hcju/proofwikips huggingface dataset:
Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 2
$f$ is '''differentiable at $x$'''  there exists a linear transformation $T: \R^n \to \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that: :$(1):\quad \map f {x + h} = \map f x + \map T h + \map r h \cdot h$ :$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$
Definition:Differentiable Mapping/Real-Valued Function/Point/Definition 1
$f$ is '''differentiable at $x$'''  there exist $\alpha_1, \ldots, \alpha_n \in \R$ and a real-valued function $r: U \setminus \set x \to \R$ such that: :$(1):\quad \map f {x + h} = \map f x + \alpha_1 h_1 + \cdots + \alpha_n h_n + \map r h\cdot h$ :$(2):\quad \displaystyle \lim_{h \mathop \to 0} \map r h = 0$
Definition:Differential/Real-Valued Function/Point
Let $U \subset \R^n$ be an open set. Let $f: U \to \R$ be a real-valued function. Let $f$ be differentiable at a point $x \in U$. :$\displaystyle \d f \left({x; h}\right) := \sum_{i \mathop = 1}^n \frac {\partial f \left({x}\right)} {\partial x_i} h_i = \frac {\partial f \left({x}\right)} {\partial x_1} h_1 + \frac {\partial f \left({x}\right)} {\partial x_2} h_2 + \cdots + \frac {\partial f \left({x}\right)} {\partial x_n} h_n$ where: : $h = \left({h_1, h_2, \ldots, h_n}\right) \in \R^n$ : $\dfrac {\partial f} {\partial x_i}$ is the partial derivative of $f$ {{WRT|Differentiation}} $x_i$.
Definition:Differential/Vector-Valued Function/Point
Let $U \subset \R^n$ be an open set. Let $f: U \to \R^m$ be a vector-valued function. Let $f$ be differentiable at a point $x \in U$. The '''differential of $f$ at $x$''' is the linear transformation $\d f \left({x}\right): \R^n \to \R^m$ defined as: :$\d f \left({x}\right) \left({h}\right) = J_f \left({x}\right) \cdot h$ where: : $J_f \left({x}\right)$ is the Jacobian matrix of $f$ at $x$.
Definition:Derivative/Vector-Valued Function/Point
Let $U \subset \R$ be an open set. Let $\mathbf f \left({x}\right) = \displaystyle \sum_{k \mathop = 1}^n f_k \left({x}\right) \mathbf e_k: U \to \R^n$ be a vector-valued function. Let $\mathbf f$ be differentiable at $u \in U$. That is, let each $f_j$ be differentiable at $u \in U$. The '''derivative of $\mathbf f$ {{WRT|Differentiation}} $x$ at $u$''' is defined as :$\dfrac {\d \mathbf f} {\d x} \left({u}\right) = \displaystyle \sum_{k \mathop = 1}^n \dfrac {\d f_k} {\d x} \left({u}\right) \mathbf e_k$ where $\dfrac {\d f_k} {\d x} \left({u}\right)$ is the derivative of $f_k$ with respect to $x$ at $u$.
Definition:Lipschitz Continuity/Point
Let $a\in A$. $f$ is a '''Lipschitz continuous at $a$'''  there exists a positive real number $K \in \R_{\ge 0}$ such that: :$\forall x \in A: d\,' \left({f \left({x}\right), f \left({a}\right)}\right) \le K d \left({x, a}\right)$
Definition:Differentiable Mapping between Manifolds
Let $M$ and $N$ be differentiable manifolds. Let $f : M \to N$ be continuous. === Definition 1 === $f$  is '''differentiable'''  for every pair of charts $(U, \phi)$ and $(V,\psi)$ of $M$ and $N$: :$\psi\circ f\circ \phi^{-1} : \phi ( U \cap f^{-1}(V)) \to \psi(V)$ is differentiable. === Definition 2 === $f$  is '''differentiable'''  $f$ is  differentiable at every point of $M$. === At a Point === {{:Definition:Differentiable Mapping between Manifolds/Point}}
Definition:Differentiable Mapping between Manifolds/Point
Let $M$ and $N$ be differentiable manifolds. Let $f: M \to N$ be continuous. Let $p \in M$. ==== Definition 1 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 1}} ==== Definition 2 ==== {{:Definition:Differentiable Mapping between Manifolds/Point/Definition 2}}
Definition:Differentiable Mapping between Manifolds/Point/Definition 1
$f$ is '''differentiable at $p$'''  for every pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$: :$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$ is differentiable at $\phi \left({p}\right)$.
Definition:Differentiable Mapping between Manifolds/Point/Definition 2
$f$ is '''differentiable at $p$'''  there exists a pair of charts $\left({U, \phi}\right)$ and $\left({V, \psi}\right)$ of $M$ and $N$ with $p \in U$ and $f \left({p}\right) \in V$ such that: :$\psi \circ f \circ \phi^{-1}: \phi \left({U \cap f^{-1} \left({V}\right)}\right) \to \psi \left({V}\right)$ is differentiable at $\phi \left({p}\right)$.
Definition:Injective Space
Let $Z = \left({S, \tau_1}\right)$ be a topological space. Then $Z$ is '''injective (space)'''  :for all topological spaces $X = \left({H, \tau_2}\right)$ :and for all continuous mappings $f:H \to S$ :and for all topological spaces $Y = \left({T, \tau_3}\right)$ such that $X$ is topological subspace of $Y$: :there exists a continuous mapping $g:T \to S$: $g \restriction H = f$
Definition:Wieferich Prime
A '''Wieferich prime''' is a prime number $p$ such that: :$p^2 \divides 2^{p − 1} − 1$ where $\divides$ denotes divisibility.
Definition:Wieferich Prime/Sequence
The sequence of Wieferich primes begins: :$1093, 3511 \ldots$ No other Wieferich primes exist below $4 \cdotp 968543 \times 10^{17}$.
Definition:Fermat's Equation
'''Fermat's equation''' is the Diophantine equation: :$x^n + y^n = z^n$
Definition:Retraction (Topology)
Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $T_2 = \left({S_2, \tau_2}\right)$ be a topological subspace of $T_1$. It means that :$S_2 \subseteq S_1$ Let $f: S_1 \to S_2$ be a mapping. Then $f$ is '''retraction''' of $T_1$  :$\forall s \in S_2: f\left({s}\right) = s$
Definition:Retract (Topology)
Let $T_1 = \left({S_1, \tau_1}\right)$ be a topological space. Let $T_2 = \left({S_2, \tau_2}\right)$ be a topological subspace of $T_1$. Then $T_2$ is a '''retract''' of $T_1$  :there exists a continuous retraction $f: S_1 \to S_2$ of $T_1$.
Definition:Sphenic Number
A '''sphenic number''' is a (positive) integer which has exactly $3$ distinct prime factors all with multiplicity of $1$.
Definition:Imperial/Volume/Cubic Inch
:$1$ '''cubic inch''' $= 16 \cdotp 38706 \, 4$ cubic centimetres $= 0 \cdotp 01638 \, 7064$ litres. This definition is exact, and is derived from the definition of the imperial unit of length as $2 \cdotp 54$ centimetres.
Definition:Imperial/Volume/Cubic Foot
The '''cubic foot''' is the FPS unit of measurement of volume. :$1$ '''cubic foot''' $= 12^3 = 1728$ cubic inches $= 28 \cdotp 31684 \, 6592$ litres $= 0 \cdotp 02831 \, 68465 \, 92$ cubic metres.
Definition:Taxicab Number
A '''taxicab number''' is a positive integer which can be expressed as the sum of $2$ cubes in $2$ different ways.
Definition:Image of Topological Space
Let $T = \left({S, \tau}\right)$ and $Q = \left({X, \tau'}\right)$ be topological spaces. Let $f:S \to X$ be a mapping. Then '''image (of the topological space $T$) of''' $f$ is equal to :$\operatorname{Im}\left({f}\right) := Q_{f\left[{S}\right]} = \left({f\left[{S}\right], \tau'_{f\left[{S}\right]} }\right)$ where $\tau'_{f\left[{S}\right]}$ denotes the subspace topology on $f\left[{S}\right]$.
Definition:Mile
There are a number of definitions for the '''mile'''. === International Mile === {{:Definition:Imperial/Length/International Mile}} === Nautical Mile === {{:Definition:Nautical Mile}} === Admiralty Mile === {{:Definition:Admiralty Mile}}
Definition:Weak Retract (Topology)
Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces. Then $T_1$ is '''a weak retract of''' $T_2$  :there exists a continuous mapping $f:S_2 \to S_2$: $f \circ f = f$ and $\operatorname{Im}\left({f}\right), T_1$ are homeomorphic.
Definition:Lower Topology
Let $T = \left({S, \preceq, \tau}\right)$ be a relational structure with topology. Then $T$ has '''lower topology'''  :$\left\{ {\complement_S\left({x^\succeq}\right): x \in S}\right\}$ is sub-basis of $T$ where $x^\succeq$ denotes the upper closure of $x$.
Definition:Ball
=== Open Ball === {{:Definition:Open Ball}} === Closed Ball === {{:Definition:Closed Ball}} === Unit Ball === {{:Definition:Unit Ball}}
Definition:Homogeneous Cyclotomic Polynomial
Let $n\geq1$ be a positive integer. The $n$th '''homogeneous cyclotomic polynomial''' is the homogenization of the $n$th cyclotomic polynomial.
Definition:Special Highly Composite Number
A '''special highly composite number''' is a highly composite number which is a divisor of all larger highly composite numbers.
Definition:Special Highly Composite Number/Sequence
The sequence of special highly composite numbers consists of: :$1, 2, 6, 12, 60, 2520$ {{OEIS|A106037}}
Definition:Modified Kaprekar Mapping
The '''modified Kaprekar mapping''' is the arithmetic function $K: \Z_{>0} \to \Z_{>0}$ defined on the positive integers  as follows: Let $n \in \Z_{>0}$ be expressed in some number base $b$ (where $b$ is usually $10$). Let $n'$ be the positive integer created by: :arranging the digits of $n$ into descending order of size then: :exchanging the last two digits. Let $n''$ be the positive integer created by: :arranging the digits of $n$ into ascending order of size then: :exchanging the first two digits. Then: :$\map {K'} n = n' - n''$ making sure to retain any leading zeroes to ensure that $\map K n$ has the same number of digits as $n$.
Definition:Modified Kaprekar Process
The '''modified Kaprekar process''' is the repeated application of the '''modified Kaprekar mapping''' to a given positive integer.
Definition:Central Tree
A tree $T$ is '''central'''  it has a center.
Definition:Bicentral Tree
A tree $T$ is '''bicentral'''  it has a bicenter.
Definition:Perfect Digit-to-Digit Invariant
A '''perfect digit-to-digit invariant''' is a number $n$ which is equal to the sum of the digits of $n$ each raised to the power of itself.
Definition:Perfect Digital Invariant
Let $n \in \Z_{>0}$ be a positive integer. $n$ is a '''perfect digital invariant'''  when expressed in decimal notation the $n$th powers of its digits add up to some positive integer $m$.
Definition:Imperial/Area/Acre
One '''acre''' is equal to an oblong measuring $1$ chain by $1$ furlong. That is, measuring $4$ rods, poles or perches by $10$ chains. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''acre''' }} {{eqn | r = 4       | c = roods }} {{eqn | r = 10       | c = square chains }} {{eqn | r = 160 = 4 \times 40       | c = square rods, poles or perches }} {{eqn | r = 4840 = 22 \times 220       | c = square yards }} {{end-eqn}} :810px
Definition:Imperial/Area/Square Yard
One '''square yard''' is equal to a square of side $1$ yard in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square yard''' }} {{eqn | r = 9 = 3^2       | c = square feet }} {{eqn | r = 1296 = 36^2       | c = square inches }} {{end-eqn}}
Definition:Imperial/Area/Square Inch
One '''square inch''' is equal to a square of side $1$ inch in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square inch''' }} {{eqn | r = 6 \cdotp 4516 = 2 \cdotp 54^2       | c = square centimetres }} {{end-eqn}} The definition is exact.
Definition:Imperial/Area/Square Foot
One '''square foot''' is equal to a square of side $1$ foot in length. {{begin-eqn}} {{eqn | o =        | r = 1       | c = '''square foot''' }} {{eqn | r = 144 = 12^2       | c = square inches }} {{end-eqn}}
Definition:Kaprekar's Constant
Let $n \in \Z_{>0}$ be a positive integer with $4$ digits, not all of which are equal. Let Kaprekar's process be applied to $n$. The number reached after no more than $8$ iterations is known as '''Kaprekar's constant''', and has the value of $6174$. {{NamedforDef|Dattathreya Ramchandra Kaprekar|cat = Kaprekar}}
Definition:Tri-Automorphic Number
A '''tri-automorphic number''' is a positive integer $n$ such that $3 n^2$ ends in a repetition of $n$.
Definition:Support of Module
Let $A$ be a commutative ring with unity. Let $M$ be a unitary $A$-module. The '''support''' $\operatorname{Supp}(M)$ of $M$ is the set of prime ideals $P$ of $A$ such that the localization of $M$ at $P$ is nonzero: :$\operatorname{Supp}(M) = \{P\in \operatorname{Spec}(A) : M_P\neq0 \}$ where $\operatorname{Spec}(A)$ is the spectrum of $A$.
Definition:Integral Ring Extension
Let $A$ be a commutative ring with unity. Let $R\subset A$ be a subring. The ring extension $R \subseteq A$ is said to be '''integral'''  for all $a \in A$, $a$ is integral over $R$.
Definition:Support of Mapping to Algebraic Structure
=== Real-Valued Function on an Abstract Set === {{Definition:Support of Mapping to Algebraic Structure/Real-Valued Function}} === General Real-Valued Function in $\R^n$ === === General Algebraic Structure === Let $(A, *)$ be an algebraic structure with an identity element $e$. Let $S$ be a set. Let $f : S \to A$ be a mapping. The '''support of $f$''' is the set: :$\operatorname{supp}(f) = \{s \in S : f(s) \neq e\}$
Definition:Weakly Locally Compact Space
Let $T = \struct {S, \tau}$ be a topological space. Then $T$ is '''weakly locally compact'''  every point of $S$ has a compact neighborhood.
Definition:Noetherian Ring/Definition 1
A commutative ring with unity $A$ is '''Noetherian'''  every ideal of $A$ is finitely generated.
Definition:Noetherian Ring/Definition 2
A commutative ring with unity $A$ is '''Noetherian'''  it satisfies the ascending chain condition on ideals.
Definition:Noetherian Ring/Definition 3
A commutative ring with unity $A$ is '''Noetherian'''  it satisfies the maximal condition on ideals.
Definition:Noetherian Ring/Definition 4
A commutative ring with unity $A$ is '''Noetherian'''  it is Noetherian as an $A$-module.
Definition:Noetherian Module/Definition 1
$M$ is a '''Noetherian module'''  every submodule of $M$ is finitely generated.
Definition:Noetherian Module/Definition 2
$M$ is a '''Noetherian module'''  it satisfies the ascending chain condition on submodules.
Definition:Noetherian Module/Definition 3
$M$ is a '''Noetherian module'''  it satisfies the maximal condition on submodules.
Definition:Sylow p-Subgroup/Definition 1
Let $\order G = k p^n$ where $p \nmid k$. A '''Sylow $p$-subgroup''' is a $p$-subgroup of $G$ which has $p^n$ elements.
Definition:Sylow p-Subgroup/Definition 2
A '''Sylow $p$-subgroup''' of $G$ is a '''maximal $p$-subgroup''' $P$ of $G$. In this context, maximality means that if $Q$ is a $p$-subgroup of $G$ and $P \le Q$, then $P = Q$.
Definition:Cover of Set/Subset
Let $A \subseteq S$ be a subset. Let $\mathcal C$ be a set of subsets of $S$. Then $\mathcal C$ is a '''cover of $A$'''  $A \subseteq \displaystyle \bigcup \mathcal C$, where $\cup$ denotes union.
Definition:Open Cover/Subset
Let $H$ be a subset of $S$. Let $\CC$ be a cover of $H$. Then $\CC$ is an '''open cover for $H$''' : :$\CC \subseteq \tau$ That is,  all the elements of $\CC$ are open sets.
Definition:Prime Number Race
The set of prime numbers $\Bbb P$ may be partitioned into subsets according to a particular property. A '''prime number race''' is a comparison of the count of the number of prime numbers in each partition with increasing $p \in \Bbb P$.
Definition:Pandigital Set
A set of integers is '''pandigital''' if it contains exactly $1$ instance of each digit. The digits may or may not contain zero, depending on context.
Definition:Pandigital Set/Integer
The element of a singleton pandigital set is itself referred to as a '''pandigital integer''' or '''pandigital number'''.
Definition:Ackermann-Péter Function
The '''Ackermann-Péter function''' $A: \Z_{\ge 0} \times \Z_{\ge 0} \to \Z_{> 0}$ is an integer-valued function defined on the set of ordered pairs of positive integers as: :$A \left({x, y}\right) = \begin{cases} y + 1 & : x = 0 \\ A \left({x - 1, 1}\right) & : x > 0, y = 0 \\ A \left({x - 1, A \left({x, y - 1}\right)}\right) & : \text{otherwise} \end{cases}$
Definition:Fermat Number/Naming Conventions
The '''Fermat number''' $F_0$ is often referred to as the ''' $1$st Fermat number''', so (confusingly) this convention dictates that $F_n$ is the '''$n + 1$th Fermat number'''. However, another convention is that $F_0$ can be referred to as the '''zeroth Fermat number''', thus bringing the appellation in line such that $F_n$ is the '''$n$th Fermat number'''. Both conventions are in place, sometimes in the same work. For example, {{AuthorRef|David Wells}}, in his {{BookLink|Curious and Interesting Numbers|David Wells|ed = 2nd|edpage = Second Edition}} of $1997$, refers to $5 = F_1$ in Section $5$ as the '''$2$nd Fermat number'''. However, in Section $257$ he defines $F_3 = 2^{2^3} + 1 = 257$ as the '''$3$rd Fermat number'''. Similarly, in Section $65,537$ he defines $F_4 = 2^{2^4} + 1 = 65 \, 537$ as the '''$4$th Fermat number''', and so on. Both of these naming conventions is more or less clumsy.  takes the position that the cat has to jump one way or the other, and so uses the second of these conventions: :$F_n$ is the '''$n$th Fermat number'''.
Definition:Unitary Divisor
Let $n \in \Z$ be an integer. Let $d \in \Z$ be such that: :$d$ is a divisor of $n$ :$d$ and $\dfrac n d$ are coprime. Then $d$ is a unitary divisor of $n$.
Definition:Unitary Perfect Number
A '''unitary perfect number''' is a positive integer which equals the sum of its positive unitary divisors apart from itself.
Definition:Prime Quadruple
A '''prime quadruple''' is a set of $4$ prime numbers of the form $n$, $n + 2$, $n + 6$, $n + 8$. That is, they are two pairs of twin primes separated by $4$.
Definition:Propositional Tableau/Construction/Infinite
An infinite labeled tree $\left({T, \mathbf H, \Phi}\right)$ is a '''propositional tableau''' : :There exists an exhausting sequence of sets $\left({T_n}\right)_{n \in \N}$ of $T$ such that for all $n \in \N$: ::$\left({T_n, \mathbf H, \Phi \restriction_{T_n}}\right)$ :is a finite propositional tableau, where $\Phi \restriction_{T_n}$ denotes the restriction of $\Phi$ to $T_n$.
Definition:Cyclic Number
A '''cyclic number''' is an integer $n$ whose cyclic permutations of its digits are integer multiples or integer divisors of $n$.
Definition:Sorted Number
A '''sorted number''' is an integer whose digits form a non-decreasing sequence, for example: :$1 \, 123 \, 334 \, 899$
Definition:Rare Number
A '''rare number''' is a non-palindromic integer $n$ which has the property that $n + r$ and $n - r$ are both square, where $r$ is the reversal of $n$.
Definition:Two-Sided Prime
A '''two-sided prime''' is a prime number which remains prime when: :any number of digits are removed from the left hand end and: :any number of digits are removed from the right hand end but, generally, not from both ends at once.
Definition:Cunningham Chain
There are $2$ types of '''Cunningham chain''': === First Kind === {{:Definition:Cunningham Chain/First Kind}} === Second Kind === {{:Definition:Cunningham Chain/Second Kind}}
Definition:Undecidable Statement
An '''undecidable statement''' is a statement which, in the context of a given deductive apparatus, cannot be assigned a definitive truth value.
Definition:Equivalence of Norms
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be norms on a vector space $V$. $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ are '''equivalent'''  there exist real constants $c$ and $C$ such that: :$\forall \mathbf x \in V: c \norm {\mathbf x}_1 \le \norm {\mathbf x}_2 \le C \norm {\mathbf x}_1$
Definition:Language of Propositional Logic/Basson-O'Connor
There are many formal languages expressing propositional logic. The formal language used on  is defined on Definition:Language of Propositional Logic. This page defines the formal language $\LL_0$ used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Explanations are omitted as this is intended for reference use only. === Alphabet === ==== Letters ==== The letters used are a non-empty set of symbols $\PP_0$. See the  definition. ==== Signs ==== ===== Brackets ===== The '''brackets''' used are '''round brackets''': {{begin-eqn}} {{eqn | ll= \bullet       | l = (       | o = :       | r = \)the '''left bracket''' sign\(       | c =  }} {{eqn | ll= \bullet       | l = )       | o = :       | r = \)the '''right bracket''' sign\(       | c =  }} {{end-eqn}} See the  definition. ===== Connectives ===== The following '''connectives''' are used: {{begin-eqn}} {{eqn | ll= \bullet       | l = .       | o = :       | r = \)the conjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \lor       | o = :       | r = \)the disjunction sign\(       | c =  }} {{eqn | ll= \bullet       | l = \supset       | o = :       | r = \)the conditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \equiv       | o = :       | r = \)the biconditional sign\(       | c =  }} {{eqn | ll= \bullet       | l = \sim       | o = :       | r = \)the negation sign\(       | c =  }} {{end-eqn}} See the  definition. === Collation System === The collation system used is that of words and concatenation. See the  definition. === Formal Grammar === The following bottom-up formal grammar is used. Let $\PP_0$ be the vocabulary of $\LL_0$. Let $Op = \left\{{., \lor, \supset, \equiv}\right\}$. The rules are: {| |- | $\mathbf W: \PP_0$ || | $:$ | If $p \in \PP_0$, then $p$ is a WFF. |- | $\mathbf W: \neg$ || | $:$ | If $\mathbf A$ is a WFF, then $\sim \mathbf A$ is a WFF. |- | $\mathbf W: Op$ || | $:$ | If $\mathbf A$ and $\mathbf B$ are WFFs and $\circ \in Op$, then $\paren {\mathbf A \circ \mathbf B}$ is a WFF. |} See the  definition.
Definition:Pea Pattern/Sequential
For the '''sequential pea pattern''', the concatenation into $p_n$ is in sequential order of how the first instance of the distinct digits appear in $p_{n - 1}$.
Definition:Pea Pattern
The first term $p_0$ is an arbitrary integer. $p_n$ is formed from $p_{n - 1}$ as follows. First, the distinct digits in $p_{n - 1}$ are counted, and the number of each is noted. The count of each distinct digit is concatenated with an instance of the digit itself. Then those concatenations are themselves concatenated into $p_n$ according to a predetermined order.
Definition:Pea Pattern/Ascending
For the '''ascending pea pattern''', the concatenation into $p_n$ is in ascending order of the distinct digits of $p_{n - 1}$.
Definition:Pea Pattern/Descending
For the '''descending pea pattern''', the concatenation into $p_n$ is in descending order of the distinct digits of $p_{n - 1}$.
Definition:Time/Unit/Great Year of Plato
{{begin-eqn}} {{eqn | o =        | r = 1       | c = '''great year of Plato''' }} {{eqn | r = 12 \, 960 \, 000       | c = days }} {{eqn | o = \approx       | r = 36 \, 000       | c = years (of $360$ days) }} {{end-eqn}}
Definition:Counterexample
Let $X$ be the universal statement: :$\forall x \in S: \map P x$ That is: :''For all the elements $x$ of a given set $S$, the property $P$ holds.'' Such a statement may or may not be true. Let $Y$ be the existential statement: :$\exists y \in S: \neg \map P y$ That is: :''There exists at least one element $y$ of the set $S$ such that the property $P$ does ''not'' hold.'' It follows immediately by De Morgan's laws that if $Y$ is true, then $X$ must be false. Such a statement $Y$ is referred to as a '''counterexample to $X$'''.
Definition:Substitution (Formal Systems)
=== Substitution for Well-Formed Part === {{:Definition:Substitution (Formal Systems)/Well-Formed Part}} === Substitution for Letter === {{:Definition:Substitution (Formal Systems)/Letter}} === Substitution of Term for Variable === {{:Definition:Substitution (Formal Systems)/Term}} === Substitution for Metasymbol === {{:Definition:Substitution (Formal Systems)/Metasymbol}}
Definition:Substitution (Formal Systems)/Letter
Let $\mathbf B$ be a well-formed formula of $\FF$. Let $p$ be a letter of $\FF$. Let $\mathbf A$ be another well-formed formula. Then the '''substitution of $\mathbf A$ for $p$ in $\mathbf B$''' is the collation resulting from $\mathbf B$ by replacing all occurrences of $p$ in $\mathbf B$ by $\mathbf A$. It is denoted as $\map {\mathbf B} {\mathbf A \mathbin {//} p}$.
Definition:Substitution (Formal Systems)/Term
{{:Definition:Substitution (Formal Systems)/Term/In WFF}}