formalContent
P19806
predicate
Indicates that the content is expressed in a formal, structured, or official manner, as opposed to casual or informal expression.
All labels observed (4)
| Label | Occurrences |
|---|---|
| formalStatement | 25 |
| UKFormalPhrase | 2 |
| formalCommunicationChannel | 1 |
| formalContent canonical | 1 |
Description generation (PDg)
The one-sentence description above was generated by prompting gpt-5.1 with the predicate name and this instruction.
Instruction
Given a predicate that represents a relationship or action between entities, generate a one-sentence description explaining its meaning. # Instructions Focus on describing the relationship, not the entities themselves. # Response Format Begin the description with \' Indicates...\'
Input
Predicate: formalContent
Generated description
Indicates that the content is expressed in a formal, structured, or official manner, as opposed to casual or informal expression.
Sample triples (29)
| Subject | Object |
|---|---|
| Burali-Forti paradox | If Ω is the set of all ordinals, then Ω itself would be an ordinal greater than every ordinal in Ω, leading to a contradiction ⓘ |
| Royal Assent | La Reyne le veult via predicate surface "UKFormalPhrase" ⓘ |
| Royal Assent | Le Roy le veult via predicate surface "UKFormalPhrase" ⓘ |
| Schauder fixed-point theorem | Every continuous compact map from a nonempty closed bounded convex subset of a Banach space into itself has a fixed point. via predicate surface "formalStatement" ⓘ |
| P versus NP problem | Does P = NP? via predicate surface "formalStatement" ⓘ |
| Stokes' theorem | ∮_{∂S} F · dr = ∬_S (∇ × F) · n dS via predicate surface "formalStatement" ⓘ |
| Archimedean property of real numbers | For every real number x, there exists a natural number n such that n > x. via predicate surface "formalStatement" ⓘ |
| Archimedean property of real numbers | For every real number x > 0, there exists a natural number n such that 1/n < x. via predicate surface "formalStatement" ⓘ |
| continuum hypothesis | There is no set A such that |ℕ| < |A| < |ℝ| via predicate surface "formalStatement" ⓘ |
| Hotelling’s lemma | partial derivative of profit with respect to an output price equals the firm’s optimal supply of that output via predicate surface "formalStatement" ⓘ |
| Hotelling’s lemma | partial derivative of profit with respect to an input price equals minus the firm’s optimal demand for that input via predicate surface "formalStatement" ⓘ |
| Chapman–Kolmogorov equation | P(X_t \in B \mid X_s = x) = \int P(X_t \in B \mid X_u = y) \, P(X_u \in dy \mid X_s = x) for s < u < t via predicate surface "formalStatement" ⓘ |
| Fundamental Theorem of Calculus | If f is continuous on [a,b] and F is an antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a) via predicate surface "formalStatement" ⓘ |
|
Fundamental Theorem of Calculus
surface form:
First Fundamental Theorem of Calculus
|
If f is integrable on [a,b] and F(x)=∫_a^x f(t) dt, then F is continuous on [a,b], differentiable on (a,b), and F′(x)=f(x) for all x in (a,b) via predicate surface "formalStatement" ⓘ |
|
Fundamental Theorem of Calculus
surface form:
Second Fundamental Theorem of Calculus
|
If f is continuous on [a,b] and F is any antiderivative of f on [a,b], then ∫_a^b f(x) dx = F(b) − F(a) via predicate surface "formalStatement" ⓘ |
|
President of Fiji opens and may address Parliament
surface form:
President of Fiji
|
address to Parliament via predicate surface "formalCommunicationChannel" ⓘ |
|
MIP equals NEXP
surface form:
MIP = NEXP
|
MIP = NEXP via predicate surface "formalStatement" ⓘ |
| Stone’s theorem on one-parameter unitary groups | If {U(t)}_{t in R} is a strongly continuous one-parameter unitary group on a Hilbert space H, then there exists a unique self-adjoint operator A such that U(t)=exp(itA) for all real t via predicate surface "formalStatement" ⓘ |
|
Axiom of Extensionality in set theory
surface form:
Axiom of Extensionality
|
∀A∀B(∀x(x ∈ A ↔ x ∈ B) → A = B) via predicate surface "formalStatement" ⓘ |
| Carathéodory measurability criterion | A set E is measurable if for every set A, μ*(A) = μ*(A ∩ E) + μ*(A \ E) via predicate surface "formalStatement" ⓘ |
| Erdős–Straus conjecture | For every integer n ≥ 2, there exist positive integers x, y, z such that 4/n = 1/x + 1/y + 1/z. via predicate surface "formalStatement" ⓘ |
| orbit-stabilizer theorem | For a group G acting on a set X and x in X, |G| = |Orb(x)| · |Stab(x)| when all sets are finite via predicate surface "formalStatement" ⓘ |
| Liouville's theorem in Hamiltonian mechanics | dρ/dt = 0 along trajectories in phase space for Hamiltonian dynamics via predicate surface "formalStatement" ⓘ |
| Liouville's theorem in Hamiltonian mechanics | ∂ρ/∂t + {ρ,H} = 0 via predicate surface "formalStatement" ⓘ |
| Liouville's theorem in Hamiltonian mechanics | ∇·v = 0 in phase space for Hamiltonian flow via predicate surface "formalStatement" ⓘ |
| univalence axiom | for types A and B, the canonical map (A = B) → (A ≃ B) is an equivalence via predicate surface "formalStatement" ⓘ |
| Lax–Milgram theorem | For every bounded coercive bilinear form a(·,·) on a Hilbert space H and every bounded linear functional f on H, there exists a unique u in H such that a(u,v)=f(v) for all v in H. via predicate surface "formalStatement" ⓘ |
| Lebesgue decomposition theorem | Given σ-finite measures μ and ν on a measurable space, there exist unique measures μ_ac and μ_s such that μ = μ_ac + μ_s, μ_ac ≪ ν, and μ_s ⟂ ν via predicate surface "formalStatement" NERFINISHED ⓘ |
| Yao’s minimax principle | the expected cost of the best randomized algorithm on the worst input is at least the expected cost of the best deterministic algorithm against some input distribution via predicate surface "formalStatement" ⓘ |