hcju/proofwikips huggingface dataset:
Inverse of Stirling's Triangle expressed as Matrix
Consider Stirling's triangle of the first kind (signed) expressed as a (square) matrix $\mathbf A$, with the top left element holding $s \left({0, 0}\right)$. :$\begin{pmatrix} 1 &     0 &       0 &      0 &      0 &     0 & \cdots \\ 0 &     1 &       0 &      0 &      0 &     0 & \cdots \\ 0 &    -1 &       1 &      0 &      0 &     0 & \cdots \\ 0 &     2 &      -3 &      1 &      0 &     0 & \cdots \\ 0 &    -6 &      11 &     -6 &      1 &     0 & \cdots \\ 0 &    24 &     -50 &     35 &    -10 &     1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots &  \ddots \\ \end{pmatrix}$ Thenconsider Stirling's triangle of the second kind expressed as a (square) matrix $\mathbf B$, with the top left element holding $\displaystyle \left \{ {0 \atop 0}\right\}$. :$\begin{pmatrix} 1 & 0 &   0 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   0 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   1 &    0 &    0 &    0 & \cdots \\ 0 & 1 &   3 &    1 &    0 &    0 & \cdots \\ 0 & 1 &   7 &    6 &    1 &    0 & \cdots \\ 0 & 1 &  15 &   25 &   10 &    1 & \cdots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{pmatrix}$ Then: : $\mathbf A = \mathbf B^{-1}$ that is: : $\mathbf B = \mathbf A^{-1}$
Reflection Rule for Gaussian Binomial Coefficients
Let $q \in \R_{\ne 1}, n \in \Z_{>0}, k \in \Z$. Then: :$\dbinom n k_q = q^{k \paren {n - k} } \dbinom n k_{q^{-1} }$ where $\dbinom n k_q$ is a Gaussian binomial coefficient.
Rooted Tree Corresponds to Arborescence
Let $T = \struct {V, E}$ be a rooted tree with root $r$. Then there is a unique orientation of $T$ which is an $r$-arborescence.
Equivalence of Definitions of Arborescence
Let $G = \struct {V, A}$ be a directed graph. Let $r \in V$. {{TFAE|def = Arborescence}}
Inductive Construction of Sigma-Algebra Generated by Collection of Subsets
Let $\EE$ be a set of sets which are subsets of some set $X$. Let $\map \sigma \EE$ be the $\sigma$-algebra generated by $\EE$. Then $\map \sigma \EE$  can be constructed inductively. The construction is as follows: Let $\Omega$ denote the minimal uncountable well-ordered set. Let $\alpha$ be an arbitrary initial segment in $\Omega$. Considering separately the cases whether or not $\alpha$ has an immediate predecessor $\beta$, we define: :$\EE_1 = \EE$ :$\EE_\alpha = \begin{cases} \set {\SS \in \powerset {\EE_\beta}: \SS \text { is countable or } \SS^\complement \text{ is countable} } & \alpha \text{ has an immediate predecessor } \beta \\ \ds \bigcup_{\beta \mathop \prec \alpha} \EE_\beta & \text { otherwise} \end{cases}$ :$\EE_\Omega = \ds \bigcup_{\alpha \mathop \in \Omega} \EE_\alpha$ Then $\map \sigma \EE = \EE_\Omega$.
Equivalence of Definitions of Set Partition
Let $S$ be a set {{TFAE|def = Set Partition}}
Summation to n of Reciprocal of k by k-1 of Harmonic Number
:$\displaystyle \sum_{1 \mathop < k \mathop \le n} \dfrac 1 {k \paren {k - 1} } H_k = 2 - \dfrac {H_n} n - \dfrac 1 n$ where $H_n$ denotes the $n$th harmonic number.
Riemann Zeta Function of 1000
To at least $100$ decimal places: :$\zeta \left({1000}\right) \approx 1$ where $\zeta$ denotes the Riemann zeta function.
Logarithmic Integral as Non-Convergent Series
The logarithmic integral can be defined in terms of a non-convergent series. That is: :$\displaystyle \operatorname {li} \left({z}\right) = \sum_{i \mathop = 0}^{+\infty} \frac {i! \, z} {\ln^{i + 1} z}  = \frac z {\ln z} \left({\sum_{i \mathop = 0}^{+\infty} \frac {i!} {\ln^i z} }\right)$
Bridge divides Graph into Two Components
Let $G$ be a connected graph. Let $e$ be a bridge of $G$. Then the edge deletion $G - e$ contains exactly $2$ components.
Connected Graph with only Even Vertices has no Bridge
Let $G$ be a connected graph whose vertices are all even. Then no edge of $G$ is a bridge.
Cut-Vertex divides Graph into Two or More Components
Let $G$ be a graph. Let $v$ be a cut-vertex of $G$. Then the vertex deletion $G - v$ contains $2$ or more components.
Median to Hypotenuse of Right Triangle equals Half Hypotenuse
Let $\triangle ABC$ be a right triangle such that $BC$ is the hypotenuse. Let $AD$ be the median to $BC$. Then $AD$ is half of $BC$.
Summation to n of Square of kth Harmonic Number
:$\displaystyle \sum_{k \mathop = 1}^n {H_k}^2 = \paren {n + 1} {H_n}^2 - \paren {2 n + 1} H_n + 2 n$ where $H_k$ denotes the $k$th harmonic number.
1+1 = 2
Define $0$ as the only element in the set $P \setminus s \left({P}\right)$, where: :$P$ is the Peano Structure :$s \left({P}\right)$ is the image of the mapping $s$ defined in Peano structure : $\setminus$ denotes the set difference. The theorem to be proven is: :$1 + 1 = 2$ where: :$1 := s \left({0}\right)$ :$2 := s \left({1}\right) = s \left({s \left({0}\right)}\right)$ :$+$ denotes addition :$=$ denotes equality :$s \left({n}\right)$ denotes the successor function as defined by Peano
Highest Power of 2 Dividing Numerator of Sum of Odd Reciprocals
Let: : $S = \dfrac p q = \displaystyle \sum_{k \mathop = 1}^n \dfrac 1 {2 k - 1}$ where $\dfrac p q$ is the canonical form of $S$. Let $n = 2^k m$ where $m$ is odd. Then the largest power of $2$ that divides $p$ is $2^{2 k}$.
Two Paths between Vertices in Cycle Graph
Let $G$ be a simple graph. Let $u, v$ be vertices in $G$ such that $u \ne v$. Then: :for any two vertices $u, v$ in $G$ such that $u \ne v$ there exists exactly two paths between $u$ and $v$ : :$G$ is a cycle graph.
Value of Radian in Degrees
The value of a radian in degrees is given by: :$1 \radians = \dfrac {180 \degrees} {\pi} \approx 57.29577 \ 95130 \ 8232 \ldots \degrees$  {{OEIS|A072097}}
Cassini's Identity/Negative Indices
Let $n \in \Z_{<0}$ be a negative integer. Let $F_n$ be the $n$th Fibonacci number (as extended to negative integers). Then Cassini's Identity: :$F_{n + 1} F_{n - 1} - F_n^2 = \left({-1}\right)^n$ continues to hold.
Sum over k of n Choose k by Fibonacci Number with index m+k
:$\displaystyle \sum_{k \mathop \ge 0} \binom n k F_{m + k} = F_{m + 2 n}$ where: : $\dbinom n k$ denotes a binomial coefficient : $F_n$ denotes the $n$th Fibonacci number.
Area of Regular Polygon
Let $P$ be a regular $n$-sided polygon whose side length is $b$. Then the area of $P$ is given by: :$\Box P = \dfrac 1 4 n b^2 \cot \dfrac \pi n$ where $\cot$ denotes cotangent.
Fibonacci Number whose Index is Plus or Minus Prime p is Multiple of p
Let $p$ be a prime number distinct from $5$. Let $F_n$ denote the $n$th Fibonacci number. Then either $F_{p - 1}$ or $F_{p + 1}$ (but not both) is a multiple of $p$.
Perimeter of Regular Polygon by Circumradius
Let $P$ be a regular $n$-gon. Let $C$ be a circumcircle of $P$. Let the radius of $C$ be $r$. Then the perimeter $\mathcal P$ of $P$ is given by: :$\mathcal P = 2 n r \sin \dfrac \pi n$
Area of Regular Polygon by Inradius
Let $P$ be a regular $n$-gon. Let $C$ be an incircle of $P$. Let the radius of $C$ be $r$. Then the area $\AA$ of $P$ is given by: :$\AA = n r^2 \tan \dfrac \pi n$
Zeckendorf Representation of Integer shifted Right
Let $f \left({x}\right)$ be the real function defined as: :$\forall x \in \R: f \left({x}\right) = \left\lfloor{x + \phi^{-1} }\right\rfloor$ where: :$\left\lfloor{\, \cdot \,}\right\rfloor$ denotes the floor function :$\phi$ denotes the golden mean. Let $n \in \Z_{\ge 0}$ be a positive integer. Let $n$ be expressed in Zeckendorf representation: :$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$ with the appropriate restrictions on $k_1, k_2, \ldots, k_r$. Then: :$F_{k_1 - 1} + F_{k_2 - 1} + \cdots + F_{k_r - 1} = f \left({\phi^{-1} n}\right)$
Linear Combination of Generating Functions
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. and $H \left({z}\right)$ be the generating function for the sequence $\left\langle{b_n}\right\rangle$. Then $\alpha G \left({z}\right) + \beta H \left({z}\right)$ is the generating function for the sequence $\left\langle{\alpha a_n + \beta b_n}\right\rangle$.
Semigroup is Group Iff Latin Square Property Holds
Let $\left({S, \circ}\right)$ be a semigroup. Then $\left({S, \circ}\right)$ is a group  for all $a, b \in S$ the Latin square property holds in $S$: :$a \circ x = b$ :$y \circ a = b$ for $x$ and $y$ each unique in $S$.
Generating Function for Even Terms of Sequence
Let $G \left({z}\right)$ be the generating function for the sequence $\left\langle{a_n}\right\rangle$. Consider the subsequence $\left\langle{b_n}\right\rangle := \left({a_0, a_2, a_4, \ldots}\right)$ Then the generating function for $\left\langle{b_n}\right\rangle$ is: :$\dfrac 1 2 \left({G \left({z}\right) + G \left({-z}\right)}\right)$
Ultraproduct is Well-Defined
'''Definition:Ultraproduct is well-defined.''' More specificly, following the definitions on Definition:Ultraproduct, we are going to prove that: :(1) $f^\mathcal M$ is well-defined :(2) $R^\mathcal M$ is well-defined
Equation of Circular Arc in Complex Plane
Let $a, b \in \C$ be complex constants representing the points $A$ and $B$ respectively in the complex plane. Let $z \in \C$ be a complex variable representing the point $Z$ in the complex plane. Let $\lambda \in \R$ be a real constant such that $-\pi < \lambda < \pi$. Then the equation: :$\arg \dfrac {z - b} {z - a} = \lambda$ represents the arc of a circle with $AB$ as a chord subtending an angle $\lambda$ at $Z$ on the circumference.
Exponential of Complex Number plus 2 pi i
:$\map \exp {z + 2 \pi i} = \map \exp z$
Sextuple Angle Formula for Sine
:$\dfrac {\map \sin {6 \theta} } {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$
Real Complex Roots of Unity for Odd Index
Let $n \in \Z_{>0}$ be a (strictly) positive integer such that $n$ is odd. Let $U_n = \set {z \in \C: z^n = 1}$ be the set of complex $n$th roots of unity. The only $x \in U_n$ such that $x \in \R$ is: :$x = 1$ That is, $1$ is the only complex $n$th root of unity which is a real number.
Unit Vectors in Complex Plane which are Vertices of Equilateral Triangle
Let $\epsilon_1, \epsilon_2, \epsilon_3$ be complex numbers embedded in the complex plane such that: :$\epsilon_1, \epsilon_2, \epsilon_3$ all have modulus $1$ :$\epsilon_1 + \epsilon_2 + \epsilon_3 = 0$ Then: :$\paren {\dfrac {\epsilon_2} {\epsilon_1} }^3 = \paren {\dfrac {\epsilon_3} {\epsilon_2} }^2 = \paren {\dfrac {\epsilon_1} {\epsilon_3} }^2 = 1$
Factorisation of x^(2n)-1 in Real Domain
Let $n \in \Z_{>0}$ be a (strictly) positive integer. Then: :$z^{2 n} - 1 = \paren {z - 1} \paren {z + 1} \displaystyle \prod_{k \mathop = 1}^n \paren {z^2 - 2 \cos \dfrac {k \pi} n + 1}$
4 Sine Pi over 10 by Cosine Pi over 5
:$4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$
Secant of Complement equals Cosecant
:$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$ where $\sec$ and $\csc$ are secant and cosecant respectively. That is, the cosecant of an angle is the secant of its complement. This relation is defined wherever $\sin \theta \ne 0$.
Cosecant of Complement equals Secant
:$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$ where $\csc$ and $\sec$ are cosecant and secant respectively. That is, the secant of an angle is the cosecant of its complement. This relation is defined wherever $\cos \theta \ne 0$.
Secant of Supplementary Angle
:$\map \sec {\pi - \theta} = -\sec \theta$ where $\sec$ denotes secant. That is, the secant of an angle is the negative of its supplement.
Absolutely Convergent Series is Convergent/Complex Numbers
Let $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ be an absolutely convergent series in $\C$. Then $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ is convergent.
Existence of Radius of Convergence of Complex Power Series/Absolute Convergence
Let $B_R \paren \xi$ denote the open $R$-ball of $\xi$. Let $z \in B_R \paren \xi$. Then $S \paren z$ converges absolutely. If $R = +\infty$, we define $B_R \paren \xi = \C$.
Imaginary Part of Complex Exponential Function
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\exp z$ denote the complex exponential function. Then: :$\Im \paren {\exp z} = e^x \sin y$ where: :$\Im z$ denotes the imaginary part of a complex number $z$ :$e^x$ denotes the real exponential function of $x$ :$\sin y$ denotes the real sine function of $y$.
Modulus of Sine of Complex Number
Let $z = x + i y \in \C$ be a complex number, where $x, y \in \R$. Let $\sin z$ denote the complex sine function. Then: :$\cmod {\sin z} = \sqrt {\sin^2 x + \sinh^2 y}$ where: :$\cmod z$ denotes the modulus of a complex number $z$ :$\sin x$ denotes the real sine function :$\sinh$ denotes the hyperbolic sine function.
Bounds for Modulus of e^z on Circle x^2 + y^2 - 2x - 2y - 2 = 0
Consider the circle $C$ embedded in the complex plane defined by the equation: :$x^2 + y^2 - 2 x - 2 y - 2 = 0$ Let $z = x + i y \in \C$ be a point lying on $C$. Then: :$e^{-1} \le \cmod {e^z} \le e^3$
Secant of Angle plus Three Right Angles
:$\map \sec {x + \dfrac {3 \pi} 2} = \csc x$
Cosecant of Angle plus Three Right Angles
:$\map \csc {x + \dfrac {3 \pi} 2} = -\sec x$
Secant of Three Right Angles less Angle
:$\map \sec {\dfrac {3 \pi} 2 - \theta} = -\csc \theta$ where $\sec$ and $\csc$ are secant and cosecant respectively.
Bell Number as Summation over Lower Index of Stirling Numbers of the Second Kind/Corollary
:$B_n = \displaystyle \sum_{k \mathop = 1}^n {n \brace k}$
Structure Induced by Commutative Ring Operations is Commutative Ring
Let $\struct {R, +, \circ}$ be a commutative ring. Let $S$ be a set. Let $\struct {R^S, +', \circ'}$ be the structure on $R^S$ induced by $+'$ and $\circ'$. Then $\struct {R^S, +', \circ'}$ is a commutative ring.
Equivalence Relation on Power Set induced by Intersection with Subset/Cardinality of Set of Equivalence Classes
Let $A$ be finite with $\card A = n$, where $\card {\, \cdot \,}$ denotes cardinality. The cardinality of the set of $\alpha$-equivalence classes is given by: :$\card {\set {\eqclass X \alpha: X \in S} } = 2^n$
Cosine to Power of Even Integer
{{begin-eqn}} {{eqn | l = \cos^{2 n} \theta       | r = \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \paren {\cos 2 n \theta + \binom {2 n} 1 \map \cos {2 n - 2} \theta + \cdots + \binom {2 n} {n - 1} \cos 2 \theta}       | c =  }} {{eqn | r = \frac 1 {2^{2 n} } \binom {2 n} n + \frac 1 {2^{2 n - 1} } \sum_{k \mathop = 0}^{n - 1} \binom {2 n} k \map \cos {2 n - 2 k} \theta       | c =  }} {{end-eqn}}
Law of Tangents
Let $\triangle ABC$ be a triangle whose sides $a, b, c$ are such that $a$ is opposite $A$, $b$ is opposite $B$ and $c$ is opposite $C$. Then: :$\dfrac {a + b} {a - b} = \dfrac {\tan \frac 1 2 \paren {A + B} } {\tan \frac 1 2 \paren {A - B} }$
Functionally Complete Singleton Sets
The only binary logical connectives that form singleton sets which are functionally complete are NAND and NOR.