hcju/proofwikips huggingface dataset:
Definition:Closed Region/Complex
Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is a '''closed region'''  $D$ the closure of an open region.
Definition:Region/Complex
Let $D \subseteq \C$ be a subset of the set of complex numbers. $D$ is a '''region of $\C$''' : :$(1): \quad$ $D$ is non-empty :$(2): \quad$ $D$ is path-connected.
Definition:Supremum of Sequence
A special case of a supremum of a mapping is a '''supremum of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\set {x_n: n \in \N}$ admit a supremum. Then the '''supremum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \sup {\sequence {x_n} } = \map \sup {\set {x_n: n \in \N} }$
Definition:Infimum of Sequence
A special case of an infimum of a mapping is an '''infimum of a sequence''', where the domain of the mapping is $\N$. Let $\struct {T, \preceq}$ be an ordered set. Let $\sequence {x_n}$ be a sequence in $T$. Let $\set {x_n: n \in \N}$ admit an infimum. Then the '''infimum''' of $\sequence {x_n}$) is defined as: :$\displaystyle \map \inf {\sequence {x_n} } = \map \inf {\set {x_n: n \in \N} }$
Definition:Semantic Tableau
'''Semantic tableaus''' are the labeled trees which can be obtained in the course of executing the Semantic Tableau Algorithm. === Semantic Tableau Algorithm === {{:Semantic Tableau Algorithm}} === Completed Tableau === {{:Definition:Semantic Tableau/Completed|Completed Tableau}} === Closed Tableau === {{:Definition:Semantic Tableau/Closed|Closed Tableau}} === Open Tableau === {{:Definition:Semantic Tableau/Open|Open Tableau}}
Definition:Semantic Tableau/Completed
Let $T$ be a semantic tableau. Then $T$ is '''completed''' iff all of its leaves are marked.
Definition:Semantic Tableau/Closed
Let $T$ be a semantic tableau. Then $T$ is '''closed''' iff all of its leaves are marked closed.
Definition:Semantic Tableau/Open
Let $T$ be a semantic tableau. Then $T$ is '''open''' iff all of its leaves are marked open.
Definition:Marked Leaf
Let $T$ be a semantic tableau. Let $t$ be a leaf of $T$. Then $t$ is '''marked''' iff it has occurred in '''Step 3''' of the Semantic Tableau Algorithm. Otherwise, it is called '''unmarked'''. === Marked Closed Leaf === {{:Definition:Marked Leaf/Closed}} === Marked Open Leaf === {{:Definition:Marked Leaf/Open}}
Definition:Alpha-Formula
Let $\mathbf A$ be a WFF of propositional logic that is not a literal. Then $\mathbf A$ is an '''$\alpha$-formula'''  either: :$\mathbf A = \neg \neg \mathbf A_1$ for some WFF $\mathbf A_1$, or: :$\mathbf A$ is semantically equivalent to a conjunction $\mathbf A_1 \land \mathbf A_2$ for some WFFs $\mathbf A_1, \mathbf A_2$. === Table of $\alpha$-Formulas === {{:Definition:Alpha-Formula/Table}}
Definition:Alpha-Formula/Table
From Classification of $\alpha$-Formulas, we obtain the following table of $\alpha$-formulas $\mathbf A$ and corresponding $\mathbf A_1$ and $\mathbf A_2$: ::$\begin{array}{ccc} \hline \mathbf A & \mathbf A_1 & \mathbf A_2\\ \hline \neg\neg \mathbf A_1 & \mathbf A_1 & \\ \mathbf A_1 \land \mathbf A_2 & \mathbf A_1 & \mathbf A_2 \\ \neg \left({\mathbf A_1 \lor \mathbf A_2}\right) & \neg \mathbf A_1 & \neg \mathbf A_2 \\ \neg \left({\mathbf A_1 \implies \mathbf A_2}\right) & \mathbf A_1 & \neg \mathbf A_2 \\ \neg \left({\mathbf A_1 \mathbin \uparrow \mathbf A_2}\right) & \mathbf A_1 & \mathbf A_2 \\ \mathbf A_1 \mathbin \downarrow \mathbf A_2 & \neg \mathbf A_1 & \neg \mathbf A_2 \\ \mathbf A_1 \iff \mathbf A_2 & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\ \neg \left({\mathbf A_1 \oplus \mathbf A_2}\right) & \mathbf A_1 \implies \mathbf A_2 & \mathbf A_2 \implies \mathbf A_1 \\ \hline \end{array}$
Definition:Beta-Formula/Table
From Classification of $\beta$-Formulas, we obtain the following table of $\beta$-formulas $\mathbf B$ and corresponding $\mathbf B_1$ and $\mathbf B_2$: ::$\begin{array}{ccc} \hline \mathbf B & \mathbf B_1 & \mathbf B_2\\ \hline \neg \left({\mathbf B_1 \land \mathbf B_2}\right) & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \mathbf B_1 \lor \mathbf B_2 & \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \implies \mathbf B_2 & \neg \mathbf B_1 & \mathbf B_2 \\ \mathbf B_1 \mathbin \uparrow \mathbf B_2 & \neg \mathbf B_1 & \neg \mathbf B_2 \\ \neg \left({\mathbf B_1 \mathbin \downarrow \mathbf B_2}\right) & \mathbf B_1 & \mathbf B_2 \\ \neg \left({\mathbf B_1 \iff \mathbf B_2}\right) & \neg \left({\mathbf B_1 \implies \mathbf B_2}\right) & \neg \left({\mathbf B_2 \implies \mathbf B_1}\right) \\ \mathbf B_1 \oplus \mathbf B_2 & \neg \left({\mathbf B_1 \implies \mathbf B_2}\right) & \neg \left({\mathbf B_2 \implies \mathbf B_1}\right) \\ \hline \end{array}$
Definition:Beta-Formula
Let $\mathbf B$ be a WFF of propositional logic that is not a literal. Then $\mathbf B$ is a '''$\beta$-formula''' iff: :$\mathbf B$ is semantically equivalent to a disjunction $\mathbf B_1 \lor \mathbf B_2$ for some WFFs $\mathbf B_1, \mathbf B_2$. === Table of $\beta$-Formulas === {{:Definition:Beta-Formula/Table}}
Definition:Triangle (Geometry)/Right-Angled/Opposite
In a right-angled triangle, for a given non-right angled vertex, the opposite side is referred to as '''the opposite'''. In the above figure: :the '''opposite''' to vertex $A$ is side $a$ :the '''opposite''' to vertex $C$ is side $c$.
Definition:Theory/Set of Formulas
Let $\mathcal F$ be a set of $\mathcal L$-formulas. Then the '''$\mathcal L$-theory of $\mathcal F$''', denoted $T \left({\mathcal F}\right)$ is the set: :$\left\{{\phi \in \mathcal L: \mathcal F \models_{\mathscr M} \phi}\right\}$ where $\models_{\mathscr M}$ denotes $\mathscr M$-semantic consequence.
Definition:Lemniscate of Bernoulli
=== Geometric Definition === {{:Definition:Lemniscate of Bernoulli/Geometric Definition}} === Cartesian Definition === {{:Definition:Lemniscate of Bernoulli/Cartesian Definition}} === Polar Definition === {{:Definition:Lemniscate of Bernoulli/Polar Definition}} === Parametric Definition === {{:Definition:Lemniscate of Bernoulli/Parametric Definition}}
Definition:Gentzen Proof System
'''Gentzen proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of few axioms and many rules of inference. * Use of formal, sequent-like notation involving the turnstile $\vdash$. * Proofs whose structure can be viewed as rooted trees.
Definition:Gentzen Proof System/Instance 1
<section notitle="true" name="tc"> This instance of a Gentzen proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\mathcal L$ be the language of propositional logic. The Gentzen system applies to sets of propositional formulae. The intuition behind the system is that a set $U$ represents its disjunction. $\mathscr G$ has the following axioms and rules of inference: === Axioms === A set $U$ of propositional formulae is an axiom of $\mathscr G$ iff $U$ contains a complementary pair of literals. The invocation of an axiom may be denoted by: :$\vdash U$ === Rules of Inference === Let $U_1, U_2$ be sets of propositional formulae. $\mathscr G$ has two rules of inference, the $\alpha$-rule and the $\beta$-rule:<ref>In {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}}, the meanings of $\alpha$-formula and $\beta$-formula are interchanged at this point compared to their earlier appearances. This is done because of a later connection with semantic tableaus, but can serve only to confuse. Caution needs to be exercised when using this source, because of the context-dependent meaning of these two terms.</ref> ==== $\alpha$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Alpha-Rule}} ==== $\beta$-Rule ==== {{:Definition:Gentzen Proof System/Instance 1/Beta-Rule}} Invocations of these rules in a proof can be denoted as: :$(\alpha) \dfrac {\vdash U_1, \mathbf A_1 \hspace{3em} \vdash U_2, \mathbf A_2}{\vdash U_1, U_2, \mathbf A} \hspace{3em}  (\beta) \dfrac {\vdash U_1, \mathbf B_1, \mathbf B_2}{\vdash U_1, \mathbf B}$ This notation suppresses the set notation as a matter of convenience. </section> {{NamedforDef|Gerhard Karl Erich Gentzen|cat = Gentzen}}
Definition:Tableau Proof (Formal Systems)
A '''tableau proof''' for a proof system is a technique for presenting a logical argument in a straightforward, standard form. On , the proof system is usually natural deduction. A '''tableau proof''' is a sequence of lines specifying the order of premises, assumptions, inferences and conclusion in support of an argument. Each line of a '''tableau proof''' has a particular format. It consists of the following parts: * '''Line:''' The line number of the proof. This is a simple numbering from 1 upwards. * '''Pool:''' The list of all the lines containing the pool of assumptions for the formula introduced on this line. * '''Formula:''' The propositional formula introduced on this line. * '''Rule:''' The justification for introducing this line. This should be the rule of inference being used to derive this line. * '''Depends on:''' The lines (if any) upon which this line ''directly'' depends. For premises and assumptions, this field will be empty. Optionally, a comment may be added to explicitly point out possible intricacies. If any assumptions are discharged on a certain line, for the sake of clarity it is preferred that such be mentioned explicitly in a comment. At the end of a tableau proof, the only lines upon which the proof depends may be those which contain the premises.
Definition:Tableau Proof (Formal Systems)/Length
The '''length''' of a tableau proof is the number of lines it has.
Definition:Variable/Complex
A '''complex variable''' is a symbol which can stand for any one of a set of complex numbers.
Definition:Independent Variable/Complex Function
Let $f: \C \to \C$ be a complex function. Let $\map f z = w$. Then $z$ is referred to as an '''independent variable (of $f$)'''.
Definition:Dependent Variable/Complex Function
Let $f: \C \to \C$ be a complex function. Let $\map f z = w$. Then $w$ is referred to as the '''dependent variable (of $f$)'''.
Definition:Left-Total Relation/Multifunction
In the field of complex analysis, a '''left-total relation''' is usually referred to as a '''multifunction'''.
Definition:Left-Total Relation/Multifunction/Branch
Let $D \subseteq \C$ be a subset of the complex numbers. Let $f: D \to \C$ be a multifunction on $D$. Let $\family {S_i}_{i \mathop \in I}$ be a partitioning of the codomain of $f$ such that: :$\forall i \in I: f \restriction_{D \times S_i}$ is a mapping. Then each $f \restriction_{D \times S_i}$ is a '''branch''' of $f$.
Definition:Distinct/Plural
Two objects $x$ and $y$ are '''distinct'''  $x \ne y$. If $x$ and $y$ are '''distinct''', then that means they can be '''distinguished''', or '''identified as being different from each other'''.
Definition:Distinct/Plural/Pairwise Distinct
A set of objects is '''pairwise distinct''' if each pair of elements of that set is distinct.
Definition:Distinct/Singular
Let $x \in S$ be an element of a set of objects $S$. $x$ is '''distinguished''' from the other elements of $S$ if is endowed with a property that the other elements of $S$ are specifically deemed not to possess. Such an element is identified as being '''distinct''' from the others.
Definition:Distinct/Indistinguishable
Two objects are '''indistinguishable''' if they can not (in a particular context) be told apart from each other. So, two objects may be '''distinct''' but (at a given level) '''indistinguishable''', like identical twins.
Definition:Left-Total Relation/Multifunction/Branch/Principal Branch
Let $D \subseteq \C$ be a subset of the complex numbers. Let $f: D \to \C$ be a multifunction on $D$. Let $\sequence {S_i}_{i \in I}$ be a partitioning of the codomain of $f$ into branches. It is usual to distinguish one such branch of $f$ from the others, and label it the '''principal branch''' of $f$.
Definition:Real Function/Two Variables
Let $S, T \subseteq \R$ be subsets of the set of real numbers $\R$. Let $f: S \times T \to \R$ be a mapping. Then $f$ is defined as a '''(real) function of two (independent) variables'''.  The expression: :$z = \map f {x, y}$ means: :(The dependent variable) $z$ is a function of (the independent variables) $x$ and $y$.
Definition:Complex Transformation
A '''complex transformation''' is a mapping on the complex plane $f: \C \to \C$ which is specifically ''not'' a multifunction. Let $z = x + i y$ be a complex variable. Let $w = u + i v = \map f z$. Then $w$ can be expressed as: :$u + i v = \map f {x + i y}$ such that: :$u = \map u {x, y}$ and: :$v = \map v {x, y}$ are real functions of two variables. Thus a point $P = \tuple {x, y}$ in the complex plane is '''transformed''' to a point $P' = \tuple {\map u {x, y}, \map v {x, y} }$ by $f$. Thus $P'$ is the image of $P$ under $f$.
Definition:Curvilinear Coordinate System/Complex Plane
Let $u + i v = \map f {x + i y}$ be a complex transformation. Let $P = \tuple {x, y}$ be a point in the complex plane. Then $\tuple {\map u {x, y}, \map v {x, y} }$ are the '''curvilinear coordinates of $P$ under $f$'''.
Definition:Complex Curvilinear Coordinates/Coordinate Curves
Let $c_1$ and $c_2$ be constants. The curves: :$\map u {x, y} = c_1$ :$\map v {x, y} = c_2$ are the '''coordinate curves''' of $f$.
Definition:Rational Function/Real
Let $P: \R \to \R$ and $Q: \R \to \R$ be polynomial functions on the set of real numbers. Let $S$ be the set $\R$ from which all the roots of $Q$ have been removed. That is: : $S = \R \setminus \left\{{x \in \R: Q \left({x}\right) = 0}\right\}$. Then the equation $y = \dfrac {P \left({x}\right)} {Q \left({x}\right)}$ defines a function from $S$ to $\R$. Such a function is a '''rational function'''.
Definition:Rational Function/Complex
Let $P: \C \to \C$ and $Q: \C \to \C$ be polynomial functions on the set of complex numbers. Let $S$ be the set $\C$ from which all the roots of $Q$ have been removed. That is: :$S = \C \setminus \set {z \in \C: \map Q z = 0}$ Then the equation $y = \dfrac {\map P z} {\map Q z}$ defines a function from $S$ to $\C$. Such a function is a '''rational (algebraic) function'''.
Definition:Power (Algebra)/Real Number/Complex
Let $x \in \R$ be a real number such that $x > 0$. Let $r \in \C$ be ''any'' complex number. Then we define $x^r$ as: :$x^r := \map \exp {r \ln x}$ where $\exp$ denotes the complex exponential function.
Definition:Natural Logarithm/Complex/Definition 1
Let $z = r e^{i \theta}$ be a complex number expressed in exponential form such that $z \ne 0$. The '''complex natural logarithm''' of $z \in \C_{\ne 0}$ is the multifunction defined as: :$\map \ln z := \set {\map \ln r + i \paren {\theta + 2 k \pi}: k \in \Z}$ where $\map \ln r$ is the natural logarithm of the (strictly) positive real number $r$.
Definition:Natural Logarithm/Notation
The notation for the natural logarithm function is misleadingly inconsistent throughout the literature. It is written variously as: :$\ln z$ :$\log z$ :$\Log z$ :$\log_e z$ The first of these is commonly encountered, and is the preferred form on . However, many who consider themselves serious mathematicians believe this notation to be unsophisticated. The second and third are ambiguous (it doesn't tell you which base it is the logarithm of). While the fourth option is more verbose than the others, there is no confusion about exactly what is meant.
Definition:Natural Logarithm/Complex/Definition 2
Let $z \in \C_{\ne 0}$ be a non-zero complex number. The '''complex natural logarithm''' of $z$ is the multifunction defined as: :$\map \ln z := \set {w \in \C: e^w = z}$
Definition:Natural Logarithm/Positive Real/Definition 1
Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\displaystyle \ln x := \int_1^x \frac {\d t} t$
Definition:Natural Logarithm/Positive Real/Definition 2
Let $x \in \R$ be a real number such that $x > 0$. The '''(natural) logarithm''' of $x$ is defined as: :$\ln x := y \in \R: e^y = x$ where $e$ is Euler's number.
Definition:Gentzen Proof System/Instance 1/Alpha-Rule
$(\alpha)$: For any $\alpha$-formula $\mathbf A$ and associated $\mathbf A_1, \mathbf A_2$ from the table of $\alpha$-formulas: ::Given $U_1 \cup \left\{{\mathbf A_1}\right\}$ and $U_2 \cup \left\{{\mathbf A_2}\right\}$, one may infer $U_1 \cup U_2 \cup \left\{{\mathbf A}\right\}$.
Definition:Gentzen Proof System/Instance 1/Beta-Rule
$(\beta)$: For any $\beta$-formula $\mathbf B$ and associated $\mathbf B_1, \mathbf B_2$ from the table of $\beta$-formulas: :: Given $U_1 \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$, one may infer $U_1 \cup \left\{{\mathbf B}\right\}$.
Definition:Gentzen Proof System/Instance 1/Alpha-Rule/Notation
In a tableau proof, the $\alpha$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \cup U_2 \cup \left\{ {\mathbf A}\right\}$. }} {{axiom|lc = '''Description:'''         |t  = $\alpha$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The lines containing $U_1 \cup \left\{ {\mathbf A_1}\right\}$ and $U_2 \cup \left\{ {\mathbf A_2}\right\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\alpha \circ$, where $\circ$ is the binary logical connective such that $\mathbf A {{=}} \mathbf A_1 \circ \mathbf A_2$ or $\mathbf A {{=}} \neg \left({\mathbf A_1 \circ \mathbf A_2}\right)$, or $\neg \neg$ in the case that $\mathbf A = \neg \neg \mathbf A_1$. }} {{end-axiom}}
Definition:Gentzen Proof System/Instance 1/Beta-Rule/Notation
In a tableau proof, the $\beta$-rule can be used as follows: {{begin-axiom}} {{axiom|lc = '''Pool:'''        |t  = Empty. }} {{axiom|lc = '''Formula:'''         |t  = $U_1 \cup \left\{ {\mathbf B}\right\}$. }} {{axiom|lc = '''Description:'''         |t  = $\beta$-Rule. }} {{axiom|lc = '''Depends on:'''         |t  = The line containing $U_1 \cup \left\{ {\mathbf B_1, \mathbf B_2}\right\}$. }} {{axiom|lc = '''Abbreviation:'''         |t  = $\beta \circ$, where $\circ$ is the binary logical connective such that $\mathbf B {{=}} \mathbf B_1 \circ \mathbf B_2$ or $\mathbf B {{=}} \neg \left({\mathbf B_1 \circ \mathbf B_2}\right)$. }} {{end-axiom}}
Definition:Natural Logarithm/Complex/Principal Branch
The principal branch of the complex natural logarithm is usually defined in one of two ways: :$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointr 0 {2 \pi}$ :$\map \Ln z = \map \ln r + i \theta$ for $\theta \in \hointl {-\pi} \pi$ It is important to specify which is in force during a particular exposition.
Definition:Principle
A '''principle''' is a theorem which is deemed to be particularly important. The word has the same force about it as does '''law''', the two words being used interchangeably.
Definition:General Logarithm/Positive Real
Let $x \in \R_{>0}$ be a strictly positive real number. Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$. The '''logarithm to the base $a$ of $x$''' is defined as: :$\log_a x := y \in \R: a^y = x$ where $a^y = e^{y \ln a}$ as defined in Powers of Real Numbers.
Definition:General Logarithm/Complex
Let $z \in \C_{\ne 0}$ be a non-zero complex number. Let $a \in \R_{>0}$ be a strictly positive real number such that $a \ne 1$. The '''logarithm to the base $a$ of $z$''' is defined as: :$\log_a z := \set {y \in \C: a^y = z}$ where $a^y = e^{y \ln a}$ as defined in Powers of Complex Numbers.
Definition:Inverse Sine/Real/Arcsine
{{:Graph of Arcsine Function|Graph}} From Shape of Sine Function, we have that $\sin x$ is continuous and strictly increasing on the interval $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$. From Sine of Half-Integer Multiple of Pi: :$\map \sin {-\dfrac {\pi} 2} = -1$ and: :$\sin \dfrac {\pi} 2 = 1$ Therefore, let $g: \closedint {-\dfrac \pi 2} {\dfrac \pi 2} \to \closedint {-1} 1$ be the restriction of $\sin x$ to $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$. Thus from Inverse of Strictly Monotone Function, $g \paren x$ admits an inverse function, which will be continuous and strictly increasing on $\closedint {-1} 1$. This function is called '''arcsine of $x$''' and is written $\arcsin x$. Thus: :The domain of $\arcsin x$ is $\closedint {-1} 1$ :The image of $\arcsin x$ is $\closedint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Definition:Inverse Sine/Real
Let $x \in \R$ be a real number such that $-1 \le x \le 1$. The '''inverse sine of $x$''' is the multifunction defined as: :$\sin^{-1} \left({x}\right) := \left\{{y \in \R: \sin \left({y}\right) = x}\right\}$ where $\sin \left({y}\right)$ is the sine of $y$.
Definition:Sine/Real Function
The real function $\sin: \R \to \R$ is defined as: {{begin-eqn}} {{eqn | l = \sin x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = x - \frac {x^3} {3!} + \frac {x^5} {5!} - \cdots       | c =  }} {{end-eqn}}
Definition:Sine/Complex Function
The complex function $\sin: \C \to \C$ is defined as: {{begin-eqn}} {{eqn | l = \sin z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!}       | c =  }} {{eqn | r = z - \frac {z^3} {3!} + \frac {z^5} {5!} - \frac {z^7} {7!} + \cdots + \paren {-1}^n \frac {z^{2 n + 1 } } {\paren {2 n + 1}!} + \cdots       | c =  }} {{end-eqn}}
Definition:Inverse Sine/Complex
{{:Definition:Inverse Sine/Complex/Definition 1}}
Definition:Hilbert Proof System
'''Hilbert proof systems''' are a class of proof systems for propositional and predicate logic. Their characteristics include: * The presence of many axioms; * Typically only Modus Ponendo Ponens as a rule of inference.
Definition:Hilbert Proof System/Instance 1
<section notitle="true" name="tc"> This instance of a Hilbert proof system is used in: * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition}} Let $\LL$ be the language of propositional logic. $\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $\mathbf A, \mathbf B, \mathbf C$ be WFFs. The following three WFFs are axioms of $\mathscr H$: {{begin-axiom}} {{axiom | lc= '''Axiom 1:'''         | m = \mathbf A \implies \paren {\mathbf B \implies \mathbf A} }} {{axiom | lc= '''Axiom 2:'''         | m = \paren {\mathbf A \implies \paren {\mathbf B \implies \mathbf C} } \implies \paren {\paren {\mathbf A \implies \mathbf B} \implies \paren {\mathbf A \implies \mathbf C} } }} {{axiom | lc= '''Axiom 3:'''         | m = \paren {\neg \mathbf B \implies \neg \mathbf A} \implies \paren {\mathbf A \implies \mathbf B} }} {{end-axiom}} === Rules of Inference === The sole rule of inference is Modus Ponendo Ponens: :From $\mathbf A$ and $\mathbf A \implies \mathbf B$, one may infer $\mathbf B$. </section> {{NamedforDef|David Hilbert}}
Definition:Derived Rule
Let $\mathcal L$ be a formal language. Let $\mathscr P$ be a proof system for $\mathcal L$. A '''derived rule''' for $\mathscr P$ is a rule of inference $R$ such that: :The proof system $\mathscr P'$, obtained by adding $R$ to $\mathscr P$, is equivalent to $\mathscr P$. That is, $R$ is a '''derived rule''' iff it does not extend the collection of $\mathscr P$-theorems.
Definition:Cosine/Real Function
The real function $\cos: \R \to \R$ is defined as: {{begin-eqn}} {{eqn | l = \cos x       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } {\paren {2 n!} }       | c =  }} {{eqn | r = 1 - \frac {x^2} {2!} + \frac {x^4} {4!} - \frac {x^6} {6!} + \cdots + \paren {-1}^n \frac {x^{2 n} } {\paren {2 n}!} + \cdots       | c =  }} {{end-eqn}}
Definition:Cosine/Complex Function
The complex function $\cos: \C \to \C$ is defined as: {{begin-eqn}} {{eqn | l = \cos z       | r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!}       | c =  }} {{eqn | r = 1 - \frac {z^2} {2!} + \frac {z^4} {4!} - \frac {z^6} {6!} + \cdots + \paren {-1}^n \frac {z^{2 n} } {\paren {2 n}!} + \cdots       | c =  }} {{end-eqn}}
Definition:Inverse Sine/Complex/Definition 1
Let $z \in \C$ be a complex number. The '''inverse sine of $z$''' is the multifunction defined as: :$\sin^{-1} \paren z := \set {w \in \C: \sin \paren w = z}$ where $\sin \paren w$ is the sine of $w$.