hcju/proofwikips huggingface dataset:
Definition:Substitution (Formal Systems)/Metasymbol
Let $S_1$ be a statement form. Let $p$ be a metasymbol which occurs one or more times in $S_1$. Let $T$ be a statement. Let $S_2$ be the string formed by replacing ''every'' occurrence of $p$ in $S_1$ with $T$. Then $S_2$ results from the '''substitution of $p$ by $T$ in $S_1$'''. $S_2$ is called a '''substitution instance''' of $S_1$.
Definition:Right-Truncatable Prime
A '''right-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the right hand end.
Definition:Hilbert Proof System/Instance 2
<section notitle="true" name="tc"> This instance of a Hilbert proof system is used in: * {{BookReference|Introduction to Symbolic Logic|1959|A.H. Basson|author2 = D.J. O'Connor|ed = 3rd|edpage = Third Edition}} Let $\mathcal L$ be the language of propositional logic. $\mathscr H$ has the following axioms and rules of inference: === Axioms === Let $p, q, r$ be propositional variables. Then the following WFFs are axioms of $\mathscr H$: {{begin-axiom}} {{axiom|n = A1        |lc = Rule of Idempotence        |m = (p \lor p) \implies p }} {{axiom|n = A2        |lc = Rule of Addition        |m = q \implies (p \lor q) }} {{axiom|n = A3        |lc = Rule of Commutation        |m = (p \lor q) \implies (q \lor p) }} {{axiom|n = A4        |lc = Factor Principle        |m = (q \implies r) \implies \left({ (p \lor q) \implies (p \lor r)}\right) }} {{end-axiom}} === Rules of Inference === ==== $RST \, 1$: Rule of Uniform Substitution ==== {{refactor|This rule should have its own dedicated page like the others. But separate from the Rule of Substitution, which is too vague at this point (might be a subpage though). Also review source flow afterwards}} Any WFF $\mathbf A$ may be substituted for any propositional variable $p$ in a $\mathscr H_2$-theorem $\mathbf B$. The resulting theorem can be denoted $\mathbf B \paren{ \mathbf A \mathbin{//} p }$. See the Rule of Substitution. ==== $RST \, 2$: Rule of Substitution by Definition ==== The following expressions are regarded definitional abbreviations: {{begin-axiom}} {{axiom|n = 1        |lc = Conjunction        |ml = \mathbf A \land \mathbf B        |mo = {{=}}_{\text{def} }        |mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom|n = 2        |lc = Conditional        |ml = \mathbf A \implies \mathbf B        |mo = {{=}}_{\text{def} }        |mr = \neg \mathbf A \lor \mathbf B }} {{axiom|n = 3        |lc = Biconditional        |ml = \mathbf A \iff \mathbf B        |mo = {{=}}_{\text{def} }        |mr = (\mathbf A \implies \mathbf B) \land (\mathbf B \implies \mathbf A) }} {{end-axiom}} ==== $RST \, 3$: Rule of Detachment ==== If $\mathbf A \implies \mathbf B$ and $\mathbf A$ are theorems of $\mathscr H$, then so is $\mathbf B$. That is, Modus Ponendo Ponens. ==== $RST \, 4$: Rule of Adjunction ==== If $\mathbf A$ and $\mathbf B$ are theorems of $\mathscr H$, then so is $\mathbf A \land \mathbf B$. That is, the Rule of Conjunction. (This rule can be proved from the other three and so is only a convenience.) </section>
Definition:Hardy-Ramanujan Number
The $n$th '''Hardy-Ramanujan number''' $\operatorname {Ta} \left({n}\right)$ is the smallest positive integer which can be expressed as the sum of $2$ cubes in $n$ different ways.
Definition:Cube Number/Sequence
The sequence of positive cube numbers begins: :$1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, \ldots$
Definition:Summation/Finite Support
Let $G$ be an abelian group. Let $S$ be a set. Let $f: S \to G$ be a mapping. Let the support $\operatorname{Supp} \left({f}\right)$ be finite. Let $g$ be the restriction of $f$ to $\operatorname{Supp}f$. The '''summation of $f$ over $S$''', denoted $\displaystyle \sum_{s \mathop \in S} f \left({s}\right)$, is the summation over the finite set $\operatorname{Supp} \left({f}\right)$ of $g$: :$\displaystyle \sum_{s \mathop \in S} f \left({s}\right) = \sum_{s \mathop \in \operatorname {Supp} \left({f}\right)} g \left({s}\right)$
Definition:Independent Proof System
Let $\mathcal L$ be a logical language. Let $\mathscr P$ be a proof system. Then $\mathscr P$ is '''independent''' if it is not possible to derive one axiom or rule of inference of $\mathscr P$ from the others. {{stub|Deliberately vague, as the context in which this definition was originally couched was equally vague. Needs to be made rigorous, or merged with pages building a more comprehensive treatment.}}
Definition:Sum of Finite Set
Let $\mathbb A$ be one of the standard number systems $\N, \Z, \Q, \R, \C$. Let $S \subset \mathbb A$ be a finite subset. The '''sum of $S$''' is the summation of the identity mapping on $S$ over $S$: :$\displaystyle \sum S = \sum_{s \mathop \in S} s$
Definition:Unital Magma
A '''unital magma''' is a magma that has an identity element.
Definition:Commutative Magma
A '''commutative magma''' is a magma whose operation is commutative.
Definition:N-Ary Operation Induced by Binary Operation
Let $\struct {G, \oplus}$ be a magma. Let $n \ge 1$ be a natural number. Let $G^n$ be the $n$th cartesian power of $G$. The '''$n$-ary operation induced by $\oplus$''' is the $n$-ary operation $\oplus_n: G^n \to G$ defined as: :$\map {\oplus_n} f = \displaystyle \bigoplus_{i \mathop = 1}^n \map f i$ where $\bigoplus$ denotes indexed iteration of $f$ from $1$ to $n$.
Definition:Constant
A '''constant''' is a name for an object (usually a number, but the concept has wider applications) which ''does not change'' during the context of a logical or mathematical argument. A constant can be considered as an operator which takes no operands. A constant can also be considered as a variable whose domain is a singleton.
Definition:Polynomial Ring/Indeterminate
==== Single indeterminate ==== Let $\left({S, \iota, X}\right)$ be a polynomial ring over $R$. The '''indeterminate''' of $\left({S, \iota, X}\right)$ is the term $X$. ==== Multiple Indeterminates ==== Let $I$ be a set. Let $\left({S, \iota, f}\right)$ be a polynomial ring over $R$ in $I$ '''indeterminates'''. The '''indeterminates''' of $\left({S, \iota, f}\right)$ are the elements of the image of the family $f$.
Definition:Polynomial Ring/Universal Property
A '''polynomial ring over $R$''' is a pointed $R$-algebra $(S, \iota, X)$ that satisfies the following universal property: :For every pointed $R$-algebra $(A, \kappa, a)$ there exists a unique pointed algebra homomorphism $h : S\to A$, called evaluation homomorphism. This is known as the '''universal property of a polynomial ring'''.
Definition:Polynomial Algebra
Let $R$ be a ring. Let $\Z \sqbrk X$ be the polynomial ring over the ring of integers $\Z$. The '''polynomial algebra in one indeterminate''' over $R$ is the tensor product $R \otimes_\Z \Z \sqbrk X$.
Definition:Unital Associative Commutative Algebra
Let $R$ be a commutative ring with unity. === Definition 1 === {{:Definition:Unital Associative Commutative Algebra/Definition 1}} === Definition 2 === {{:Definition:Unital Associative Commutative Algebra/Definition 2}} == Also known as == A '''unital associative commutative algebra''' over $R$ is also known as an '''$R$-algebra''', as is a general algebra over $R$. == Equivalence of Definitions == While, strictly speaking, the above definitions do define different objects, they are equivalent in the following sense: * An algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary, is identified with the ring under $R$ equal to its underlying ring together with its canonical mapping $R \to A$. The algebra is '''viewed as a ring'''. * A ring under $R$, $(A, f)$, is identified with the algebra defined by $f$. The algebra is '''viewed as an algebra'''. For the detailed statements, see Equivalence of Definitions of Unital Associative Commutative Algebra.
Definition:Unital Ring Homomorphism
Let $R$ and $S$ be rings with unities $1_R$ and $1_S$. A '''unital ring homomorphism''' is a ring homomorphism $f: R \to S$ such that $\map f {1_R} = 1_S$.
Definition:Unital Associative Commutative Algebra/Definition 1
A '''unital associative commutative algebra''' over $R$ is an algebra $\left({A, *}\right)$ over $R$ that is unital, associative and commutative and whose underlying module is unitary.
Definition:Unital Associative Commutative Algebra/Definition 2
A '''unital associative commutative algebra''' over $R$ is a ring under $A$, that is, an ordered pair $(A, f)$ where: :$A$ is a commutative ring with unity :$f : R \to A$ is a unital ring homomorphism <!--whose image is contained in the center of $A$-->
Definition:Algebra Defined by Ring Homomorphism
Let $R$ be a commutative ring. Let $\struct {S, +, *}$ be a ring. Let $f : R \to S$ be a ring homomorphism. Let the image of $f$ be a subset of the center of $S$. Let $S_R$ be the module defined by the ring homomorphism $f$. The '''algebra defined by $f$''' is the algebra over $R$ defined as: :$\struct {S_R, *}$
Definition:Underlying Ring of Associative Algebra
Let $R$ be a commutative ring. Let $(A, *)$ be an associative algebra over $R$. Let $A = (M, +, \circ)$ be the underlying module of $(A, *)$. The '''underlying ring of $(A, *)$''' is the ring $(A, +, *)$.
Definition:Pointed Algebra over Ring
Let $R$ be a commutative ring. A '''pointed algebra over $R$''' is an ordered pair $(A, a)$ where: :$A$ is an algebra over $R$ :$a$ is an element of $A$
Definition:Polynomial Ring/Embedding
Let the ordered triple $(S, \iota, X)$ be a polynomial ring over $R$ in one indeterminate $X$. The unital ring homomorphism $\iota$ is called the '''canonical embedding into the polynomial ring'''. === Multiple Indeterminates === Let $I$ be a set. Let $(S, \iota, X)$ be a polynomial ring over $R$ in $I$ indeterminates. The unital ring homomorphism $\iota$ is called the '''canonical embedding into the polynomial ring'''.
Definition:Monoid Ring/Canonical Mapping
Let $e_1$ be the canonical basis element. The '''canonical mapping to $R \left[{G}\right]$''' is the mapping $R \to R \left[{G}\right]$ which sends $r$ to $r * e_1$.
Definition:Unital Subalgebra/Definition 1
The subalgebra $\left({B_R, *}\right)$ is a '''unital subalgebra''' of $A_R$  $1_A \in B$.
Definition:Unital Subalgebra/Definition 2
The subalgebra $\left({B_R, *}\right)$ is a '''unital subalgebra''' of $A_R$ : :$(1) \quad B_R$ is unital :$(2) \quad$Its unit is $1_A$. That is, a '''unital subalgebra''' of $A_R$ must not only have a unit, but that unit must also be the same unit as that of $A_R$.
Definition:Field of Rational Fractions
Let $R$ be an integral domain.
Definition:Continuous Mapping (Topology)/Point/Open Sets
The mapping $f$ is '''continuous at (the point) $x$''' (with respect to the topologies $\tau_1$ and $\tau_2$) : :For every neighborhood $N$ of $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.
Definition:Continuous Mapping (Topology)/Point/Filters
The mapping $f$ is '''continuous at (the point) $x$'''  for any filter $\mathcal F$ on $T_1$ that converges to $x$, the corresponding image filter $f \left({\mathcal F}\right)$ converges to $f \left({x}\right)$.
Definition:Continuous Mapping (Topology)/Everywhere/Pointwise
The mapping $f$ is '''continuous everywhere''' (or simply '''continuous''')  $f$ is continuous at every point $x \in S_1$.
Definition:Open Neighborhood/Topology
Let $T = \struct {S, \tau}$ be a topological space. Let $A \subseteq S$ be a subset of $S$. Let $N_A$ be a neighborhood of $A$. If $N_A \in \tau$, that is, if $N_A$ is itself open in $T$, then $N_A$ is called an '''open neighborhood'''.
Definition:Continuous Real Function/Everywhere
Let $f: \R \to \R$ be a real function. Then $f$ is '''everywhere continuous'''  $f$ is continuous at every point in $\R$.
Definition:Continuous Real Function/Closed Interval
Let $f$ be a real function defined on a closed interval $\left[{a \,.\,.\, b}\right]$. ==== Definition 1 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 1}} ==== Definition 2 ==== {{Definition:Continuous Real Function/Closed Interval/Definition 2}}
Definition:Continuous Real Function/Closed Interval/Definition 2
The function $f$ is '''continuous on $\left[{a \,.\,.\, b}\right]$'''  it is continuous at every point of $\left[{a \,.\,.\, b}\right]$.
Definition:Indexed Iterated Binary Operation
Let $\struct {G, *}$ be a magma. Let $a, b \in \Z$ be integers. Let $\closedint a b$ be the integer interval between $a$ and $b$. Let $f: \closedint a b \to G$ be a mapping. The '''indexed iteration of $*$ of $f$ from $a$ to $b$''' is recursively defined and denoted: :$\ds \prod_{k \mathop = a}^b \map f k = \begin {cases} \map f a & : b = a \\ \paren {\ds \prod_{k \mathop = a}^{b - 1} \map f k} * \map f b & : b > a \end {cases}$ For each ordered $n$-tuple $\tuple {a_1, a_2, \ldots, a_n} \in S^n$, the '''composite''' of $\tuple {a_1, a_2, \ldots, a_n}$ for $\oplus$ is the value at $\tuple {a_1, a_2, \ldots, a_n}$ of the $n$-ary operation defined by $\oplus$. This '''composite''' is recursively defined and denoted: {{begin-eqn}} {{eqn | l = \bigoplus_{k \mathop = 1}^n a_k       | r = \map {\oplus_n} {a_1, a_2, \ldots, a_n}       | c =  }} {{eqn | r = \begin {cases} a & : n = 1 \\ \map {\oplus_m} {a_1, \ldots, a_m} \oplus a_{m + 1} & : n = m + 1 \end {cases}       | c =  }} {{eqn | r = \paren {\paren {\cdots \paren {\paren {a_1 \oplus a_2} \oplus a_3} \oplus \cdots} \oplus a_{n - 1} } \oplus a_n       | c =  }} {{end-eqn}}
Definition:Iterated Binary Operation over Finite Set
Let $\struct {G, *}$ be a commutative semigroup. Let $S$ be a finite non-empty set. Let $f: S \to G$ be a mapping. Let $n \in \N$ be the cardinality of $S$. Let $g: \N_{<n} \to S$ be a bijection, where $\N_{<n}$ is an initial segment of the natural numbers. The '''iteration of $*$ of $f$ over $S$''', denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the indexed iteration of $*$ of the composition $f \circ g$ over $\N_{<n}$: :$\displaystyle \prod_{s \mathop \in S} \map f s = \displaystyle \prod_{i \mathop = 0}^{n - 1} \map f {\map g i}$
Definition:Iterated Binary Operation over Set with Finite Support
Let $\struct {G, *}$ be a commutative monoid. Let $S$ be a set. Let $f: S \to G$ be a mapping. Let the support $\operatorname {Supp} f$ be finite. The '''iteration of $*$ of $f$ over $S$''', denoted $\displaystyle \prod_{s \mathop \in S} \map f s$, is the iteration over the finite set $\operatorname{Supp} f$ of $f$: :$\displaystyle \prod_{s \mathop \in S} \map f s = \prod_{s \mathop \in \operatorname {Supp} f} \map f s$
Definition:Ring of Sequences of Finite Support
Let $\struct {R, +, \circ}$ be a ring. Let $P \sqbrk R$ be the set of all sequences in $R$ whose domain is $\N$: :$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$ and whose support is finite. Let the operations $\oplus$ and $\odot$ on $P \sqbrk R$ be defined as follows: {{:Definition:Operations on Polynomial Ring of Sequences}} The '''ring of sequences of finite support over $R$''' is the ring $\struct {P \sqbrk R, \oplus, \odot}$.
Definition:Consistent (Logic)/Proof System
Then $\mathscr P$ is '''consistent''' : :There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$ That is, some logical formula $\phi$ is '''not''' a theorem of $\mathscr P$.
Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 1
Then $\mathscr P$ is '''consistent''' : :There exists a logical formula $\phi$ such that $\not \vdash_{\mathscr P} \phi$ That is, some logical formula $\phi$ is '''not''' a theorem of $\mathscr P$.
Definition:Consistent (Logic)/Proof System/Propositional Logic/Definition 2
Suppose that in $\mathscr P$, the Rule of Explosion (Variant 3) holds. Then $\mathscr P$ is '''consistent''' : :For every logical formula $\phi$, not ''both'' of $\phi$ and $\neg \phi$ are theorems of $\mathscr P$
Definition:Consistent (Logic)/Set of Formulas
Let $\FF$ be a collection of logical formulas. Then $\FF$ is '''consistent for $\mathscr P$''' : :There exists a logical formula $\phi$ such that $\FF \nvdash_{\mathscr P} \phi$. That is, some logical formula $\phi$ is '''not''' a provable consequence of $\FF$.
Definition:Left-Truncatable Prime
A '''left-truncatable prime''' is a prime number which remains prime when any number of digits are removed from the left hand end. Zeroes are excluded, in order to eliminate, for example, prime numbers of the form $10^n + 3$ for arbitrarily large $n$.
Definition:Irreducible Component
Let $T = \left({S, \tau}\right)$ be a topological space. A subspace $Y \subset T$ is an '''irreducible component of $T$''' : : $Y$ is irreducible : $Y$ is not a proper subset of an irreducible subspace of $T$. That is,  $Y$ is maximal in the ordered set of irreducible subspaces of $T$, ordered by inclusion.
Definition:Irreducible Space/Definition 1
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  there exists no cover of $T$ by two proper closed sets of $T$.
Definition:Irreducible Space/Definition 2
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  there is no finite cover of $T$ by proper closed sets of $T$.
Definition:Irreducible Space/Definition 3
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  every two non-empty open sets of $T$ have non-empty intersection: :$\forall U, V \in \tau: U, V \ne \O \implies U \cap V \ne \O$
Definition:Irreducible Space/Definition 4
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  every non-empty open set of $T$ is (everywhere) dense in $T$.
Definition:Irreducible Space/Definition 5
A topological space $T = \struct {S, \tau}$ is '''irreducible'''  the interior of every proper closed set of $T$ is empty.
Definition:Krull Dimension
=== Krull Dimension of a Ring === {{Definition:Krull Dimension of Ring}} === Krull Dimension of a Topological Space === {{Definition:Krull Dimension of Topological Space}} {{NamedforDef|Wolfgang Krull|cat = Krull}}
Definition:Length of Chain
Let $T$ be a chain in $S$. Let $T$ be finite and non-empty. The '''length''' of the chain $T$ is its cardinality minus $1$.
Definition:Krull Dimension of Topological Space
Let $T$ be a topological space. Its '''Krull dimension''' $\map {\dim_{\mathrm {Krull} } } T$ is the supremum of lengths of chains of closed irreducible sets of $T$, ordered by inclusion. Thus, the '''Krull dimension''' is $\infty$ if there exist arbitrarily long chains.
Definition:Noetherian Topological Space
=== Definition 1 === {{:Definition:Noetherian Topological Space/Definition 1}} === Definition 2 === {{:Definition:Noetherian Topological Space/Definition 2}}
Definition:Set Ordered by Inclusion
Let $S$ be a set. Then '''$S$ ordered by inclusion''' is the ordered set $\struct {S, \subseteq}$, where $\subseteq$ is the inclusion relation on $S$.
Definition:Total Ring of Fractions
Let $A$ be a commutative ring with unity. The '''total ring of fractions''' of $A$ is the localization of $A$ at the set of its regular elements.
Definition:Regular Element of Ring
Let $A$ be a commutative ring with unity. Let $a\in A$. Then $a$ is '''regular'''  it is not a zero divisor.
Definition:Monomial of Polynomial Ring
=== Multiple Variables === {{Definition:Monomial of Polynomial Ring/Multiple Variables}}
Definition:Pointed Set
A '''pointed set''' is an ordered pair $\struct {A, a}$ where: :$A$ is a set :$a \in A$ is an element.
Definition:Pointed Mapping
Let $\struct {A, a}$ and $\struct {B, b}$ be pointed sets. A '''pointed mapping''' $f: \struct {A, a} \to \struct {B, b}$ is a mapping $f : A \to B$ such that $\map f a = b$.
Definition:Multiple Pointed Set
Let $I$ be a set. An '''$I$-pointed set''' is an ordered pair $\left({A, f}\right)$ where: :$A$ is a set :$f : I \to A$ is a mapping
Definition:Multiple Pointed Algebra
Let $R$ be a commutative ring with unity. Let $I$ be a set. An '''$I$-pointed algebra over $R$''' is an ordered pair $(A, f)$ where: :$A$ is an algebra over $R$ :$f : I \to A$ is a mapping
Definition:Square Pyramorphic Number
A '''square pyramorphic number''' is a square pyramidal number $P_n$ whose decimal representation ends in $n$.
Definition:Square Pyramorphic Number/Sequence
The sequence of square pyramorphic numbers, for $n \in \Z_{\ge 0}$, begins: {{begin-eqn}} {{eqn | l = P_1       | r = 1 }} {{eqn | l = P_5       | r = 55 }} {{eqn | l = P_{25}       | r = 5525 }} {{eqn | l = P_{40}       | r = 22 \, 140 }} {{eqn | l = P_{65}       | r = 93 \, 665 }} {{eqn | l = P_{80}       | r = 1 \, 043 \, 280 }} {{end-eqn}}
Definition:Additive Arithmetic Function
Let $f : \N \to \C$ be an arithmetic function. Then $f$ is '''additive''' : :$m \perp n \implies \map f {m n} = \map f m + \map f n$
Definition:Multiplicative Function on UFD
Let $R$ be a unique factorization domain. Let $f : R \to \C$ be a complex-valued function. Then $f$ is '''multiplicative''' : :For all coprime $x, y\in R$: $f \left({x y}\right) = f \left({x}\right) f \left({y}\right)$
Definition:Constructed Semantics
Let $\mathcal L$ be a formal language. A '''constructed semantics''' for $\mathcal L$ is a formal semantics which is invented ''solely'' for proving a property about $\mathcal L$ or other entities related to $\mathcal L$.
Definition:Additive Function on UFD
Let $R$ be a unique factorization domain. Let $f : R \to \C$ be a complex-valued function. Then $f$ is '''additive''' : :For all coprime $x,y\in R$, $f(xy) = f(x) + f(y)$
Definition:Multiplicative Arithmetic Function
Let $f : \N \to \C$ be an arithmetic function. Then $f$ is '''multiplicative''' : :$m \perp n \implies \map f {m n} = \map f m \map f n$ where $m \perp n$ denotes that $m$ is coprime to $n$.
Definition:Constructed Semantics/Instance 1
Let $\mathcal L_0$ be the language of propositional logic. The constructed semantics $\mathscr C_1$ for $\mathcal L_0$ is used for the following results: * Hilbert Proof System Instance 2 is Consistent === Structures === Define the structures of $\mathscr C_1$ as mappings $v$ by the Principle of Recursive Definition, as follows. Let $\mathcal P_0$ be the vocabulary of $\mathcal L_0$. Let a mapping $v: \mathcal P_0 \to \{ 1, 2 \}$ be given. Next, regard the following as definitional abbreviations: {{begin-axiom}} {{axiom | n  = 1         | lc = Conjunction         | ml = \mathbf A \land \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \left({ \neg \mathbf A \lor \neg \mathbf B }\right) }} {{axiom | n  = 2         | lc = Conditional         | ml = \mathbf A \implies \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \neg \mathbf A \lor \mathbf B }} {{axiom | n  = 3         | lc = Biconditional         | ml = \mathbf A \iff \mathbf B         | mo = {{=}}_{\text{def} }         | mr = \left({\mathbf A \implies \mathbf B}\right) \land \left({\mathbf B \implies \mathbf A}\right) }} {{end-axiom}} It only remains to define $v \left({ \neg \phi }\right)$ and $v \left({ \phi \lor \psi}\right)$ recursively, by: {{begin-eqn}} {{eqn | l = v \left({\neg \phi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = 2$} \\ 2 & : \text{if $v \left({\phi}\right) = 1$}\end{cases} }} {{eqn | l = v \left({ \phi \lor \psi }\right)       | o = :=       | r = \begin{cases} 1 & : \text{if $v \left({\phi}\right) = v \left({\psi}\right) = 1$} \\ 2 & : \text{otherwise}\end{cases} }} {{end-eqn}} === Validity === Define validity in $\mathscr C_1$ by declaring: :$\models_{\mathscr C_1} \phi$  $v \left({\phi}\right) = 2$
Definition:Floor Function/Definition 2
The '''floor function of $x$''', denoted $\floor x$, is defined as the greatest element of the set of integers: :$\set {m \in \Z: m \le x}$ where $\le$ is the usual ordering on the real numbers.
Definition:Ceiling Function/Definition 1
The '''ceiling function of $x$''' is defined as the infimum of the set of integers no smaller than $x$: :$\ceiling x := \inf \set {m \in \Z: x \le m}$ where $\le$ is the usual ordering on the real numbers.
Definition:Ceiling Function/Definition 2
The '''ceiling function of $x$''', denoted $\ceiling x$, is defined as the smallest element of the set of integers: :$\set {m \in \Z: x \le m}$ where $\le$ is the usual ordering on the real numbers.
Definition:Ceiling Function/Definition 3
The '''ceiling function of $x$''' is the unique integer $\ceiling x$ such that: :$\ceiling x - 1 < x \le \ceiling x$