typicalProofUses
P27215
predicate
Indicates that a proof characteristically or commonly employs a particular method, technique, or component.
All labels observed (9)
| Label | Occurrences |
|---|---|
| usedInProofOf | 29 |
| proofUses | 23 |
| provedUsing | 21 |
| typicalProofUses canonical | 17 |
| typicalProofMethod | 9 |
| proofTechniques | 6 |
| canBeProvedUsing | 5 |
| isUsedToProve | 3 |
| proofIdea | 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: typicalProofUses
Generated description
Indicates that a proof characteristically or commonly employs a particular method, technique, or component.
Sample triples (114)
| Subject | Object |
|---|---|
| Noether normalization lemma | induction on the number of generators ⓘ |
| Noether normalization lemma | linear changes of variables ⓘ |
| Noether normalization lemma | integral dependence and minimal polynomials ⓘ |
| Ricci flow | Poincaré conjecture via predicate surface "usedInProofOf" ⓘ |
| Ricci flow | geometrization conjecture via predicate surface "usedInProofOf" ⓘ |
| Whitney approximation theorem | partitions of unity ⓘ |
| Whitney approximation theorem | local coordinate charts ⓘ |
| Whitney approximation theorem | smoothing by convolution in Euclidean space ⓘ |
| Cantor’s theorem | assume a surjection from S to P(S) and derive a contradiction using a specially constructed subset via predicate surface "proofIdea" ⓘ |
| Fermat's Last Theorem | elliptic curves via predicate surface "proofUses" ⓘ |
| Fermat's Last Theorem | modular forms via predicate surface "proofUses" ⓘ |
| Fermat's Last Theorem | Galois representations via predicate surface "proofUses" ⓘ |
| Poincaré lemma | Poincaré duality via predicate surface "usedInProofOf" ⓘ |
| Poincaré lemma |
de Rham theorem
via predicate surface "usedInProofOf"
ⓘ
linked to:
de Rham cohomology
|
|
cyclotomic field
linked to:
cyclotomic fields
|
Kronecker–Weber theorem via predicate surface "usedInProofOf" ⓘ |
| NP-completeness | polynomial-time reduction from known NP-complete problem via predicate surface "typicalProofMethod" ⓘ |
| Chebotarev density theorem | existence of infinitely many primes in given conjugacy classes of Galois groups via predicate surface "usedInProofOf" ⓘ |
| Cauchy–Hadamard theorem | properties of limsup via predicate surface "proofUses" ⓘ |
| Cauchy–Hadamard theorem | root test for series via predicate surface "proofUses" ⓘ |
| Cauchy–Hadamard theorem | estimates on |a_n z^n| via predicate surface "proofUses" ⓘ |
| Weil conjectures | étale cohomology via predicate surface "provedUsing" ⓘ |
| Weil conjectures | ℓ-adic cohomology via predicate surface "provedUsing" ⓘ |
| Weil conjectures | Grothendieck’s theory of schemes via predicate surface "provedUsing" ⓘ |
| Weil conjectures | Grothendieck’s theory of weights via predicate surface "provedUsing" ⓘ |
| Weil conjectures |
Lefschetz trace formula
via predicate surface "provedUsing"
ⓘ
linked to:
Lefschetz fixed-point theorem
|
| Weil conjectures | Deligne’s theory of weights via predicate surface "provedUsing" ⓘ |
| Schwarz lemma | Riemann mapping theorem via predicate surface "usedInProofOf" ⓘ |
| Schwarz lemma |
Schwarz–Pick theorem
via predicate surface "usedInProofOf"
ⓘ
linked to:
Schwarz lemma
|
| Schwarz lemma | application of the maximum modulus principle via predicate surface "typicalProofMethod" ⓘ |
| Schwarz lemma | consideration of auxiliary function f(z)/z via predicate surface "typicalProofMethod" ⓘ |
| Hurwitz bound on automorphism groups of curves | Riemann–Hurwitz formula via predicate surface "proofUses" ⓘ |
| Hurwitz bound on automorphism groups of curves | ramified coverings of the Riemann sphere via predicate surface "proofUses" ⓘ |
| Hurwitz bound on automorphism groups of curves | orbifold Euler characteristic via predicate surface "proofUses" ⓘ |
| Kolmogorov zero–one law | independence and invariance arguments ⓘ |
| Kolmogorov zero–one law | properties of conditional expectation ⓘ |
| Picard theorem |
Casorati–Weierstrass theorem (converse directions and related results)
via predicate surface "usedInProofOf"
ⓘ
linked to:
Picard theorem
|
| Ramanujan–Petersson conjecture | Weil conjectures via predicate surface "proofUses" ⓘ |
| Ramanujan–Petersson conjecture | étale cohomology via predicate surface "proofUses" ⓘ |
| Tychonoff theorem for products of compact spaces | ultrafilters via predicate surface "canBeProvedUsing" ⓘ |
| Tychonoff theorem for products of compact spaces | nets via predicate surface "canBeProvedUsing" ⓘ |
| Tychonoff theorem for products of compact spaces | Alexander subbase theorem via predicate surface "canBeProvedUsing" ⓘ |
| Young inequality for convolutions | Hölder inequality via predicate surface "proofUses" ⓘ |
| Young inequality for convolutions |
Minkowski integral inequality
via predicate surface "proofUses"
ⓘ
linked to:
Minkowski inequality
|
| Young's inequality |
Hölder's inequality
via predicate surface "isUsedToProve"
ⓘ
linked to:
Hölder inequality
|
| Young's inequality |
Minkowski's inequality
via predicate surface "isUsedToProve"
ⓘ
linked to:
Minkowski inequality
|
| Young's inequality |
Young's convolution inequality
via predicate surface "isUsedToProve"
ⓘ
linked to:
Young inequality for convolutions
|
| Banach limit | mean ergodic theorems in some formulations via predicate surface "usedInProofOf" ⓘ |
| Banach–Saks theorem | weak compactness in reflexive spaces ⓘ |
| Banach–Saks theorem | diagonal subsequence arguments ⓘ |
| Banach–Saks theorem | properties of Cesàro averages ⓘ |