%%ANKI
Basic
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What is the result of the following? $$(x < y \land ( \forall i : 0 \leq i < n : b [ i ] < y )) _z ^ y $$
Back: $$(x < z \land ( \forall i : 0 \leq i < n : b [ i ] < z ))$$
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Reference: Gries, David.
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<!-- ID: 1707762304140 -->
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END%%
%%ANKI
Basic
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What is the result of the following? $$(x < y \land ( \forall i : 0 \leq i < n : b [ i ] < y )) _z ^ i $$
Back: $$(x < y \land ( \forall i : 0 \leq i < n : b [ i ] < y ))$$
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Reference: Gries, David.
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<!-- ID: 1707762304141 -->
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END%%
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* $(E_u^x)_v^x = E_{u_v^x}^x$
* The only possible free occurrences of $x$ that may appear after the first of the sequential substitutions occur in $u$.
* If $y$ is not free in $E$, then $(E_u^x)_v^y = E_{u_v^y}^x$.
* $y$ 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
Basic
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How do we simplify $(E_u^x)_v^x$?
Back: As $E_{u_v^x}^x$
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Reference: Gries, David.
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<!-- ID: 1707762304143 -->
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END%%
%%ANKI
Basic
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How is $E_{u_v^x}^x$ rewritten as sequential substitution?
Back: As $(E_u^x)_v^x$
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Reference: Gries, David.
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<!-- ID: 1707762304145 -->
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END%%
%%ANKI
Basic
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Why is $(E_u^x)_v^x = E_{u_v^x}^x$ an equivalence?
Back: After the first substitution, the only possible free occurrences of $x$ are in $u$.
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Reference: Gries, David.
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<!-- ID: 1707762304146 -->
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END%%
%%ANKI
Basic
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In what two scenarios is $(E_u^x)_v^y = E_{u_v^y}^x$ always an equivalence?
Back: $x = y$ or $y$ is not free in $E$.
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Reference: Gries, David.
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%%ANKI
Basic
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If $x \neq y$, when is $(E_u^x)_v^y = E_{u_v^y}^x$?
Back: When $y$ is not free in $E$.
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Reference: Gries, David.
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%%ANKI
Basic
2024-02-12 18:27:16 +00:00
Why should $y$ not be free in $E$ for $(E_u^x)_v^y = E_{u_v^y}^x$ to be an equivalence?
Back: If it were, a $v$ would exist in the LHS that doesn't in the RHS.
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Reference: Gries, David.
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## References
* Avigad, Jeremy. ‘ ’
* Gries, David.
