hcju/proofwikips huggingface dataset:
Common Factor Cancelling in Congruence/Corollary 1
== Corollary to Common Factor Cancelling in Congruence == Let $a, b, x, y, m \in \Z$. Let: :$a x \equiv b y \pmod m$ and $a \equiv b \pmod m$ where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$.  If $a$ is coprime to $m$, then: :$x \equiv y \pmod m$  == Proof == Let $a \perp m$. Then by definition of coprime: :$\gcd \set {a, m} = 1$ The result follows immediately from Common Factor Cancelling in Congruence.  == Warning == {{:Common Factor Cancelling in Congruence/Corollary 1/Warning}} == Sources == * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Modulo Multiplication is Well-Defined|next = Congruence by Product of Moduli}}: $\S 1.2.4$: Integer Functions and Elementary Number Theory: Law $\text{B}$ * {{BookReference|The Art of Computer Programming: Volume 1: Fundamental Algorithms|1997|Donald E. Knuth|ed = 3rd|edpage = Third Edition|prev = Law of Inverses (Modulo Arithmetic)/Corollary 1|next = Fundamental Theorem of Arithmetic}}: $\S 1.2.4$: Integer Functions and Elementary Number Theory: Exercise $20$
Sequence of Powers of Reciprocals is Null Sequence/Corollary
== Corollary to Sequence of Powers of Reciprocals is Null Sequence ==  Let $\sequence {x_n}$ be the sequence in $\R$ defined as: :$x_n = \dfrac 1 n$ Then $\sequence {x_n}$ is a null sequence.  == Proof == $n = n^1$ from the definition of power. As $1 \in \Q_{>0}$ the result follows from Sequence of Powers of Reciprocals is Null Sequence.
Integer Less One divides Power Less One/Corollary
== Corollary to Integer Less One divides Power Less One ==  Let $m, n, q \in \Z_{>0}$. Let: :$m \divides n$ where $\divides$ denotes divisibility. Then: :$\paren {q^m - 1} \divides \paren {q^n - 1}$  === Converse to Corollary === {{:Integer Less One divides Power Less One/Corollary/Converse}} == Proof == By hypothesis: :$m \divides n$ By definition of divisibility: :$\exists k \in \Z: k m = n$ Thus: :$q^n = q^{k m} = \paren {q^m}^k$ {{explain|Find the exponent combination laws to justify the above}} Then by Integer Less One divides Power Less One: :$\paren {q^m - 1} \divides \paren {\paren {q^m}^k - 1}$ Hence the result.  == Sources == * {{BookReference|Algebra Volume 1|1982|P.M. Cohn|edpage = Second Edition|ed = 2nd|prev = Integer Less One divides Power Less One|next = Integer Less One divides Power Less One/Corollary/Converse}}: $\S 2.4$: The rational numbers and some finite fields: Further Exercises $5$
Axiom:Axiom of Choice/Formulation 1
== Axiom ==  For every set of non-empty sets, it is possible to provide a mechanism for choosing one element of each element of the set. :$\displaystyle \forall s: \paren {\O \notin s \implies \exists \paren {f: s \to \bigcup s}: \forall t \in s: \map f t \in t}$  That is, one can always create a choice function for selecting one element from each element of the set.  == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Axiom:Axiom of Choice/Formulation 2|next = Equivalence of Versions of Axiom of Choice/Formulation 2 implies Formulation 1}}: $\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Definition:Choice Function|next = Axiom:Axiom of Choice/Historical Note}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents * {{BookReference|The Axiom of Choice|1973|Thomas J. Jech|next = Definition:Choice Function}}: $1.1$ The Axiom of Choice Choice 34dmh7pd26iip1t65a6xykrb6fqsnr2
Axiom:Axiom of Choice/Formulation 2
== Axiom ==  Let $\family {X_i}_{i \mathop \in I}$ be an indexed family of sets all of which are non-empty, indexed by $I$ which is also non-empty. Then there exists an indexed family $\family {x_i}_{i \mathop \in I}$ such that: :$\forall i \in I: x_i \in X_i$ That is, the Cartesian product of a non-empty family of sets which are non-empty is itself non-empty.  == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{BookReference|Naive Set Theory|1960|Paul R. Halmos|prev = Definition:Lexicographic Order|next = Axiom:Axiom of Choice/Formulation 1}}: $\S 15$: The Axiom of Choice * {{BookReference|Point Set Topology|1964|Steven A. Gaal|prev = Well-Ordering Theorem|next = Equivalence of Versions of Axiom of Choice}}: Introduction to Set Theory: $3$. The Axiom of Choice and Its Equivalents Choice s59iex6vzqt7v2z3rdl05apu6t2kaw0
Axiom:Axiom of Choice/Formulation 3
== Axiom ==  Let $\SS$ be a set of non-empty pairwise disjoint sets. Then there is a set $C$ such that for all $S \in \SS$, $C \cap S$ has exactly one element. Symbolically: :$\forall s: \paren {\paren {\O \notin s \land \forall t, u \in s: t = u \lor t \cap u = \O} \implies \exists c: \forall t \in s: \exists x: t \cap c = \set x}$  == Also see == * Equivalence of Versions of Axiom of Choice {{LinkToCategory|Axiom of Choice|the Axiom of Choice}} == Sources == * {{citation|date = 1908|title = Neuer Beweis für die Möglichkeit einer Wohlordnung|trans = A new proof of the possibility of well-ordering|journal = Mathematische Annalen|abbr = Math. Ann.|volume = 65|startpage = 107|endpage = 128|author = Ernst Zermelo}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Definition:Axiomatic System|next = Definition:Axis|entry = axiom of choice}} Choice aa2hjgpjv581bfh9pq3d6z8ccjnxs4g
Image of Empty Set is Empty Set/Corollary
== Corollary of Image of Empty Set is Empty Set ==  Let $f: S \to T$ be a mapping. The image of the empty set is the empty set: :$f \sqbrk \O = \O$  == Proof == By definition, a mapping is a relation. Thus Image of Empty Set is Empty Set applies.  == Sources == * {{BookReference|Undergraduate Topology|1971|Robert H. Kasriel|prev = Definition:Image of Subset under Mapping/Definition 1|next = Image of Domain of Mapping is Image Set}}: $\S 1.10$: Functions: Remark $10.8 \ \text{(a)}$
Axiom:Propositions of Incidence/Line in Plane
== Axiom ==  A line defined by any two distinct points in a plane lies entirely within that plane.  == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Axiom:Propositions of Incidence/Plane|next = Plane contains Infinite Number of Lines}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence
Axiom:Propositions of Incidence/Plane and Line
== Axiom ==  A plane and a (straight) line which does not lie in that plane have exactly one point in common.  That is, a point is completely determined by any two of the infinite number of distinct (straight) lines which intersect at that point. == Sources == * {{BookReference|Projective Geometry|1952|T. Ewan Faulkner|ed = 2nd|edpage = Second Edition|prev = Definition:Coplanar Points|next = Two Planes have Line in Common}}: Chapter $1$: Introduction: The Propositions of Incidence: $1.2$: The projective method: The propositions of incidence
Primitive of Reciprocal/Corollary 2
{{mergeto|Derivative of Natural Logarithm Function}} {{refactor|What would probably be best would be for this to be a corollary of the above page.}} == Corollary to Primitive of Reciprocal ==  :$\displaystyle \frac {\d} {\d x} \ln \size x = \frac 1 x$ for $x \ne 0$.  == Proof == Follows directly from Primitive of Reciprocal.  == Sources == * {{BookReference|Calculus|2005|Roland E. Larson|author2 = Robert P. Hostetler|author3 = Bruce H. Edwards|ed = 8th|edpage = Eighth Edition}}: $\S 5.1$
Comparison Test/Corollary
== Corollary to Comparison Test ==  Let $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ be a convergent series of positive real numbers. Let $\left \langle {a_n} \right \rangle$ be a sequence in sequence $\R$ or sequence in $\C$. Let $H \in \R$. Let $\exists M: \forall n > M: \left\vert {a_n}\right\vert \le H b_n$. Then the series $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges.  == Proof == Let $\epsilon > 0$. Then $\dfrac \epsilon H > 0$. As $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ converges, its tail tends to zero. So: : $\displaystyle \exists N: \forall n > N: \sum_{k \mathop = n+1}^\infty b_k < \frac \epsilon H$ Let $\left \langle s_n \right \rangle$ be the sequence of partial sums of $\displaystyle \sum_{n \mathop = 1}^\infty a_n$. Then $\forall n > m > \max \left\{{M, N}\right\}$: {{begin-eqn}} {{eqn | l = \left\vert{s_n - s_m}\right\vert       | r = \left\vert{\left({a_1 + a_2 + \cdots + a_n}\right) - \left({a_1 + a_2 + \cdots + a_m}\right)}\right\vert       | c =  }} {{eqn | r = \left\vert{a_{m+1} + a_{m+2} + \cdots + a_n}\right\vert       | c =  }} {{eqn | o = \le       | r = \left\vert { a_{m+1} } \right\vert + \left\vert { a_{m+2} } \right\vert + \cdots + \left\vert { a_n } \right\vert       | c = Triangle Inequality }} {{eqn | o = \le       | r = H b_{m+1} + H b_{m+2} + \cdots + H b_n       | c =  }} {{eqn | o = \le       | r = H \sum_{k \mathop = n+1}^\infty b_k }} {{eqn | o = <       | r = H \frac \epsilon H }} {{eqn | r = \epsilon       | c =  }} {{end-eqn}} So $\left \langle s_n \right \rangle$ is a Cauchy sequence and the result follows from: : Real Number Line is Complete Metric Space or: : Complex Plane is Complete Metric Space.  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Comparison Test|next = Ratio Test}}: $\S 6.15$
Sum of Sequence of Products of Consecutive Reciprocals/Corollary
== Corollary to Sum of Sequence of Products of Consecutive Reciprocals ==  :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = 1$  == Proof == From Sum of Sequence of Products of Consecutive Reciprocals: :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = \frac n {n + 1} = 1 - \frac 1 {n + 1}$ We have that: :$\dfrac 1 {n + 1} < \dfrac 1 n$ and that $\sequence {\dfrac 1 n}$ is a basic null sequence. Thus by the Squeeze Theorem: :$\displaystyle \lim_{n \mathop \to \infty} \sum_{j \mathop = 1}^n \frac 1 {j \paren {j + 1} } = 1$  == Sources == * {{BookReference|Mathematical Analysis: A Straightforward Approach|1977|K.G. Binmore|prev = Sum of Sequence of Products of Consecutive Reciprocals/Proof 2|next = Harmonic Series is Divergent/Proof 1}}: $\S 6.3$ * {{BookReference|Special Functions of Mathematics for Engineers|1992|Larry C. Andrews|ed = 2nd|edpage = Second Edition|prev = Definition:Divergent Series|next = Divergent Sequence may be Bounded}}: $\S 1.2$: Infinite Series of Constants: Example $1$
Newton's Method
 == Proof Technique == '''Newton's method''' is a method of solving an equation expressed as a real function for which there may be no convenient closed form solution. The derivative of the function has to be known in order to use '''Newton's method'''. Let the equation to be solved be of the form: :$y = \map f x$ Let the value of $x$ be required for a given $y$. Then an iterative improvement on an initial guess is of the form :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$ where $\map {f'} {x_1}$ is the derivative of $f$ {{WRT|Differentiation}} $x$ evaluated at $x_1$. == Proof == The function $\map f x$ can be expanded using Taylor's Theorem: :$\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1} + \dfrac 1 2 \map {f''} {x_1} \paren {x_2 - x_1}^2 + \dotsb$ As $x_2$ gets closer to $x_1$, this series can be truncated to: :$\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1}$ {{explain|Not sure but that assumptions about convergence need to be made?}} Let $x_\infty$ be the exact solution where: :$\map f {x_\infty} = y$ Let $\epsilon \in \R_{>0}$ be the difference from the new estimate to the solution: :$x_\infty = x_2 + \epsilon$ Then the function expanded around the new estimate is: :$y = \map f {x_2} + \map {f'} {x_2} \epsilon$ Solving for $x_2$ produces: :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} } - \epsilon \dfrac {\map {f'} {x_2} } {\map {f'} {x_1} }$ For $\epsilon$ small enough the final term can be neglected: :$x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$ {{explain|Greater linguistic and mathematical precision may be needed in the above statement.}}  == Example == {{:Newton's Method/Example}} {{Namedfor|Isaac Newton|cat = Newton}} == Sources == {{SourceReview|Expressed in the below in a slightly different form. Need to check whether it is equivalent to the one given here.}} * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Newton's Third Law of Motion|next = Mathematician:Jerzy Neyman|entry = Newton's method}}
Negated Upper Index of Binomial Coefficient/Corollary 1
== Corollary to Negated Upper Index of Binomial Coefficient ==  Let $r \in \R, k \in \Z$. Then: :$\dbinom {-r} k = \paren {-1}^k \dbinom {r + k - 1} k$ where $\dbinom {-r} k$ is a binomial coefficient.  == Proof 1 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 1}} == Proof 2 == {{:Negated Upper Index of Binomial Coefficient/Corollary 1/Proof 2}}
Set of Integer Combinations includes Zero
== Lemma == Let $a, b \in \Z$ be integers. Let $S = \set {a x + b y: x, y \in \Z}$ be the set of integer combinations of $a$ and $b$. Then $0 \in S$. == Proof == By setting $x = 0$ and $y = 0$: :$a \cdot 0 + b \cdot 0 = 0$  == Sources == * {{BookReference|Computability and Unsolvability|1982|Martin Davis|ed = 2nd|edpage = Second Edition|prev = Definition:Coprime Integers|next = Set of Integer Combinations includes those Integers}}: Appendix $1$: Some Results from the Elementary Theory of Numbers: Lemma $1$
Euler's Formula/Real Domain/Corollary
== Corollary to Euler's Formula: Real Domain == Let $\theta \in \R$ be a real number. Then:  :$e^{-i \theta} = \cos \theta - i \sin \theta$  where: : $e^{-i \theta}$ denotes the complex exponential function : $\cos \theta$ denotes the real cosine function : $\sin \theta$ denotes the real sine function : $i$ denotes the imaginary unit. == Proof == {{begin-eqn}} {{eqn | l = e^{-i \theta}       | r = \cos \paren {-\theta} + i \sin \paren {-\theta}       | c = Euler's Formula }} {{eqn | r = \cos \theta + i \sin \paren {-\theta}       | c = Cosine Function is Even }} {{eqn | r = \cos \theta - i \sin \theta       | c = Sine Function is Odd }} {{end-eqn}}  == Sources == * {{BookReference|Complex Numbers|1960|Walter Ledermann|prev = Definition:Exponential Form of Complex Number|next = Cosine Exponential Formulation/Real Domain/Proof 3}}: $\S 2$. Geometrical Representations * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Euler's Formula/Real Domain|next = Sine Exponential Formulation/Real Domain}}: $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.16$
Hyperbolic Sine of Sum/Corollary
== Corollary of Hyperbolic Sine of Sum ==  :$\map \sinh {a - b} = \sinh a \cosh b - \cosh a \sinh b$  where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \sinh {a - b}       | r = \sinh a \map \cosh {-b} + \cosh a \map \sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \sinh a \cosh b - \cosh a \sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $\S 8$: Hyperbolic Functions: $8.20$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Sine of Sum|next = Hyperbolic Cosine of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Hyperbolic Cosine of Sum/Corollary
== Corollary of Hyperbolic Cosine of Sum ==  :$\map \cosh {a - b} = \cosh a \cosh b - \sinh a \sinh b$  where $\sinh$ denotes the hyperbolic sine and $\cosh$ denotes the hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn | l = \map \cosh {a - b}       | r = \cosh a \map \cosh {-b} + \sinh a \map \sinh {-b}       | c = Hyperbolic Sine of Sum }} {{eqn | r = \cosh a \cosh b - \sinh a \sinh b       | c = Hyperbolic Cosine Function is Even and Hyperbolic Sine Function is Odd }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $\S 8$: Hyperbolic Functions: $8.21$: Addition Formulas * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Hyperbolic Cosine of Sum|next = Hyperbolic Tangent of Sum}}: $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
Double Angle Formulas/Hyperbolic Cosine/Corollary 1
== Corollary to Double Angle Formula for Hyperbolic Cosine ==  : $\cosh 2 x = 2 \cosh^2 x - 1$  where $\cosh$ denotes hyperbolic cosine. == Proof == {{begin-eqn}} {{eqn|l = \cosh 2 x      |r = \cosh^2 x + \sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \cosh^2 x + \left({\cosh^2 x - 1}\right)      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 2 \ \cosh^2 x - 1 }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formula for Hyperbolic Cosine|next=Double Angle Formulas/Hyperbolic Cosine/Corollary 2}}: $\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas
Double Angle Formulas/Hyperbolic Cosine/Corollary 2
== Corollary to Double Angle Formula for Hyperbolic Cosine ==  : $\cosh 2 x = 1 + 2 \sinh^2 x$  where $\cosh$ and $\sinh$ denote hyperbolic cosine and hyperbolic sine respectively. == Proof == {{begin-eqn}} {{eqn|l = \cosh 2 x      |r = \cosh^2 x + \sinh^2 x      |c = Double Angle Formula for Hyperbolic Cosine }} {{eqn|r = \left({1 + \sinh^2 x}\right) + \sinh^2 \theta      |c = Difference of Squares of Hyperbolic Cosine and Sine }} {{eqn|r = 1 + 2 \sinh^2 x }} {{end-eqn}}  == Sources == * {{BookReference|Mathematical Handbook of Formulas and Tables|1968|Murray R. Spiegel|prev=Double Angle Formulas/Hyperbolic Cosine/Corollary 1|next=Double Angle Formula for Hyperbolic Tangent}}: $\S 8$: Hyperbolic Functions: $8.25$: Double Angle Formulas
Complex Multiplication as Geometrical Transformation/Corollary
== Corollary to Complex Multiplication as Geometrical Transformation ==  Let $z = \polar {r, \theta}$ be a complex number expressed in polar form. Let $z$ be represented on the complex plane $\C$ in vector form. The effect of multiplying $z$ by $e^{i \alpha}$ is to rotate it about the origin of $\C$ by $\alpha$ in the positive direction  == Proof == From Complex Multiplication as Geometrical Transformation, the effect of multiplying a complex number by $r e^{i \alpha}$ is: :to rotate it about the origin of $\C$ by $\alpha$ in the positive direction :to multiply its modulus by $r$. In this instance $r = 1$. Hence the result.  == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Power Reduction Formulas/Cosine to 4th/Proof 2|next = Periodicity of Complex Exponential Function}}: $1$: Complex Numbers: Solved Problems: De Moivre's Theorem: $24$
Semantic Tableau Algorithm
== Algorithm ==  Let $\mathbf A$ be a WFF of propositional logic. The purpose of this algorithm is to create a completed semantic tableau $T$ for $\mathbf A$. :'''Step 1''': Start with a labeled tree $T$ comprising only a root node labeled with the singleton $\left\{{\mathbf A}\right\}$. :'''Step 2''': Choose a leaf node $t$ of $T$ that has not yet been marked. Let $U \left({t}\right)$ be the set that labels $t$. :'''Step 3''': If $U \left({t}\right)$ is a set of literals, mark $t$ as follows: ::'''a)''': If $U \left({t}\right)$ contains a complementary pair, mark it closed, $\times$, and go to '''Step 2'''. ::'''b)''': Otherwise, mark it open, $\odot$, and go to '''Step 2'''. :'''Step 4''': Choose a formula $\mathbf B \in U \left({t}\right)$ that is not a literal. Classify $\mathbf B$ as an $\alpha$-formula or a $\beta$-formula, and determine $\mathbf B_1, \mathbf B_2$ accordingly. ::'''a)''': If $\mathbf B$ is an $\alpha$-formula, add a child $t'$ to $t$, labeled $U \left({t'}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_1, \mathbf B_2}\right\}$. ::'''b)''': If $\mathbf B$ is a $\beta$-formula, add two children $t', t''$ to $t$ with labels: ::::$U \left({t'}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_1}\right\}$ ::::$U \left({t''}\right) = \left({U \left({t}\right) \setminus \left\{{\mathbf B}\right\}}\right) \cup \left\{{\mathbf B_2}\right\}$ :'''Step 5''': Go to '''Step 2''' if there remain leaves that are not marked. Otherwise, '''stop'''.  === Finiteness === The finiteness of this algorithm is demonstrated on Semantic Tableau Algorithm Terminates.  === Heuristics === {{:Semantic Tableau Algorithm/Heuristics}} == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed=3rd|edpage=Third Edition|prev=Definition:Beta-Formula|next=Definition:Marked Leaf}}: $\S 2.6.2$: Algorithm $2.64$
Quadruple Angle Formulas/Sine/Corollary 1
== Corollary to Quadruple Angle Formula for Sine ==  For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ : $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 \cos \theta$ where $\sin$ denotes sine and $\cos$ denotes cosine.  == Proof == First note that when $\theta = 0, \pm \pi, \pm 2 \pi \ldots$: :$\sin \theta = 0$ so $\dfrac {\sin 4 \theta} {\sin \theta}$ is undefined. Therefore for the rest of the proof it is assumed that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$ {{begin-eqn}} {{eqn | l = \sin 4 \theta       | r = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta       | c = Quadruple Angle Formula for Sine }} {{eqn | ll= \leadsto       | l = \dfrac {\sin 4 \theta} {\sin \theta}       | r = 4 \cos \theta - 8 \paren {1 - \cos^2 \theta} \cos \theta       | c = Sum of Squares of Sine and Cosine }} {{eqn | r = 8 \cos^3 \theta - 4 \cos \theta       | c = multiplying out and gathering terms }} {{end-eqn}}  == Sources == * {{BookReference|Theory and Problems of Complex Variables|1981|Murray R. Spiegel|ed = SI|edpage = SI (Metric) Edition|prev = Cosine of 72 Degrees|next = Quadruple Angle Formulas/Sine/Corollary 2}}: $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $93 \ \text{(a)}$ ::(although see Quadruple Angle Formulas/Sine/Mistake for analysis of an error in that work)
Soundness and Completeness of Semantic Tableaus/Corollary 2
== Corollary to Soundness and Completeness of Semantic Tableaus == Let $\mathbf A$ be a WFF of propositional logic. Then $\mathbf A$ is a tautology  $\neg \mathbf A$ has a closed tableau.  == Proof == By Tautology iff Negation is Unsatisfiable, $\mathbf A$ is a tautology  $\neg \mathbf A$ is unsatisfiable. By the Soundness and Completeness of Semantic Tableaus, this amounts to the existence of a closed tableau for $\neg \mathbf A$.  == Sources == * {{BookReference|Mathematical Logic for Computer Science|2012|M. Ben-Ari|ed = 3rd|edpage = Third Edition|prev = Soundness and Completeness of Semantic Tableaus/Corollary 1|next = Semantic Tableau Algorithm is Decision Procedure for Tautologies}}: $\S 2.7$: Corollary $2.69$
Exponential of Sum/Real Numbers/Corollary
== Corollary to Exponential of Sum: Real Numbers ==  Let $x, y \in \R$ be real numbers. Let $\exp x$ be the exponential of $x$. Then: :$\map \exp {x - y} = \dfrac {\exp x} {\exp y}$  == Proof == By Exponential of Sum: Real Numbers: :$\map \exp {x - y} = \exp x \, \map \exp {-y}$ {{improve|No real counterpart exists to link to}} By Reciprocal of Complex Exponential: :$\dfrac 1 {\exp y} = \map \exp {-y}$ Combining these two, we obtain the result: :$\map \exp {x - y} = \dfrac {\exp x} {\exp y}$