notebook/notes/formal-system/proof-system/equiv-trans.md

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Equivalence Transformation Obsidian::STEM formal-system::equiv-trans
equiv-trans
logic
programming

Overview

Equivalence-transformation is a proof system used alongside classical truth-functional pred-logic. It is the foundation upon which pred-trans are based.

%%ANKI Basic Who is the author of "The Science of Programming"? Back: David Gries Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze Gries replaces logical operator {\Leftrightarrow} in favor of {=}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What Lean theorem justifies Gries' choice of = over \Leftrightarrow? Back: propext Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981. Tags: lean

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

A prop-logic 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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is tautology e written equivalently with a quantifier? Back: For free identifiers i_1, \ldots, i_n in e, as \forall (i_1, \ldots, i_n), e. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic The term "equivalent" refers to a comparison between what two objects? Back: Expressions. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is an equivalence? Back: Given propositions E1 and E2, tautology E1 = E2. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What distinguishes an equality from an equivalence? Back: An equivalence is an equality that is also a tautology. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Axioms

Commutativity

For propositions E1 and E2:

  • (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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the commutative laws allow us to do? Back: Reorder operands. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the commutative law of e.g. \land? Back: E1 \land E2 = E2 \land E1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Associativity

For propositions E1, E2, and E3:

  • 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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the associative laws allow us to do? Back: Remove parentheses. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Distributivity

For propositions E1, E2, and E3:

  • 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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What do the distributive laws allow us to do? Back: "Factor" propositions. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

De Morgan's

For propositions E1 and E2:

  • \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 involve? Back: \neg, \land, and \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is De Morgan's law (distributing \land) expressed as an equivalence? Back: \neg (E1 \land E2) = \neg E1 \lor \neg E2 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Law of Negation

For any proposition E1, it follows that \neg (\neg E1) = E1.

%%ANKI Basic How is the law of negation expressed as an equivalence? Back: \neg (\neg E1) = E1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Law of Excluded Middle

For any proposition E1, it follows that E1 \lor \neg E1 = T.

%%ANKI Basic Which of the basic logical operators does the law of excluded middle involve? Back: \lor Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is the law of excluded middle expressed as an equivalence? Back: E1 \lor \neg E1 = T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic "This sentence is false" questions which classical principle? Back: The law of excluded middle. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Law of Contradiction

For any proposition E1, it follows that E1 \land \neg E1 = F.

%%ANKI Basic Which of the basic logical operators does the law of contradiction involve? Back: \land Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is the law of contradiction expressed as an equivalence? Back: E1 \land \neg E1 = F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze The law of {1:excluded middle} is to {2:\lor} whereas the law of {2:contradiction} is to {1:\land}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does the principle of explosion state? Back: That any statement can be proven from a contradiction. Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic How is the principle of explosion stated in first-order logic? Back: \forall P, F \Rightarrow P Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic What does the law of contradiction say? Back: For any proposition P, it holds that \neg (P \land \neg P). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does the principle of explosion relate to the law of contradiction? Back: If a contradiction could be proven, then anything can be proven. Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Basic Suppose P and \neg P. Show schematically how to use the principle of explosion to prove Q. Back: \begin{align*} P \ \neg P \ P \lor Q \ \hline Q \end{align*}$$Reference: “Principle of Explosion,” in Wikipedia, July 3, 2024, https://en.wikipedia.org/w/index.php?title=Principle_of_explosion.

END%%

%%ANKI Cloze The law of {contradiction} and law of {excluded middle} create a dichotomy in "logical space". Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which property of partitions is analagous to the law of contradiction on "logical space"? Back: Disjointedness. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which property of partitions is analagous to the law of excluded middle on "logical space"? Back: Exhaustiveness. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Cloze The law of {1:contradiction} is to "{2:mutually exclusive}" whereas the law of {2:excluded middle} is "{1:jointly exhaustive}". Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

%%ANKI Basic Which logical law proves equivalence of the law of contradiction and excluded middle? Back: De Morgan's law. Reference: “Law of Noncontradiction,” in Wikipedia, June 14, 2024, https://en.wikipedia.org/w/index.php?title=Law_of_noncontradiction.

END%%

Law of Implication

For any propositions E1 and E2, it follows that E1 \Rightarrow E2 = \neg E1 \lor E2.

Law of Equality

For any propositions E1 and E2, it follows that (E1 = E2) = (E1 \Rightarrow E2) \land (E2 \Rightarrow E1).

Law of Or-Simplification

For any propositions E1 and E2, it follows that:

  • E1 \lor E1 = E1
  • E1 \lor T = T
  • E1 \lor F = E1
  • E1 \lor (E1 \land E2) = E1

Law of And-Simplification

For any propositions E1 and E2, it follows that:

  • E1 \land E1 = E1
  • E1 \land T = E1
  • E1 \land F = F
  • E1 \land (E1 \lor E2) = E1

Law of Identity

For any proposition E1, E1 = E1.

Inference 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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Which of the two inference rules that make up the equivalence-transformation formal system is redundant? Back: Transitivity. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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}

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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}

END%%

%%ANKI Cloze The system of evaluation has {equivalences} whereas the formal system has {theorems}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is a "theorem" in the equivalence-transformation formal system? Back: An equivalence derived from the axioms and inference rules. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is e.g. the Law of Implication proven in the system of evaluation? Back: With truth tables. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Selectors

A selector denotes a finite sequence of subscript expressions, each enclosed in brackets. \epsilon denotes the empty selector. For example, variable x is equivalently denoted as x \circ \epsilon whereas for array b, b[i] is equivalently denoted as b \circ [i].

Selector update syntax allows specifying a new value with previous subscripted values overridden. For instance, (b; i{:}e) denotes b with b[i] now referring to e. More formally, for any j \in \mathop{domain}(b), (b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \ i \neq j \rightarrow b[j] \end{cases}

%%ANKI Basic Let b be an array. What does b.lower denote? Back: The lower subscript bound of the array. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What does b.upper denote? Back: The upper subscript bound of the array. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is domain(b) defined in set-theoretic notation? Back: \{i \mid b.lower \leq i \leq b.upper\} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b[0{:}2] be an array. What is b.lower? Back: 0 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b[0{:}2] be an array. What is b.upper? Back: 2 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Execution of b[i] := e of array b in state s yields what new value of b? Back: b = (b; i{:}s(e)) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let s be a state. What is x in (s; x{:}e)? Back: An identifier found in s. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let s be a state. What is e in (s; x{:}e)? Back: An expression. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let s be a state. What is e's type in (s; x{:}e)? Back: A type matching x's declaration. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What is x in (b; x{:}e)? Back: An expression that evaluates to a member of domain(b). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What is e's type in (b; x{:}e)? Back: A type matching b's member declaration. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What case analysis does (b; i{:}e)[j] evaluate to? Back: (b; i{:}e)[j] = \begin{cases} i = j \rightarrow e \ i \neq j \rightarrow b[j] \end{cases} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is (((b; i{:}e_1); j{:}e_2); k{:}e_3) rewritten without nesting? Back: As (b; i{:}e_1; j{:}e_2; k{:}e_3) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is (b; (i{:}e_1; (j{:}e_2; (k{:}e_3)))) rewritten without nesting? Back: N/A. This is invalid syntax. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is (b; i{:}e_1; j{:}e_2; k{:}e_3) rewritten with nesting? Back: As (((b; i{:}e_1); j{:}e_2); k{:}e_3). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What does (b; i{:}2; i{:}3; i{:}4)[i] evaluate to? Back: 4 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is (b; 0{:}8; 2{:}9; 0{:}7)[1] simplified? Back: As b[1]. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic According to Gries, what is the traditional interpretation of an array? Back: As a collection of subscripted independent variables (with a common name). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic According to Gries, what is the new interpretation of an array? Back: As a function. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What propositional expression results from eliminating (b; \ldots) notation from (b; i{:}5)[j] = 5? Back: (i = j \land 5 = 5) \lor (i \neq j \land b[j] = 5) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What logical axiom is used when eliminating (b; \ldots) notation from e.g. (b; i{:}5)[j] = 5? Back: The Law of the Excluded Middle. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze For state s and array b, {(s; x{:}s(x))} is analagous to {(b; i{:}b[i])}. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the simplification of (b; i{:}b[i]; j{:}b[j]; k{:}b[j])? Back: (b; k{:}b[j]) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b, what terminology does Gries use to describe i{:}j in e.g. b[i{:}j]? Back: A section. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b, how many elements are in section b[i{:}j]? Back: j - i + 1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b and fixed j, what is the largest possible value of i in b[i{:}j]? Back: j + 1 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b, how many elements are in b[j{+}1{:}j]? Back: 0 Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b, what is b[1{:}5] = x an abbreviation for? Back: \forall i, 1 \leq i \leq 5 \Rightarrow b[i] = x Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given array b, what is b[1{:}k{-}1] < x < b[k{:}n{-}1] an abbreviation for? Back: (\forall i, 1 \leq i < k \Rightarrow b[i] < x) \land (\forall i, k \leq i < n \Rightarrow x < b[i]) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Generalizing further to all variable types x, \begin{align*} (x; \epsilon{:}e) & = e \ (x; [i] {\circ} s{:}e)[j] & = \begin{cases} i \neq j \rightarrow x[j] \ i = j \rightarrow (x[j]; s{:}e) \end{cases} \end{align*}

%%ANKI Basic What is a selector? Back: A finite sequence of subscript expressions. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given valid expression (x; [i]{\circ}s{:}e), what can be said about i? Back: i is in the domain of x. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the base case of selector update syntax? Back: (x; \epsilon{:}e) = e Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is the null selector usually denoted? Back: \epsilon Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic The null selector is the identity element of what operation? Back: Subscript sequence concatenation. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is assignment x := e rewritten with a selector? Back: x \circ \epsilon := e Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is x \circ \epsilon := e rewritten using selector update syntax? Back: x := (x; \epsilon{:}e) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is command x := (x; \epsilon{:}e) more compactly rewritten? Back: x := e Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What two assignments (i.e. using :=) are used to prove e = (x; \epsilon{:}e)? Back: x := e and x \circ \epsilon := e. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How do assignments x := e and x \circ \epsilon := e prove e = (x; \epsilon{:}e)? Back: The assignments have the same effect and the latter can be written as x := (x; \epsilon{:}e). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. How is b[i][j] := e rewritten using selector update syntax? Back: b := (b; [i][j]{:}e) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array. What does (b; \epsilon{:}g) evaluate to? Back: g Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array and i = j. What does (b; [i]{\circ}s{:}e)[j] evaluate to? Back: (b[j]; s{:}e) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Let b be an array and i \neq j. What does (b; [i]{\circ}s{:}e)[j] evaluate to? Back: b[j] Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Maintaining selector update syntax, how is (c; 1{:}3)[1] more explicitly written with a selector? Back: (c; [1]{:}3)[1] Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Maintaining selector update syntax, how is (c; 1{:}3)[1] rewritten with [1] commuted as leftward as possible? Back: (c[1]; \epsilon{:}3) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Consider selector update syntax. Is precedence left-to-right or right-to-left? Back: Right-to-left. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does it mean for selector update syntax to have right-to-left precedence? Back: Rightmost selectors overwrite duplicate selectors. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is (b; s_1{:}e_1; s_2{:}e_2; s_1{:}e_3) simplified? Back: As (b; s_2{:}e_2; s_1{:}e_3). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Substitution

Textual substitution refers to the replacement of a pred-logic#Identifiers identifier with an expression, introducing parentheses as necessary. This concept amounts to the #Equivalence Rules with different notation.

%%ANKI Basic Textual substitution is derived from what equivalence rule? Back: The substitution rule. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Simple

If x denotes a variable and e an expression, substitution of x by e is denoted as \large{E_e^x}

%%ANKI Basic What term refers to x in textual substitution E_e^x? Back: The reference. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What term refers to e in textual substitution E_e^x? Back: The expression. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What term refers to both x and e together in textual substitution E_e^x? Back: The reference-expression pair. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What identifier is guaranteed to not occur freely in E_e^x? Back: N/A. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What identifier is guaranteed to not occur freely in E_{s(e)}^x? Back: x Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Why does x not occur freely in E_{s(e)}^x? Back: Because s(e) evaluates to a constant proposition. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the role of E in textual substitution E_e^x? Back: It is the expression in which free occurrences of x are replaced. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the role of e in textual substitution E_e^x? Back: It is the expression that is evaluated and substituted into E. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the role of x in textual substitution E_e^x? Back: It is the identifier matching free occurrences in E that are replaced. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is textual substitution E_e^x interpreted as a function? Back: As E(e), where E is a function of x. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Why does Gries prefer notation E_e^x over e.g. E(e)? Back: The former indicates the identifier to replace. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What two scenarios ensure E_e^x = E is an equivalence? Back: x = e or no free occurrences of x exist in E. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If x \neq e, why might E_e^x = E be an equivalence despite x existing in E? Back: The only occurrences of x in E may be bound. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is required for E_e^x to be valid? Back: Substitution must result in a syntactically valid expression. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic 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)) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic 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)) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic 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)) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

General

We can generalize textual substitution to operate on a vector of reference-expression pairs, where each reference corresponds to some identifier concatenated with a selector. Let \bar{x} = \langle x_1, \ldots, x_n \rangle denote a vector of identifiers concatenated with selectors and \bar{e} = \langle e_1, \ldots, e_n \rangle denote a vector of expressions. Then textual substitition of \bar{x} with \bar{e} in expression E is denoted as \large{E_{\bar{e}}^{\bar{x}}}

Substitution is defined recursively as follows:

  1. If each x_i is a distinct identifier with a null selector, then E_{\bar{e}}^{\bar{x}} is the simultaneous substitution of \bar{x} with \bar{e}.
  2. Adjacent reference-expression pairs may be permuted as long as they begin with different identifiers. That is, for all distinct b and c, \Large{E_{\bar{e}, ,f, ,h, ,\bar{g}}^{\bar{x}, ,b, ,c, ,\bar{y}} = E_{\bar{x}, ,h, ,f, ,\bar{g}}^{\bar{x}, ,c, ,b, ,\bar{y}}}
  3. Multiple assignments to subparts of an object b can be viewed as a single assignment to b. That is, provided b does not begin any of the x_i, \Large{E_{e_1, ,\ldots, ,e_m, ,\bar{g}}^{b ,\circ, s_1, ,\ldots, ,b ,\circ, s_m, ,\bar{x}} = E_{(b; ,s_1{:}e_1; ,\cdots; ,s_m{:}e_m), ,\bar{g}}^{b, ,\bar{x}}}

Note that simultaneous substitution is different from sequential substitution.

%%ANKI Basic Consider E_{\bar{e}}^{\bar{x}}. What is each x_i in \bar{x}? Back: An identifier concatenated with a selector. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Consider E_{\bar{e}}^{\bar{x}}. What is each e_i in \bar{e}? Back: An expression. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the base case in the evaluation of E_{\bar{e}}^{\bar{x}}? Back: If \bar{x} are distinct identifiers with null selectors, direct simultaneous substitution. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Which of E_{\bar{e}}^{\bar{x}}'s reference-expression pairs may be moved? Back: Adjacent pairs with distinct identifiers. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is b_1 \circ s_1 and b_2 \circ s_2 said to have distinct identifiers? Back: When b_1 \neq b_2. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is b_1 \circ s_1 and b_2 \circ s_2 said to have distinct selectors? Back: When s_1 \neq s_2. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x and y are distinct. Is the following a tautology? \large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{y, x}} Back: Yes. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is the following a tautology? \large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{y, x}} Back: When x and y refer to distinct identifiers. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x = y. When is the following a tautology? \large{E_{e_1, e_2}^{x, y} = E_{e_2, e_1}^{x, y}} Back: When e_1 = e_2. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x, y, and z are distinct. What is the result of a single evaluation step? \large{E_{e_1, e_2, e_3}^{x, y, z}} Back: N/A. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x \neq y. What is the result of a single evaluation step? \large{E_{e_1, e_2, e_3}^{x, y, x}} Back: \large{E_{e_1, e_3, e_2}^{x, x, y}} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x \neq y. What is the result of a single evaluation step? \large{E_{e_1, e_3, e_2}^{x, x, y}} Back: \large{E_{(x; ,\epsilon{:}e_1; ,\epsilon{:}e_3), e_2}^{x, y}} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x \neq y. What is the result of a single evaluation step? \large{E_{(x; ,\epsilon{:}e_1; ,\epsilon{:}e_3), e_2}^{x, y}} Back: \large{E_{e_3, e_2}^{x, y}} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x \neq y. Why isn't the following a tautology? \large{E_{e_1, e_2, e_3}^{x, y, x}} = E_{(x; \epsilon{:}e_1), e_2, e_3}^{x, y, x} Back: N/A. It is. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Suppose x \neq y. Why isn't the following a tautology? \large{E_{e_1, e_2, e_3, e_4}^{x, x, y, x}} = E_{(x; \epsilon{:}e_1; \epsilon{:}e_2), e_3, e_4}^{x, y, x} Back: Because not every x was made adjacent before grouping. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Consider array b and i \in \mathop{domain}(b). What is the result of a single evaluation step? \large{E_{e}^{b[i]}} Back: \large{E_{(b; [i]{:}e)}^{b}} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Consider identifier x, array b and i \in \mathop{domain}(b). What is the result of a single evaluation step? \large{E_{b[i]}^{x}} Back: N/A. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Theorems

  • (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 substitutions occur in u.

%%ANKI Basic How do we simplify (E_u^x)_v^x? Back: As E_{u_v^x}^x Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is E_{u_v^x}^x rewritten as sequential substitution? Back: As (E_u^x)_v^x Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic 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. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • If y \not\in FV(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.

%%ANKI Basic 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. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If x \neq y, when is (E_u^x)_v^y = E_{u_v^y}^x? Back: When y is not free in E. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Why does y \not\in FV(E) ensure (E_u^x)_v^y = E_{u_v^y}^x is an equivalence? Back: If it were, a v would exist in the LHS that doesn't in the RHS. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How does Gries denote state s with identifer x set to value v? Back: (s; x{:}v) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How is (s; x{:}v) written instead using set-theoretical notation? Back: (s - \{\langle x, s(x) \rangle\}) \cup \{\langle x, v \rangle\} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Execution of x := e in state s terminates in what new state? Back: (s; x{:}s(e)) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given state s, what statement does (s; x{:}s(e)) derive from? Back: x := e Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What missing value guarantees state s satisfies equivalence s = (s; x{:}\_)? Back: s(x) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given state s, why is it that s = (s; x{:}s(x))? Back: Evaluating x in state s yields s(x). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • s(E_e^x) = s(E_{s(e)}^x)
    • Substituting x with e and then evaluating is the same as substituting x with the evaluation of e.

%%ANKI Basic How can we simplify s(E_{s(e)}^x)? Back: As s(E_e^x) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given state s, what equivalence relates E_e^x with E_{s(e)}^x? Back: s(E_e^x) = s(E_{s(e)}^x) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Let s be a state and s' = (s; x{:}s(e)). Then s'(E) = s(E_e^x).

%%ANKI Cloze Let s be a state and s' = ({s; x{:}s(e)}). Then s'({E}) = s({E_e^x}). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If s' = (s; x{:}s(e)), then s'(E) = s(E_e^x). Why do we not say s' = (s; x{:}e) instead? Back: The value of a state's identifier must always be a constant proposition. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic How do you define s' such that s(E_e^x) = s'(E)? Back: s' = (s; x{:}s(e)). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given defined value v \neq s(x), when is s(E) = (s; x{:}v)(E)? Back: When x is not free in E. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

  • Given identifiers \bar{x} and fresh identifiers \bar{u}, (E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E.

%%ANKI Basic When is (E_{\bar{u}}^{\bar{x}})_{\bar{x}}^{\bar{u}} = E guaranteed to be an equivalence? Back: When \bar{x} and \bar{u} refer to distinct identifiers (concatenated with selectors). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

States

A state is a function that maps identifiers to T or F. A proposition can be equivalently seen as a representation of the set of states in which it is true.

%%ANKI Basic What is a state? Back: A function mapping identifiers to values. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \land c) well-defined in \{\langle b, T \rangle, \langle c, F \rangle\}? Back: Yes. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (b \lor d) well-defined in \{\langle b, T \rangle, \langle c, F \rangle\}? Back: No. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic A proposition is well-defined with respect to what? Back: A state to evaluate against. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What proposition represents states \{(b, T), (c, T)\} and \{(b, F), (c, F)\}? Back: (b \land c) \lor (\neg b \land \neg c) Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does proposition a \land b represent? Back: \{\{\langle a, T \rangle, \langle b, T \rangle\}\} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does proposition a \lor b represent? Back: \{\{\langle a, T \rangle, \langle b, T \rangle\}, \{\langle a, T \rangle, \langle b, F \rangle\}, \{\langle a, F \rangle, \langle b, T \rangle\}\} Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the weakest proposition? Back: T Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does T represent? Back: The set of all states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is the strongest proposition? Back: F Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What set of states does F represent? Back: The set of no states. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What does a proposition represent? Back: The set of states in which it is true. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p stronger than q? Back: When p \Rightarrow q. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If p \Rightarrow q, which of p or q is considered stronger? Back: p Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic When is p weaker than q? Back: When q \Rightarrow p. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic If p \Rightarrow q, which of p or q is considered weaker? Back: q Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

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. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Given sets a and b, a = b is equivalent to the conjunction of what two expressions? Back: a \subseteq b and b \subseteq a. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze {a \Rightarrow b} is to propositional logic as {a \subseteq b} is to sets. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Cloze {a \Leftrightarrow b} is to propositional logic as {a = b} is to sets. Reference: Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (i \geq 0) well-defined in \{(i, -10)\}? Back: Yes. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic Is (i \geq 0) well-defined in \{(j, -10)\}? Back: No. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What predicate represents states \{(i, 0), (i, 2), (i, 4), \ldots\}? Back: i \geq 0 is even. Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

%%ANKI Basic What is sloppy about phrase "the states in i + j = 0"? Back: i + j = 0 is not a set but a representation of a set (of states). Reference: Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.

END%%

Bibliography

  • Avigad, Jeremy. Theorem Proving in Lean, n.d.
  • Gries, David. The Science of Programming. Texts and Monographs in Computer Science. New York: Springer-Verlag, 1981.