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The dataset generation failed because of a cast error
Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 10 new columns ({'full_name', 'q1c_natural', 'q4_special_case', 'q1a_lean', 'file', 'difficulty', 'q3_nickname', 'q2_slogan', 'kind', 'q1b_latex'}) and 5 missing columns ({'premise_group', 'NL_main_result', 'pr', 'formal_statement', 'formal_main_result'}).
This happened while the json dataset builder was generating data using
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'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:0f3fca4e.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:14569f29.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:29f874d6.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:570db61a.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:5ee95a40.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:60e0413b.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:88c25164.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:98836c9b.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:acc4ef74.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:bb325665.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:c52cd205.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_notequiv2i2jasqbsqdiv8:c8bd4647.jsonl', 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'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:9658d30e.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:9df5fcd3.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:bfef6768.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:cae4c2a4.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:d3158e14.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:dcba212a.jsonl', 'hf://datasets/CamiloS1111/ragsolver-index@94275fcf88cec5d2dd6f25cc7e40b5b8a1a01293/eval/runs/eval-v1/traces/numbertheory_sqmod3in01d:e393d33b.jsonl']
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1848, in _prepare_split_single
writer.write_table(table)
~~~~~~~~~~~~~~~~~~^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 765, in write_table
self._write_table(pa_table, writer_batch_size=writer_batch_size)
~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/arrow_writer.py", line 773, in _write_table
pa_table = table_cast(pa_table, self._schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2378, in table_cast
return cast_table_to_schema(table, schema)
File "/usr/local/lib/python3.14/site-packages/datasets/table.py", line 2306, in cast_table_to_schema
raise CastError(
...<3 lines>...
)
datasets.table.CastError: Couldn't cast
id: string
full_name: string
file: string
difficulty: string
kind: string
q1a_lean: string
q1b_latex: string
q1c_natural: string
q2_slogan: string
q3_nickname: string
q4_special_case: string
to
{'id': Value('string'), 'pr': Value('int64'), 'formal_main_result': Value('string'), 'NL_main_result': Value('string'), 'formal_statement': Value('string'), 'premise_group': List({'kind': Value('string'), 'docs': List(Value('string'))})}
because column names don't match
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1369, in compute_config_parquet_and_info_response
parquet_operations, partial, estimated_dataset_info = stream_convert_to_parquet(
~~~~~~~~~~~~~~~~~~~~~~~~~^
builder, max_dataset_size_bytes=max_dataset_size_bytes
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
)
^
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 948, in stream_convert_to_parquet
builder._prepare_split(split_generator=splits_generators[split], file_format="parquet")
~~~~~~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1694, in _prepare_split
for job_id, done, content in self._prepare_split_single(
~~~~~~~~~~~~~~~~~~~~~~~~~~^
gen_kwargs=gen_kwargs, job_id=job_id, **_prepare_split_args
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
):
^
File "/usr/local/lib/python3.14/site-packages/datasets/builder.py", line 1850, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
...<4 lines>...
)
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 10 new columns ({'full_name', 'q1c_natural', 'q4_special_case', 'q1a_lean', 'file', 'difficulty', 'q3_nickname', 'q2_slogan', 'kind', 'q1b_latex'}) and 5 missing columns ({'premise_group', 'NL_main_result', 'pr', 'formal_statement', 'formal_main_result'}).
This happened while the json dataset builder was generating data using
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Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
id string | pr int64 | formal_main_result string | NL_main_result string | formal_statement string | premise_group list |
|---|---|---|---|---|---|
mathlibmpr_003 | 27,229 | DoubleCoset.finite_quotient_iff_exists_finset_iUnion_eq_univ | For subgroups `H, K ≤ G`, the double-coset quotient `H \ G / K` is finite if and only if there exists a finite collection of cosets in this quotient whose double-coset preimages cover `G`. | import Mathlib.GroupTheory.DoubleCoset
open DoubleCoset
variable {G : Type*} [Group G]
/-- **Main result.** The double-coset quotient `H \ G / K` is finite if and
only if there exists a finite collection of cosets in this quotient whose
double-coset preimages cover `G`. -/
lemma finite_quotient_iff_exists_finset_iUn... | [
{
"kind": "original",
"docs": [
"DoubleCoset.union_quotToDoubleCoset",
"DoubleCoset.mk_out_eq_mul"
]
}
] |
mathlibmpr_005 | 29,235 | ClosedSubmodule.closure_map_eq_mapEquiv_closure | For a continuous `R`-linear equivalence `f : M ≃L[R] N` and any submodule `s : Submodule R M`, the closure of the image `s.map f` equals the image under `mapEquiv f` of the closure of `s`. In other words, "closure of image = image of closure" through a continuous linear equivalence. | import Mathlib
namespace ClosedSubmodule
variable {R M N : Type*} [Ring R]
[TopologicalSpace M] [AddCommGroup M] [Module R M]
[TopologicalSpace N] [AddCommGroup N] [Module R N]
(f : M ≃L[R] N)
/-- Action of a continuous `R`-linear equivalence on closed submodules. -/
def mapEquiv : ClosedSubmodule R M ≃ Closed... | [
{
"kind": "original",
"docs": [
"ContinuousLinearEquiv.image_closure",
"IsClosedMap.closure_image_eq_of_continuous"
]
},
{
"kind": "alternative",
"docs": [
"Homeomorph.image_closure",
"IsClosedMap.closure_image_eq_of_continuous"
]
}
] |
mathlibmpr_006 | 30,112 | ProbabilityTheory.Kernel.exists_measurable_map_eq_unitInterval | Let `Y` be a standard Borel space and `κ : Kernel X Y` a Markov kernel. Then there exists a jointly measurable map `f : X → I → Y` such that for every `a : X`, the pushforward of the uniform measure `volume` on the unit interval `I` by `f a` equals `κ a`. | import Mathlib
open MeasureTheory unitInterval
namespace ProbabilityTheory.Kernel
variable {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y] [StandardBorelSpace Y]
[Nonempty Y]
/-- **Main result.** Kallenberg's "isolation of randomness": for any Markov
kernel `κ : Kernel X Y` with `Y` a standard Borel space, ... | [
{
"kind": "original",
"docs": [
"measurableEmbedding_sigmoid_comp_embeddingReal",
"MeasurableEmbedding.comp"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.ext_of_Iic"
]
},
{
"kind": "alternative",
"docs": [
"MeasureTheory.exists_subset_re... |
mathlibmpr_008 | 31,121 | GradedObject.eulerChar_eq_sum_finSet_of_finrankSupport_subset | Let `c` be a complex shape on an index type `ι` with an `EulerCharSigns` instance, and let `X` be a graded `R`-module object with `finrank R (X i) = 0` for every `i` outside a given finite set `s`. Then the `finsum` Euler characteristic equals the explicit finite sum: `eulerChar c X = ∑ i ∈ s, (χ i : ℤ) · finrank R (X ... | import Mathlib
open CategoryTheory
namespace ComplexShape
variable {ι : Type*}
/-- Alternating-sign data on the index type of a complex shape. -/
class EulerCharSigns (c : ComplexShape ι) where
χ : ι → ℤˣ
χ_neg_of_rel : ∀ ⦃i j⦄, c.Rel i j → χ j = -χ i
end ComplexShape
namespace GradedObject
variable {ι R : T... | [
{
"kind": "original",
"docs": [
"finsum_eq_sum_of_support_subset",
"finsum_eq_finset_sum_of_support_subset",
"finsum_eq_sum_of_support_subset_of_finite"
]
}
] |
mathlibmpr_009 | 31,219 | IsBaseChange.of_left_exact | Let `S` be a flat `R`-algebra and consider a commutative diagram with exact rows ``` M₁ --f--> M₂ --g--> M₃ | | | i₁ i₂ i₃ v v v N₁ --f'-> N₂ --g'-> N₃ ``` where `f, f'` are injective. If `i₂` and `i₃` exhibit `Nₖ` as the base change of `Mₖ` from `R` to `S`, then so does `i₁`. | import Mathlib.RingTheory.IsTensorProduct
import Mathlib.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Tower
variable {R : Type*} [CommRing R] (S : Type*) [CommRing S] [Algebra R S]
variable {M₁ M₂ M₃ N₁ N₂ N₃ : Type*}
[AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃]
[AddCommGroup N₁] [Add... | [
{
"kind": "original",
"docs": [
"Module.Flat.lTensor_exact",
"Module.Flat.iff_lTensor_exact",
"Module.Flat.iff_lTensor_exact'",
"Module.Flat.rTensor_exact"
]
},
{
"kind": "original",
"docs": [
"Module.Flat.lTensor_preserves_injective_linearMap",
"Module.Fl... |
mathlibmpr_010 | 31,388 | exists_isExtrOn_uIoo_of_tendsto | Let `α` be a densely-ordered linearly-ordered topological space and `β` a `T₂` regular space. If `f : α → β` is continuous on the unordered open interval `uIoo a b` (where `a ≠ b`) and tends, within `uIoo a b`, to the same limit `l` as the argument approaches either endpoint, then `f` attains a global extremum on `uIoo... | import Mathlib
open Set Filter Topology
variable {α β : Type*}
[ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [DenselyOrdered α] [OrderTopology α]
[LinearOrder β] [TopologicalSpace β] [OrderTopology β]
{f : α → β} {a b : α} {l : β}
/-- **Main result.** If `f` is continuous on the unordered open
inte... | [
{
"kind": "original",
"docs": [
"exists_Ioo_extr_on_Icc",
"exists_isLocalExtr_Ioo"
]
}
] |
mathlibmpr_011 | 31,411 | ConvexOn.sSup_affine_eq | Let `E` be a real locally convex topological vector space (with `𝕜 = ℝ` or `ℂ` providing the affine functionals), `s ⊆ E` a closed convex subset, and `φ : E → ℝ` a function that is convex and lower-semicontinuous on `s`. Then the family ``` 𝓕 := { f : s → ℝ | f ≤ φ|_s ∧ f = (Re ∘ l)|_s + c for some l ∈ E →L[𝕜] 𝕜, c... | import Mathlib
open Function Set RCLike ContinuousLinearMap
namespace ConvexOn
variable {𝕜 E : Type*} {s : Set E} {φ : E → ℝ} [RCLike 𝕜]
[AddCommGroup E] [Module ℝ E] [TopologicalSpace E] [AddGroup E]
[ContinuousSMul ℝ E] [LocallyConvexSpace ℝ E] [Module 𝕜 E] [ContinuousSMul 𝕜 E]
/-- **Main result.** Let `E... | [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_point_closed",
"geometric_hahn_banach_closed_point",
"RCLike.geometric_hahn_banach_point_closed"
]
},
{
"kind": "original",
"docs": [
"Convex.linear_preimage",
"Convex.is_linear_preimage"
]
},
{
... |
mathlibmpr_012 | 31,719 | Complex.borelCaratheodory_zero | Let `f : ℂ → ℂ` be holomorphic on the open ball `‖z‖ < R` with `f 0 = 0`, and assume `Re (f z) < M` for every `z` in the ball (with `0 < M`, `0 < R`). Then for every `z` with `‖z‖ < R`, ``` ‖f z‖ ≤ 2 M ‖z‖ / (R − ‖z‖). ``` (The general Borel–Carathéodory bound `borelCaratheodory` follows by applying this to `f − f 0` a... | import Mathlib.Analysis.Complex.Schwarz
open Metric Set
namespace Complex
variable {f : ℂ → ℂ} {M R : ℝ} {z : ℂ}
/-- **Main result.** Borel–Carathéodory (case `f 0 = 0`). If `f` is
holomorphic on the open ball `‖z‖ < R` with `f 0 = 0`, and
`Re (f z) < M` for every `z` in the ball (with `0 < M`, `0 < R`), then
for e... | [
{
"kind": "original",
"docs": [
"Complex.dist_le_div_mul_dist_of_mapsTo_ball"
]
}
] |
mathlibmpr_014 | 32,186 | norm_sub_le_integral_of_norm_deriv_le_of_le | Let `f : ℝ → E` be continuous on `[a, b]` and differentiable on `(a, b)`, with `‖f'(t)‖ ≤ B(t)` for almost every `t ∈ (a, b)` and `B` interval-integrable on `[a, b]`. Then | import Mathlib
open MeasureTheory intervalIntegral
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E} {B : ℝ → ℝ} {a b : ℝ}
/-- **Main result.** If `f : ℝ → E` is continuous on `[a,b]` and differentiable
on `(a,b)` with `‖deriv f t‖ ≤ B t` a.e., and `B` is interval-integrable on
`[a,b]`, the... | [
{
"kind": "original",
"docs": [
"intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le",
"intervalIntegral.integral_eq_sub_of_hasDeriv_right"
]
},
{
"kind": "original",
"docs": [
"intervalIntegral.norm_integral_le_integral_norm",
"MeasureTheory.norm_integral_le_int... |
mathlibmpr_015 | 32,260 | EuclideanGeometry.angle_eq_angle_div_two_of_oangle_eq_of_sSameSide | Let `p₁, p₂, p₃, p₄` be points in a real Euclidean affine space with `p₁ ≠ p₂`. If `p₃` bisects the oriented angle `∡p₁p₂p₄` (i.e. `∡p₁p₂p₃ = ∡p₃p₂p₄`) and `p₃, p₄` lie on the same (strictly defined) side of the line through `p₁` and `p₂`, then the unoriented angle satisfies | import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
namespace EuclideanGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P]
[NormedAddTorsor V P] [Fact (Module.finrank ℝ V = 2)] [Module.Oriented ℝ V (Fin 2)]
/-- **Main result.** Let `p₁ p₂ p₃ p₄` be points with `p₁ ≠ p... | [
{
"kind": "original",
"docs": [
"AffineSubspace.SSameSide.oangle_sign_eq"
]
},
{
"kind": "original",
"docs": [
"EuclideanGeometry.angle_eq_abs_oangle_toReal",
"Orientation.angle_eq_abs_oangle_toReal"
]
}
] |
mathlibmpr_016 | 32,744 | Set.Definable.preimage_map | Let `L` be a first-order language, `M` an `L`-structure, `A ⊆ M` a parameter set, `β` a finite type, and `F : (α → M) → (β → M)` a *definable map* over `A` (each coordinate `v ↦ F(v)(i)` is a definable function over `A`). For every `A`-definable set `S ⊆ (β → M)`, the preimage `F⁻¹(S)` is `A`-definable. | import Mathlib.ModelTheory.Definability
namespace Set
open FirstOrder Language
variable {M : Type*} {L : Language} [L.Structure M]
{α β : Type*} (A : Set M)
/-- A function whose tuple-graph is definable over `A`. -/
def DefinableFun (f : (α → M) → M) : Prop :=
Definable A L { vw : α ⊕ Unit → M | vw (Sum.inr ())... | [
{
"kind": "original",
"docs": [
"Set.Definable.preimage_comp"
]
},
{
"kind": "original",
"docs": [
"Set.definable_finset_inf",
"Set.definable_biInter_finset",
"Set.definable_finset_biInter"
]
},
{
"kind": "original",
"docs": [
"FirstOrder.Languag... |
mathlibmpr_017 | 33,189 | ModelWithCorners.isOpen_interior | Let `M` be a C¹ manifold modelled on `(E, H, I : ModelWithCorners 𝕜 E H)` with `n ≠ 0`. Then `I.interior M = { x : M | extChartAt I x x ∈ interior (range I) }` is open. | import Mathlib
namespace ModelWithCorners
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{H : Type*} [TopologicalSpace H]
(I : ModelWithCorners 𝕜 E H)
(M : Type*) [TopologicalSpace M] [ChartedSpace H M]
{n : WithTop ℕ∞} [IsManifold I n M]
/-- **Mai... | [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_point_open",
"geometric_hahn_banach_open_point",
"RCLike.geometric_hahn_banach_point_open",
"RCLike.geometric_hahn_banach_open_point"
]
},
{
"kind": "original",
"docs": [
"IsLocalMax.fderiv_eq_zero",
"... |
mathlibmpr_018 | 33,275 | Polynomial.Chebyshev.eval_iterate_derivative_le_of_forall_abs_le_one | Let `P : ℝ[X]` have `deg P ≤ n` and satisfy `|P(x)| ≤ 1` for all `x ∈ [-1, 1]`. Then for every `k ∈ ℕ` and every `x ≥ 1`, `(P^{(k)})(x) ≤ (T_n^{(k)})(x)`, where `T_n` is the `n`-th Chebyshev polynomial of the first kind. | import Mathlib
namespace Polynomial.Chebyshev
variable {n k : ℕ} {x : ℝ}
/-- **Main result.** Let `P : ℝ[X]` have `degree P ≤ n` and satisfy
`|P(x)| ≤ 1` for all `x ∈ [-1, 1]`. Then for every `k ∈ ℕ` and every `x ≥ 1`,
`(P^{(k)})(x) ≤ (T_n^{(k)})(x)`. -/
theorem eval_iterate_derivative_le_of_forall_abs_le_one
{P... | [
{
"kind": "original",
"docs": [
"Lagrange.eval_iterate_derivative_eq_sum",
"Lagrange.iterate_derivative_interpolate"
]
},
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.sumNodes_le_sumNodes_T",
"Polynomial.Chebyshev.sumNodes_eq_sumNodes_T_iff"
]
}
] |
mathlibmpr_019 | 33,372 | MeasureTheory.IsStoppingTime.iInf | Let `F = (F_t)` be a right-continuous filtration on `ι` (a conditionally complete linearly-ordered first-countable densely-ordered space with no maximum), and let `(τ_n)_{n : κ}` be a countable family of stopping times for `F` with values in `WithTop ι`. Then `ω ↦ ⨅_n τ_n(ω)` is again a stopping time. | import Mathlib.Probability.Process.Stopping
namespace MeasureTheory
open ProbabilityTheory
variable {Ω ι κ : Type*} [Countable κ]
[ConditionallyCompleteLinearOrderBot ι] [TopologicalSpace ι] [OrderTopology ι]
[DenselyOrdered ι] [NoMaxOrder ι] [FirstCountableTopology ι]
{m : MeasurableSpace Ω} {F : Filtration ι... | [
{
"kind": "original",
"docs": [
"MeasureTheory.Filtration.IsRightContinuous"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.IsStoppingTime.measurableSet_ge'"
]
}
] |
mathlibmpr_020 | 33,449 | ProbabilityTheory.tendsto_choose_mul_pow_of_tendsto_mul_atTop | Fix `k : ℕ`. Let `p : ℕ → ℝ` be a sequence with `n · p_n → r`. Then `(n choose k) · p_n^k · (1 − p_n)^{n − k} → e^{−r} · r^k / k!`. | import Mathlib.Probability.Distributions.Poisson
import Mathlib.Analysis.SpecialFunctions.Exp
open Filter
namespace ProbabilityTheory
/-- **Main result.** Poisson limit theorem: if `n · p_n → r`, then for each
fixed `k`, `(n choose k) · p_n^k · (1 − p_n)^{n − k} → e^{−r} · r^k / k!`. -/
theorem tendsto_choose_mul_po... | [
{
"kind": "original",
"docs": [
"Real.tendsto_one_add_pow_exp_of_tendsto",
"Complex.tendsto_one_add_pow_exp_of_tendsto",
"Real.tendsto_one_add_rpow_exp_of_tendsto"
]
},
{
"kind": "original",
"docs": [
"isEquivalent_choose"
]
}
] |
mathlibmpr_021 | 33,463 | Polynomial.norm_coeff_le_choose_mul_mapMahlerMeasure | Let `A` be a normed ring and `v : A →+* ℂ` an isometry. For every polynomial `p ∈ A[X]` and every `n : ℕ`, `‖p.coeff n‖ ≤ binomial(deg p, n) · M_v(p)`, where `M_v(p) := mahlerMeasure (p.map v)` is the generalised Mahler measure. | import Mathlib.Analysis.Polynomial.MahlerMeasure
namespace Polynomial
variable {A : Type*} [NormedRing A]
/-- The Mahler measure of `p ∈ A[X]` computed via a ring hom `v : A →+* ℂ`. -/
noncomputable def mapMahlerMeasure (p : Polynomial A) (v : A →+* ℂ) : ℝ :=
(p.map v).mahlerMeasure
/-- **Main result.** Let `v : ... | [
{
"kind": "original",
"docs": [
"Polynomial.norm_coeff_le_choose_mul_mahlerMeasure"
]
}
] |
mathlibmpr_022 | 33,479 | EuclideanGeometry.cospherical_of_mul_dist_eq_mul_dist_of_angle_eq_pi | Let `p₁, p₂, p₃, p₄, p` be points in a real inner-product affine space `P`. Suppose `p` lies strictly between `p₁` and `p₂` and strictly between `p₃` and `p₄` (so `∠ p₁ p p₂ = ∠ p₃ p p₄ = π`), and that `p₁, p, p₃` are not collinear. If `dist p₁ p · dist p₂ p = dist p₃ p · dist p₄ p`, then `{p₁, p₂, p₃, p₄}` is cospheri... | import Mathlib
namespace EuclideanGeometry
variable {V P : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
[MetricSpace P] [NormedAddTorsor V P]
/-- **Main result.** (Converse of intersecting-chords theorem.) If
`∠ p₁ p p₂ = π` and `∠ p₃ p p₄ = π` (i.e. `p` lies on lines `p₁p₂` and `p₃p₄`),
`p₁, p, p₃` are n... | [
{
"kind": "original",
"docs": [
"EuclideanGeometry.cospherical_of_two_zsmul_oangle_eq_of_not_collinear",
"EuclideanGeometry.concyclic_of_two_zsmul_oangle_eq_of_not_collinear",
"EuclideanGeometry.cospherical_or_collinear_of_two_zsmul_oangle_eq",
"EuclideanGeometry.concyclic_or_colline... |
mathlibmpr_023 | 33,507 | Polynomial.sum_sq_norm_coeff_eq_circleAverage | For any complex polynomial `p ∈ ℂ[X]`, `Σ_{i ∈ supp p} ‖p.coeff i‖² = (1 / 2π) ∫₀^{2π} ‖p(e^{iθ})‖² dθ`. That is, Parseval's identity on the unit circle, read coefficient-wise. | import Mathlib
open Complex MeasureTheory Set AddCircle
open scoped Real
namespace Polynomial
variable (p : ℂ[X])
local instance instTwoPiPos : Fact (0 < 2 * Real.pi) := Fact.mk Real.two_pi_pos
@[fun_prop, continuity]
theorem continuous_toCircle {T : ℝ} : Continuous (@toCircle T) :=
continuous_coinduced_dom.mp... | [
{
"kind": "original",
"docs": [
"tsum_sq_fourierCoeff",
"hasSum_sq_fourierCoeff"
]
},
{
"kind": "original",
"docs": [
"HilbertBasis.repr_apply_apply",
"OrthonormalBasis.repr_apply_apply"
]
},
{
"kind": "alternative",
"docs": [
"orthonormal_fourie... |
mathlibmpr_024 | 33,520 | FirstOrder.Language.MeetsDefinable.isElementary_closure | Let `M` be a structure for a first-order language `L`, and let `A ⊆ M` be a subset that meets every nonempty `L`-definable subset of `M`. Then the substructure of `M` generated by `A` is an elementary substructure. | import Mathlib.ModelTheory.ElementarySubstructures
import Mathlib.ModelTheory.Definability
namespace FirstOrder.Language
variable {L : Language} {M : Type*} [L.Structure M]
/-- A set meets definable sets if it meets every nonempty definable subset. -/
def MeetsDefinable (A : Set M) : Prop :=
∀ (D : Set M), D.Nonem... | [
{
"kind": "original",
"docs": [
"FirstOrder.Language.Substructure.isElementary_of_exists",
"FirstOrder.Language.Substructure.toElementarySubstructure"
]
},
{
"kind": "original",
"docs": [
"FirstOrder.Language.Substructure.coe_closure_eq_range_term_realize",
"FirstOrde... |
mathlibmpr_025 | 33,676 | Polynomial.Chebyshev.integral_eval_T_real_mul_eval_T_real_measureT_of_ne | For `n, m : ℕ` with `n ≠ m`, `∫ (T_n(x) · T_m(x)) / √(1 − x²) dx = 0`, where the integral is taken on `(−1, 1]` against the weighted Lebesgue measure `μ_T = 1_{(−1,1]} · (1 / √(1 − x²)) dx`. | import Mathlib
namespace Polynomial.Chebyshev
open Real intervalIntegral MeasureTheory
/-- Lebesgue measure scaled by `√(1 - x²)⁻¹`, restricted to `(-1, 1]`. -/
noncomputable def measureT : Measure ℝ :=
(volume.withDensity
fun x ↦ ENNReal.ofNNReal ⟨Real.sqrt (1 - x ^ 2)⁻¹, by positivity⟩).restrict (Set.Ioc (-1... | [
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.T_mul_T",
"Polynomial.Chebyshev.C_mul_C"
]
},
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.T_real_cos",
"Polynomial.Chebyshev.T_complex_cos",
"Polynomial.Chebyshev.C_two_mul_real_cos"
]
},
{
... |
mathlibmpr_027 | 33,736 | Polynomial.gaussNorm_mul | Let `R` be a ring, `v : R → ℝ` a non-archimedean absolute value, and `c > 0` a real number. Then for any `p, q ∈ R[X]`, `gaussNorm v c (p · q) = gaussNorm v c p · gaussNorm v c q`. | import Mathlib.RingTheory.Polynomial.GaussNorm
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Ring.IsNonarchimedean
namespace Polynomial
variable {R : Type*} [Ring R] {v : AbsoluteValue R ℝ} {c : ℝ}
(hna : IsNonarchimedean v) (hc : 0 < c)
include hna hc in
/-- **Main result.** If `v`... | [
{
"kind": "original",
"docs": [
"IsNonarchimedean",
"Valuation.map_add"
]
},
{
"kind": "original",
"docs": [
"Polynomial.le_gaussNorm",
"PowerSeries.le_gaussNorm"
]
},
{
"kind": "original",
"docs": [
"Polynomial.gaussNorm_nonneg",
"PowerSer... |
mathlibmpr_028 | 33,802 | Representation.IsIrreducible.bijective_or_eq_zero | Let `ρ : Representation k G V` and `σ : Representation k G W` be irreducible monoid representations of `G` on `k`-vector spaces. Then any intertwining map `f : ρ → σ` is either the zero map or a bijection. | import Mathlib.RepresentationTheory.Irreducible
import Mathlib.RepresentationTheory.Intertwining
namespace Representation.IsIrreducible
variable {G k V W : Type*}
[Monoid G] [Field k]
[AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W]
{ρ : Representation k G V} {σ : Representation k G W}
[IsIrreducib... | [
{
"kind": "original",
"docs": [
"LinearMap.bijective_or_eq_zero",
"LinearMap.bijective_of_ne_zero",
"CategoryTheory.isIso_of_hom_simple"
]
},
{
"kind": "original",
"docs": [
"Representation.irreducible_iff_isSimpleModule_asModule",
"Representation.is_simple_modu... |
mathlibmpr_029 | 33,860 | Probability.integral_cauchyPDFReal | For every location `x₀ : ℝ` and positive scale `γ > 0`, the Cauchy probability density `x ↦ (πγ)⁻¹ · (1 + ((x − x₀)/γ)²)⁻¹` integrates to `1` over `ℝ`. | import Mathlib
open MeasureTheory Measure Real
open scoped NNReal
namespace Probability
/-- The pdf of the Cauchy distribution depending on its location `x₀` and scale `γ`. -/
noncomputable def cauchyPDFReal (x₀ : ℝ) (γ : ℝ≥0) (x : ℝ) : ℝ :=
π⁻¹ * γ * ((x - x₀) ^ 2 + γ ^ 2)⁻¹
/-- **Main result.** For positive sca... | [
{
"kind": "original",
"docs": [
"integral_one_div_one_add_sq",
"integral_univ_inv_one_add_sq",
"integral_inv_one_add_sq"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.integral_sub_right_eq_self"
]
},
{
"kind": "original",
"docs": [
"Measure... |
mathlibmpr_030 | 33,941 | ScottContinuousOn.comp | Let `f : α → β` and `g : β → γ` be Scott-continuous on a domain family `D` of subsets of `α` (resp. `D'` of `β`). Suppose `D` is closed under two-element subsets `{a,b}` (with `a ≤ b`) and that `f` sends `D`-sets to `D'`-sets after image. Then `g ∘ f` is Scott continuous on `D`. | import Mathlib.Order.ScottContinuity
open Set
variable {α β γ : Type*}
section ScottContinuous
variable [Preorder α] [Preorder β] [Preorder γ] {D : Set (Set α)}
{f : α → β}
/-- **Main result.** Composition of Scott-continuous-on-a-domain functions
is Scott continuous (on the doubleton-closed family `D`, given tha... | [
{
"kind": "original",
"docs": [
"ScottContinuousOn.monotone",
"ScottContinuous.monotone"
]
},
{
"kind": "alternative",
"docs": [
"Topology.IsScott.scottContinuousOn_iff_continuous"
]
}
] |
mathlibmpr_031 | 33,972 | Polynomial.zero_lt_eval_of_roots_lt_of_leadingCoeff_nonneg | Let `P ∈ ℝ[X]` and `x ∈ ℝ`. If every (real) root of `P` is strictly less than `x` and `P` has a non-negative leading coefficient, then `P(x) > 0`. | import Mathlib
open Real Polynomial
namespace Polynomial
variable {P : ℝ[X]} {x : ℝ}
/-- **Main result.** A real polynomial is strictly positive at `x` if all
its (real) roots are strictly less than `x` and its leading coefficient is
non-negative. -/
theorem zero_lt_eval_of_roots_lt_of_leadingCoeff_nonneg
(hroo... | [
{
"kind": "original",
"docs": [
"intermediate_value_Icc",
"intermediate_value_uIcc",
"intermediate_value_Icc'"
]
},
{
"kind": "original",
"docs": [
"Polynomial.tendsto_atTop_of_leadingCoeff_nonneg"
]
}
] |
mathlibmpr_032 | 34,006 | Submodule.CoFG.inf | Over a Noetherian ring `R`, if `S` and `T` are CoFG submodules of an `R`-module `M`, then so is their intersection `S ⊓ T`. | import Mathlib.RingTheory.Noetherian.Basic
import Mathlib.RingTheory.Finiteness.Basic
namespace Submodule
variable {R : Type*} [Ring R] {M : Type*} [AddCommGroup M] [Module R M]
/-- A submodule `S` of a module `M` is co-finitely generated (CoFG) if the quotient
space `M ⧸ S` is finitely generated. -/
abbrev CoFG (... | [
{
"kind": "original",
"docs": [
"LinearMap.ker_prod",
"ContinuousLinearMap.ker_prod"
]
},
{
"kind": "original",
"docs": [
"LinearMap.quotKerEquivRange"
]
}
] |
mathlibmpr_033 | 34,109 | Nat.infinite_odd_abundant | The set of odd abundant natural numbers is infinite. | import Mathlib
namespace Nat
/-- **Main result.** The set of odd abundant natural numbers is infinite. -/
theorem infinite_odd_abundant : {n : ℕ | Odd n ∧ n.Abundant}.Infinite := sorry
end Nat | [
{
"kind": "original",
"docs": [
"Set.infinite_iff_exists_gt"
]
},
{
"kind": "original",
"docs": [
"Finset.sum_image"
]
},
{
"kind": "original",
"docs": [
"Finset.sum_le_sum_of_subset"
]
}
] |
mathlibmpr_034 | 34,255 | Submodule.exists_eq_colon_of_mem_minimalPrimes | Let `R` be a Noetherian (commutative) ring, `M` an `R`-module, `N ⊆ M` a submodule, and `x ∈ M`. If `I` is a minimal prime over the annihilator-style ideal `N : {x}` (`= colon N {x}`), then there exists `x' ∈ M` with `I = N : {x'}`. | import Mathlib.RingTheory.Ideal.MinimalPrime.Noetherian
import Mathlib.RingTheory.Finiteness.Ideal
import Mathlib.RingTheory.Noetherian.Basic
namespace Submodule
variable {R M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M] {N : Submodule R M}
{I : Ideal R} {x : M}
/-- **Main result.** In a Noetherian rin... | [
{
"kind": "original",
"docs": [
"Ideal.exists_radical_pow_le_of_fg",
"Ideal.exists_pow_le_of_le_radical_of_fg",
"Ideal.exists_pow_le_of_le_radical_of_fg_radical"
]
},
{
"kind": "original",
"docs": [
"Ideal.sInf_minimalPrimes",
"Ideal.radical_eq_sInf"
]
},
... |
mathlibmpr_035 | 34,260 | IsCompact.powerset_vietoris | For every compact subset `K` of a topological space `α`, its powerset `K.powerset = {t : Set α | t ⊆ K}`, equipped with the (subspace of the) Vietoris topology on `Set α`, is compact. | import Mathlib.Topology.Sets.VietorisTopology
namespace IsCompact
variable {α : Type*} [TopologicalSpace α] {K : Set α}
attribute [local instance] TopologicalSpace.vietoris
/-- **Main result.** For every compact subset `K` of a topological space `α`,
its powerset `K.powerset = {t : Set α | t ⊆ K}`, equipped with th... | [
{
"kind": "original",
"docs": [
"isCompact_generateFrom",
"isCompact_generateFrom'",
"compactSpace_generateFrom",
"compactSpace_generateFrom'"
]
},
{
"kind": "original",
"docs": [
"IsCompact.elim_finite_subcover_image",
"isCompact_iff_finite_subcover"
... |
mathlibmpr_036 | 34,263 | HasMFDerivAt.smul | Let `M` be a smooth manifold, `𝕜` a non-trivially normed field, and `V` a `𝕜`-normed space. Suppose `f : M → 𝕜` and `g : M → V` are maps having `HasMFDerivAt`-derivatives `f' : T_x M → 𝕜` and `g' : T_x M → V` at `x` (with both targets canonically identified with the corresponding tangent spaces via `NormedSpace.fro... | import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
open Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{H : Type*} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H}
{M : Type*} [Topological... | [
{
"kind": "original",
"docs": [
"HasFDerivWithinAt.smul",
"HasFDerivAt.smul",
"fderivWithin_smul",
"fderiv_smul"
]
},
{
"kind": "original",
"docs": [
"ContinuousLinearMap.toSpanSingleton"
]
},
{
"kind": "alternative",
"docs": [
"contMDiff_s... |
mathlibmpr_037 | 34,297 | exteriorPower.ιMulti_family_span_fixedDegree_of_span | Let `M` be an `R`-module, `n : ℕ`, and `v : I → M` an indexed family with `I` linearly ordered. If the range of `v` spans `M`, then the family `(v(i₁) ∧ ⋯ ∧ v(iₙ))` indexed by ordered subsets `{i₁ < ⋯ < iₙ} ⊆ I` of size `n` spans the `n`th exterior power `⋀ⁿ M ⊆ ⋀M` (viewed as a submodule of the exterior algebra). | import Mathlib.LinearAlgebra.ExteriorPower.Basic
namespace exteriorPower
variable {R M I : Type*} [CommRing R] [AddCommGroup M] [Module R M]
[LinearOrder I] (n : ℕ) (v : I → M)
/-- **Main result.** If the range of `v : I → M` spans `M`, then the family of
ordered wedges `(v i₁) ∧ ⋯ ∧ (v iₙ)` indexed by ordered `n`... | [
{
"kind": "original",
"docs": [
"Submodule.span_pow",
"Submodule.pow_eq_span_pow_set"
]
},
{
"kind": "original",
"docs": [
"AlternatingMap.map_perm",
"AlternatingMap.map_congr_perm"
]
},
{
"kind": "alternative",
"docs": [
"exteriorPower.alternati... |
mathlibmpr_038 | 34,298 | MeasureTheory.tendsto_integral_Ioi_zero | Let `f : ℝ → E` be a function and `μ` a measure on `ℝ`. For a function `b : ι → ℝ` and a filter `l` on `ι`, if `b → +∞` along `l`, then `∫_{Ioi (b i)} f dμ → 0` along `l`. | import Mathlib
namespace MeasureTheory
variable {ι E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
{f : ℝ → E} {μ : Measure ℝ}
/-- **Main result.** For `b : ι → ℝ` tending to `+∞` along filter `l`,
`∫_{Ioi (b i)} f dμ → 0`. -/
theorem tendsto_integral_Ioi_zero
{l : Filter ι} {b : ι → ℝ} (hb : Filter.Tendst... | [
{
"kind": "original",
"docs": [
"MeasureTheory.intervalIntegral_tendsto_integral_Ioi"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.setIntegral_union",
"MeasureTheory.integral_union_ae"
]
},
{
"kind": "alternative",
"docs": [
"MeasureTheory.tends... |
mathlibmpr_039 | 34,364 | Polynomial.Chebyshev.abs_iterate_derivative_T_real_le | For every integer `n`, every `k ∈ ℕ`, and every real `x` with `|x| ≤ 1`, `|T_n^{(k)}(x)| ≤ T_n^{(k)}(1)` (where `T_n` is the `n`th Chebyshev polynomial of the first kind, real-valued). | import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev
namespace Polynomial.Chebyshev
/-- **Main result.** For every integer `n`, every `k ∈ ℕ`, and every real `x`
with `|x| ≤ 1`, `|T_n^{(k)}(x)| ≤ T_n^{(k)}(1)`. -/
theorem abs_iterate_derivative_T_real_le (n : ℤ) (k : ℕ) {x : ℝ} (hx : |x| ≤ 1) :
|((Pol... | [
{
"kind": "original",
"docs": [
"Polynomial.Chebyshev.abs_eval_T_real_le_one"
]
},
{
"kind": "alternative",
"docs": [
"Polynomial.Chebyshev.T_real_cos",
"Polynomial.Chebyshev.T_complex_cos"
]
},
{
"kind": "alternative",
"docs": [
"DifferentiableOn.circ... |
mathlibmpr_040 | 34,594 | AddTorsor.convexCombination_assoc | Let `P` be an affine space over a strictly ordered ring `R` with model module `V`, and let `f : StdSimplex R (StdSimplex R P)` be a "simplex of simplices" (a probability distribution over probability distributions over `P`). Then the convex combination of the inner combinations equals the convex combination of the join... | import Mathlib
namespace AddTorsor
variable {R V P : Type*} [Ring R] [PartialOrder R] [IsStrictOrderedRing R]
[AddCommGroup V] [Module R V] [AddTorsor V P]
/-- The convex combination of points in an affine space, given a probability distribution. -/
noncomputable def convexCombination (s : StdSimplex R P) : P :=
... | [
{
"kind": "original",
"docs": [
"Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one",
"Finset.weightedVSubOfPoint_vadd_eq_of_sum_eq_one"
]
}
] |
mathlibmpr_042 | 34,632 | SimpleGraph.completeBipartiteGraph_isContained_bipartiteDoubleCover | Let `G : SimpleGraph V` and `α, β` non-empty finite types. Then `K_{α,β}` (the complete bipartite graph) is contained as a subgraph of the bipartite double cover `G.bipartiteDoubleCover` iff it is contained in `G`. | import Mathlib
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
/-- The bipartite double cover of `G`: a graph on `V ⊕ V` with `inl v ∼ inr w`
iff `v ∼ w` in `G`. -/
def bipartiteDoubleCover : SimpleGraph (V ⊕ V) where
Adj
| Sum.inl v, Sum.inr w => G.Adj v w
| Sum.inr v, Sum.inl w => G.Adj v ... | [
{
"kind": "original",
"docs": [
"SimpleGraph.completeBipartiteGraph_isContained_iff"
]
}
] |
mathlibmpr_043 | 34,636 | integral_preimage | For any Banach space `E` and any function `f : UnitAddTorus d → E`, and for any base point `a : d → ℝ`, `∫ x : UnitAddTorus d, f x = ∫ x in {x | ∀ i, x i ∈ (a i, a i + 1]}, f (fun i ↦ ↑(x i))`. | import Mathlib.Analysis.Fourier.AddCircleMulti
open MeasureTheory
variable {d : Type*} [Fintype d] {E : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E]
/-- **Main result.** For any function `f : UnitAddTorus d → E` and any base
point `a : d → ℝ`,
`∫ x : UnitAddTorus d, f x = ∫ x in {x | ∀ i, x i ∈ Ioc (a i) (a i +... | [
{
"kind": "original",
"docs": [
"MeasureTheory.integral_map_equiv",
"MeasureTheory.lintegral_map_equiv",
"MeasureTheory.setIntegral_map_equiv"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.integral_subtype_comap"
]
},
{
"kind": "original",
"docs"... |
mathlibmpr_044 | 34,709 | discrete_gronwall | Let `u, b, c : ℕ → ℝ` with `0 ≤ u n₀`, `0 ≤ b n`, `0 ≤ c n` for all `n ≥ n₀`, and assume the recurrence inequality `u (n+1) ≤ (1 + c n) · u n + b n` for `n ≥ n₀`. Then for every `n ≥ n₀`, `u n ≤ (u n₀ + ∑_{k = n₀}^{n−1} b k) · exp (∑_{i = n₀}^{n−1} c i)`. | import Mathlib
/-- **Main result.** Discrete Grönwall inequality: if `u (n+1) ≤ (1 + c n)·u n + b n`
for all `n ≥ n₀`, with `0 ≤ u n₀`, `0 ≤ b n`, `0 ≤ c n` for `n ≥ n₀`, then for
every `n ≥ n₀`,
`u n ≤ (u n₀ + ∑_{k = n₀}^{n−1} b k) · exp (∑_{i = n₀}^{n−1} c i)`. -/
theorem discrete_gronwall {u b c : ℕ → ℝ} {n₀ : ℕ}
... | [
{
"kind": "original",
"docs": [
"Real.add_one_le_exp",
"Real.add_one_lt_exp"
]
}
] |
mathlibmpr_045 | 34,735 | IsRelPrime.moebius_mul | Let `α` be a commutative monoid with zero that is a unique factorization monoid. For any `a, b : α` with `a` and `b` coprime (`IsRelPrime a b`), `μ(a · b) = μ(a) · μ(b)`. | import Mathlib
noncomputable section
namespace UniqueFactorizationMonoid
variable {α : Type*} [CommMonoidWithZero α] [UniqueFactorizationMonoid α]
[DecidableEq α] [NormalizedGCDMonoid α]
open Classical
/-- The Möbius function on a unique factorization monoid:
`μ(a) = (−1)^(factors a).card` if `a` is squarefree, a... | [
{
"kind": "original",
"docs": [
"squarefree_mul_iff",
"Nat.squarefree_mul_iff"
]
},
{
"kind": "original",
"docs": [
"UniqueFactorizationMonoid.factors_mul"
]
},
{
"kind": "original",
"docs": [
"Multiset.card_eq_card_of_rel"
]
}
] |
mathlibmpr_046 | 34,858 | Affine.Simplex.setInterior_map | Let `f : P →ᵃ[k] P₂` be an injective affine map, `s` an affine simplex in `P`, and `I ⊆ k` a weight-set defining the "interior" of a simplex. Then the interior of the image simplex `s.map f hf` (with weights in `I`) equals the image under `f` of the interior of `s`: `(s.map f hf).setInterior I = f '' s.setInterior I`. | import Mathlib
namespace Affine.Simplex
variable {k V P V₂ P₂ : Type*}
[Ring k]
[AddCommGroup V] [Module k V] [AddTorsor V P]
[AddCommGroup V₂] [Module k V₂] [AddTorsor V₂ P₂]
{n : ℕ} (s : Simplex k P n)
(f : P →ᵃ[k] P₂) (hf : Function.Injective f)
(I : Set k)
/-- **Main result.** The interior of a simpl... | [
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_interior_iff",
"Affine.Simplex.affineCombination_mem_setInterior_iff"
]
},
{
"kind": "original",
"docs": [
"Finset.affineCombination_map",
"Finset.map_affineCombination"
]
},
{
"kind": "... |
mathlibmpr_048 | 34,915 | RestrictedProduct.isUnit_iff | Let `R = (R i)` be a family of monoids and `B = (B i)` a family of submonoids (one in each `R i`), and let `𝓕` be a filter on the index set. An element `x : Πʳ i, [R i, B i]_[𝓕]` is a unit if and only if (i) `x i` is a unit in `R i` for every `i`, and (ii) eventually along `𝓕`, the value `x i` lies in `B i` and is a... | import Mathlib
namespace RestrictedProduct
variable {ι : Type*} {R : ι → Type*} [(i : ι) → Monoid (R i)]
{S : ι → Type*} [(i : ι) → SetLike (S i) (R i)] [∀ (i : ι), SubmonoidClass (S i) (R i)]
{B : (i : ι) → S i} {𝓕 : Filter ι}
/-- **Main result.** An element `x` of the restricted product is a unit iff
each `x i` i... | [
{
"kind": "original",
"docs": [
"MulEquiv.piUnits",
"ContinuousMulEquiv.piUnits"
]
}
] |
mathlibmpr_049 | 34,969 | HopfAlgebra.antipode_mul | Let `R` be a commutative semiring and `A` an `R`-Hopf algebra with antipode `S` and multiplication `μ`. Then for all `a, b ∈ A`, `S(a · b) = S(b) · S(a)`. | import Mathlib.RingTheory.HopfAlgebra.Basic
namespace HopfAlgebra
variable {R A : Type*} [CommSemiring R] [Semiring A] [HopfAlgebra R A]
/-- **Main result.** The antipode of a Hopf algebra is an antihomomorphism:
`S(a · b) = S(b) · S(a)`. -/
theorem antipode_mul (a b : A) :
HopfAlgebra.antipode (R := R) (a * b)
... | [
{
"kind": "original",
"docs": [
"HopfAlgebra.mul_antipode_rTensor_comul_apply",
"HopfAlgebra.mul_antipode_rTensor_comul",
"HopfAlgebra.sum_mul_antipode_eq_smul",
"HopfAlgebra.sum_mul_antipode_eq_algebraMap_counit",
"HopfAlgebra.mul_antipode_lTensor_comul_apply"
]
},
{
... |
mathlibmpr_050 | 34,977 | CategoryTheory.isContinuous_toGrothendieck_of_pullbacksPreservedBy | Let `F : C ⥤ D` be a functor between sites equipped with precoverages `J` on `C` and `K` on `D` that are stable under base change and have pullbacks. If `F` preserves the (pairwise) pullbacks of every `J`-covering and the image of every `J`-covering under `F` is a `K`-covering, then `F` is continuous for the Grothendie... | import Mathlib
namespace CategoryTheory
open CategoryTheory CategoryTheory.Limits
variable {C D : Type*} [Category C] [Category D]
(F : C ⥤ D) (J : Precoverage C) (K : Precoverage D)
/-- A functor `F` preserves pairwise pullbacks of a presieve `R` if for every pair
of morphisms `f` and `g` in `R`, the pullback of... | [
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.isSheaf_coverage",
"CategoryTheory.Presieve.isSheaf_pretopology"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.isSheafFor_arrows_iff_pullbacks"
]
},
{
"kind": "original",
"docs": [
... |
mathlibmpr_051 | 35,014 | ProbabilityTheory.rnDeriv_compProd | For finite measures `μ, ν` on `α`, finite kernels `κ, η : Kernel α β`, with `μ ⊗ₘ κ ≪ μ ⊗ₘ η`, the Radon–Nikodym derivative of `μ ⊗ₘ κ` with respect to `ν ⊗ₘ η` (as measures on `α × β`) is, almost everywhere with respect to `ν ⊗ₘ η`, equal to the product `p ↦ (dμ/dν)(p.1) · (d(μ ⊗ₘ κ)/d(μ ⊗ₘ η))(p)`. | import Mathlib
open MeasureTheory ProbabilityTheory
/-- **Main result.** For finite measures `μ, ν` on `α`, finite kernels
`κ, η : Kernel α β`, with `μ ⊗ₘ κ ≪ μ ⊗ₘ η`, the Radon–Nikodym derivative of
`μ ⊗ₘ κ` with respect to `ν ⊗ₘ η` is `(ν ⊗ₘ η)`-a.e. equal to
`p ↦ (dμ/dν)(p.1) · (d(μ ⊗ₘ κ)/d(μ ⊗ₘ η))(p)`. -/
theore... | [
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.rnDeriv_mul_rnDeriv",
"MeasureTheory.Measure.rnDeriv_mul_rnDeriv'"
]
},
{
"kind": "original",
"docs": [
"MeasureTheory.Measure.haveLebesgueDecomposition_add",
"MeasureTheory.Measure.singularPart_add_rnDeriv"
]... |
mathlibmpr_052 | 35,041 | Finset.rado_selection | Let `α` be any type and `β : α → Type*` with each `β a` finite. Suppose for every finite set `s : Finset α` we have a function `g s : (a : α) → β a`. Then there is a single function `χ : (a : α) → β a` such that for every finite set `s`, there exists `t ⊇ s` with `χ x = g t x` for every `x ∈ s`. Informally: the family ... | import Mathlib.Topology.Compactness.Compact
import Mathlib.Data.Set.Finite.Basic
variable {α : Type*} {β : α → Type*} [∀ a, Finite (β a)]
/-- **Main result.** Rado selection lemma: a family of "local
approximations" (one per finite set) can be stitched into a global
function that agrees with some member of the family... | [
{
"kind": "original",
"docs": [
"CompactSpace.iInter_nonempty",
"IsCompact.inter_iInter_nonempty"
]
},
{
"kind": "alternative",
"docs": [
"exists_orderEmbedding_covby_of_forall_covby_finite_of_bot"
]
}
] |
mathlibmpr_053 | 35,084 | SimpleGraph.mulCayley_gc | For a magma `M`, the assignment `s ↦ mulCayley s` from `Set M` to `SimpleGraph M` is a left adjoint (a Galois connection): for every `S ⊆ M` and every simple graph `G` on `M`, `mulCayley S ≤ G ↔ S ⊆ {g ∈ M | ∀ a, a * g ≠ a → G.Adj (a * g) a}`. Equivalently, `mulCayley` is the smallest simple graph in which every elemen... | import Mathlib
namespace SimpleGraph
variable {M : Type*} [Mul M]
/-- The Cayley graph on `M` with generator set `s`: distinct vertices `u, v`
are adjacent iff some `g ∈ s` satisfies `u * g = v` or `v * g = u`. -/
def mulCayley (s : Set M) : SimpleGraph M :=
SimpleGraph.fromRel (fun u v => ∃ g ∈ s, u * g = v)
/--... | [
{
"kind": "original",
"docs": [
"SimpleGraph.le_iff_adj"
]
}
] |
mathlibmpr_055 | 35,257 | eq_of_zeroHypercover_target | Let `J` be a precoverage on `C` such that the morphism property "is an isomorphism" is local at the target for `J`. Let `f, g : X ⟶ Y` be two morphisms over `S` and let `𝒰 = {Uᵢ → S}` be a `J`-cover of `S`. If for every `i` the base-changed maps `f ×_S Uᵢ = g ×_S Uᵢ : X ×_S Uᵢ → Y ×_S Uᵢ` agree, then `f = g`. | import Mathlib
open CategoryTheory CategoryTheory.Limits
namespace CategoryTheory
universe v u
variable {C : Type u} [Category.{v} C]
/-- **Main result.** Let `J` be a precoverage on `C` for which the
property "is an isomorphism" is local at the target. Let `f, g : X ⟶ Y`
be two morphisms over `S`, and let `𝒰` be ... | [
{
"kind": "original",
"docs": [
"CategoryTheory.MorphismProperty.IsLocalAtTarget.iff_of_zeroHypercover"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.isIso_limit_cone_parallelPair_of_epi"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Mor... |
mathlibmpr_056 | 35,365 | Affine.Simplex.closedInterior_eq_interior_union | For an `n`-simplex `s` in an affine space over a linearly ordered field (with `n ≥ 1`), the closed interior of `s` is the union of its open interior and the closed interiors of its `n + 1` opposite faces: `s.closedInterior = s.interior ∪ ⋃ i, (s.faceOpposite i).closedInterior`. | import Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic
namespace Affine.Simplex
variable {k V P : Type*} [Field k] [LinearOrder k] [IsStrictOrderedRing k]
[AddCommGroup V] [Module k V] [AddTorsor V P]
/-- **Main result.** For an `n`-simplex `s` (with `n ≥ 1`), the closed
interior decomposes as the union of the ope... | [
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_closedInterior_iff"
]
},
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_interior_iff"
]
},
{
"kind": "original",
"docs": [
"Affine.Simplex.affineCombination_mem_clos... |
mathlibmpr_057 | 35,497 | CategoryTheory.Triangulated.extensionProduct_assoc | In a triangulated category `C`, for any three object properties `P, Q, R` we have `extensionProduct (extensionProduct P Q) R = extensionProduct P (extensionProduct Q R)`, i.e. the extension product is associative as an operation on `ObjectProperty C`. | import Mathlib
namespace CategoryTheory.Triangulated
open CategoryTheory CategoryTheory.Pretriangulated CategoryTheory.ObjectProperty
universe v u
variable {C : Type u} [Category.{v} C] [Limits.HasZeroObject C] [HasShift C ℤ] [Preadditive C]
[∀ n : ℤ, (CategoryTheory.shiftFunctor C n).Additive] [Pretriangulated C]... | [
{
"kind": "original",
"docs": [
"CategoryTheory.Triangulated.someOctahedron",
"CategoryTheory.Triangulated.someOctahedron'",
"CategoryTheory.IsTriangulated.octahedron_axiom"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Pretriangulated.distinguished_cocone_tria... |
mathlibmpr_058 | 35,508 | AlgebraicGeometry.preservesLimitsOfShape_discrete_of_isSheaf_zariskiTopology | Let `F : Schemeᵒᵖ ⥤ Type v` be a Zariski sheaf. Then for any small index type `ι`, `F` preserves limits of shape `Discrete ι` — i.e. `F` carries arbitrary coproducts of schemes to products of types. | import Mathlib.AlgebraicGeometry.Sites.BigZariski
import Mathlib.CategoryTheory.Sites.Preserves
open CategoryTheory CategoryTheory.Limits Opposite
namespace AlgebraicGeometry
universe v u
/-- **Main result.** Any Zariski sheaf `F : Schemeᵒᵖ ⥤ Type v`
preserves arbitrary (small) products: for every small index type ... | [
{
"kind": "original",
"docs": [
"CategoryTheory.Presieve.preservesProduct_of_isSheafFor",
"CategoryTheory.Presieve.isSheafFor_iff_preservesProduct"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.Limits.preservesLimitsOfShape_of_discrete"
]
},
{
"kind": "or... |
mathlibmpr_060 | 35,582 | ContinuousLinearMap.ker_adjoint_comp_self | For a continuous `𝕜`-linear map `T : E →L[𝕜] F` between Hilbert spaces, `ker (T† ∘L T) = ker T`. (Axler 7.64(b).) | import Mathlib
namespace ContinuousLinearMap
variable {𝕜 E F : Type*} [RCLike 𝕜]
[NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E]
[NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
/-- **Main result.** For a continuous `𝕜`-linear map `T : E →L[𝕜] F` between
Hilbert spaces, `ke... | [
{
"kind": "original",
"docs": [
"ContinuousLinearMap.adjoint_inner_left",
"ContinuousLinearMap.adjoint_inner_right"
]
},
{
"kind": "alternative",
"docs": [
"ContinuousLinearMap.orthogonal_range"
]
},
{
"kind": "alternative",
"docs": [
"Submodule.orthog... |
mathlibmpr_062 | 35,745 | Asymptotics.IsBigOWith.sum_congr | Let `(Aᵢ)_{i ∈ s}` and `(Bᵢ)_{i ∈ s}` be finite families of functions on a filter `l`. If for each `i` we have `Aᵢ =O_{Cᵢ}[l] Bᵢ` (i.e. eventually `‖Aᵢ x‖ ≤ Cᵢ · ‖Bᵢ x‖`), then the sum `∑ᵢ Aᵢ` is `O_{sup Cᵢ}[l]` of the function `∑ᵢ ‖Bᵢ‖`. The constant `sup Cᵢ` is the supremum of the individual constants over the index ... | import Mathlib
namespace Asymptotics
variable {α ι E F : Type*}
[NormedAddCommGroup E] [NormedAddCommGroup F]
{l : Filter α} {s : Finset ι} {C : ι → ℝ} {A : ι → α → E} {B : ι → α → F}
/-- **Main result.** Termwise big-O for finite sums: if `A i =O_{C i}[l] B i`
for each `i ∈ s`, then `∑ᵢ A i =O_{s.sup' _ C}[l] (... | [
{
"kind": "original",
"docs": [
"Finset.sup'_eq_csSup_image"
]
}
] |
mathlibmpr_063 | 35,767 | ValueDistribution.logCounting_isBigO_one_iff_analyticOnNhd | Let `f : 𝕜 → E` be meromorphic. Then the pole counting function `r ↦ N(r, f, ⊤)` is `O(1)` (i.e. uniformly bounded for large `r`) if and only if the normal-form `toMeromorphicNFOn f univ` is analytic on all of `𝕜` — equivalently, every singularity of `f` is removable. | import Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic
open Asymptotics Filter Function Real Set
namespace ValueDistribution
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜] [ProperSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
/-- **Main result.** A meromorphic function has only... | [
{
"kind": "original",
"docs": [
"MeromorphicNFAt.meromorphicOrderAt_nonneg_iff_analyticAt"
]
},
{
"kind": "original",
"docs": [
"MeromorphicOn.divisor_apply"
]
},
{
"kind": "original",
"docs": [
"MeromorphicOn.divisor_of_toMeromorphicNFOn"
]
},
{
... |
mathlibmpr_064 | 35,810 | ArithmeticFunction.isUnit_iff_isUnit_apply_one | Let `f : ArithmeticFunction R` over a commutative ring `R`. Then `f` is a unit in the Dirichlet convolution ring iff `f 1` is a unit in `R`. | import Mathlib.NumberTheory.ArithmeticFunction
namespace ArithmeticFunction
variable {R : Type*} [CommRing R]
/-- **Main result.** An arithmetic function over a commutative ring is
a unit (under Dirichlet convolution) iff its value at `1` is a unit. -/
theorem isUnit_iff_isUnit_apply_one {f : ArithmeticFunction R} :... | [
{
"kind": "original",
"docs": [
"ArithmeticFunction.mul_apply"
]
},
{
"kind": "original",
"docs": [
"ArithmeticFunction.mul_apply_one"
]
},
{
"kind": "original",
"docs": [
"Nat.cons_self_properDivisors",
"Nat.insert_self_properDivisors"
]
}
] |
mathlibmpr_065 | 35,918 | Group.fg_of_descent | Let `G` be a group, `f : G →* G` an endomorphism such that `f(U) ⊆ U` for every subgroup `U`, and `h : G → ℝ` a height function. Suppose: a finite set `s` surjects onto `G ⧸ f(G)`; constants `0 ≤ a < b` and `c ≥ 0` satisfy `h x ≤ a · h(g·x) + c` for `g ∈ s`, `x ∈ G`, and `b · h x − c ≤ h(f x)` for all `x`; and every he... | import Mathlib
open scoped Pointwise
/-- **Main result.** Abstract descent theorem: if `G` carries a
subgroup-preserving endomorphism `f` and a height function `h` with
quasi-multiplicativity bounds, a finite quotient generating set, and
Northcott-finiteness of bounded sublevel sets, then `G` is finitely
generated. -... | [
{
"kind": "original",
"docs": [
"Set.exists_min_image",
"Finset.exists_min_image",
"Finite.exists_min"
]
},
{
"kind": "original",
"docs": [
"Subgroup.mem_closure_of_mem",
"Subgroup.subset_closure"
]
},
{
"kind": "original",
"docs": [
"Subgr... |
mathlibmpr_066 | 35,923 | Height.mulHeight_mul_le | For tuples `x, y : ι → K` over an admissible-absolute-value field `K`, the multiplicative height of the pointwise product satisfies `H(x · y) ≤ H(x) · H(y)`. | import Mathlib.NumberTheory.Height.Basic
namespace Height
open AdmissibleAbsValues Real Function
variable {K : Type*} [Field K] [AdmissibleAbsValues K] {ι : Type*} [Finite ι]
/-- **Main result.** Submultiplicativity of the multiplicative height
on tuples: `H(x · y) ≤ H(x) · H(y)`. -/
lemma mulHeight_mul_le (x y : ι... | [
{
"kind": "original",
"docs": [
"Real.iSup_fun_mul_eq_iSup_mul_iSup_of_nonneg",
"Cardinal.ciSup_mul_ciSup"
]
},
{
"kind": "original",
"docs": [
"Finite.le_ciSup",
"le_iSup"
]
},
{
"kind": "original",
"docs": [
"finprod_le_finprod",
"single_... |
mathlibmpr_068 | 36,091 | geometric_hahn_banach_of_nonempty_interior | Let `E` be a real topological vector space, and let `s`, `t ⊆ E` be convex sets with `interior s` nonempty and disjoint from `t`, and `t` itself nonempty. Then there exist a *nonzero* continuous linear functional `f : E → ℝ` and a real number `u` such that `f a ≤ u` for all `a ∈ s` and `u ≤ f b` for all `b ∈ t`. | import Mathlib.Analysis.LocallyConvex.Separation
import Mathlib.Analysis.Convex.Topology
open Set
variable {E : Type*} [AddCommGroup E] [Module ℝ E] [TopologicalSpace E]
[IsTopologicalAddGroup E] [ContinuousSMul ℝ E] {s t : Set E}
/-- **Main result.** Eidelheit's separation theorem: two convex subsets
`s, t` of a ... | [
{
"kind": "original",
"docs": [
"geometric_hahn_banach_open",
"geometric_hahn_banach_open_open",
"RCLike.geometric_hahn_banach_open"
]
},
{
"kind": "original",
"docs": [
"Convex.closure_interior_eq_closure_of_nonempty_interior",
"Convex.interior_closure_eq_inter... |
mathlibmpr_069 | 36,192 | ArithmeticFunction.tendsTo_eulerProduct_of_tendsTo | Let `R` be a commutative semiring. Equip `ArithmeticFunction R` with the topology of pointwise convergence (the discrete topology on each coefficient slot). Let `f : ι → ArithmeticFunction R` be a family such that for every `n`, `(f i) n = (1 : ArithmeticFunction R) n` for all but finitely many `i`. Then for every `n`,... | import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
namespace ArithmeticFunction
open Filter
variable {ι R : Type*} [CommSemiring R]
/-- A private uniform space on `ArithmeticFunction R` (discrete on `R`,
product on `ℕ → R`) us... | [
{
"kind": "original",
"docs": [
"multipliable_iff_cauchySeq_finset"
]
},
{
"kind": "original",
"docs": [
"isClosed_setOf_map_zero"
]
}
] |
mathlibmpr_070 | 36,320 | Quiver.SchreierGraph.labelling_isCovering | Let a group `M` act on a set `V`, let `S` be a generator-index type with map `ι : S → M`, and form the Schreier graph `Γ := SchreierGraph V ι`. The labelling prefunctor `labelling : Γ ⥤q SingleObj S` is a *covering of quivers*: at every vertex `x : Γ`, the induced maps on stars and costars `Star_x → Star_{∗}` and `Cost... | import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Basic
import Mathlib.Combinatorics.Quiver.Covering
import Mathlib.Combinatorics.Quiver.SingleObj
namespace Quiver
/-- A Schreier graph for a monoid `M` acting on `V` with generators
`ι : S → M`: vertices `V`, with an edge `x → ι(s) • x` for each `s... | [
{
"kind": "original",
"docs": [
"Prefunctor.IsCovering",
"Prefunctor.isCovering_of_bijective_star",
"Prefunctor.isCovering_of_bijective_costar"
]
}
] |
mathlibmpr_071 | 36,405 | Derivative.normalizedDerivOfComplex_mul | The normalised derivative `D` satisfies the Leibniz rule on the upper half-plane: for `MDifferentiable` functions `F, G : ℍ → ℂ`, `D (F · G) = (D F) · G + F · (D G)`. | import Mathlib
open UpperHalfPlane hiding I
open Real Complex
open scoped Manifold
namespace Derivative
noncomputable section
/-- The normalized derivative `D = (2π i)⁻¹ · d/dz` on functions on `ℍ`. -/
def normalizedDerivOfComplex (F : ℍ → ℂ) (z : ℍ) : ℂ :=
(2 * π * I)⁻¹ * deriv (F ∘ ofComplex) z
scoped notation... | [
{
"kind": "original",
"docs": [
"deriv_mul"
]
},
{
"kind": "original",
"docs": [
"UpperHalfPlane.mdifferentiableAt_iff",
"deriv_mul"
]
},
{
"kind": "alternative",
"docs": [
"HasMFDerivAt.mul"
]
},
{
"kind": "alternative",
"docs": [
... |
mathlibmpr_072 | 36,408 | mem_unitary_iff_isStarNormal_and_realPart_sq_add_imaginaryPart_sq_eq_one | For a complex unital `*`-algebra `A` (with a `StarModule ℂ` and the obvious tower / commutation typeclasses), an element `x : A` is unitary iff it is star-normal and `(ℜ x)² + (ℑ x)² = 1`. (When one writes `x = a + i·b` with `a := ℜ x`, `b := ℑ x`, this is the familiar "`x` is unitary iff `a` and `b` commute and `a² + ... | import Mathlib
variable {A : Type*}
/-- **Main result.** For a complex unital `*`-algebra `A` (a `StarRing`
with a `StarModule ℂ` and the obvious tower / commutation typeclasses),
an element `x : A` is unitary iff it is star-normal and
`(ℜ x)² + (ℑ x)² = 1`. -/
theorem mem_unitary_iff_isStarNormal_and_realPart_sq_add... | [
{
"kind": "original",
"docs": [
"isStarNormal_iff"
]
},
{
"kind": "original",
"docs": [
"Unitary.mem_iff",
"Unitary.mem_iff_self_mul_star",
"Unitary.mem_iff_star_mul_self"
]
}
] |
mathlibmpr_073 | 36,447 | FiniteDimensional.of_totallyBounded_nhds_zero | Let `E` be a Hausdorff topological vector space over a complete non-trivial normed field `𝕜`. If `E` admits a totally bounded neighbourhood of `0`, then `E` is finite-dimensional. | import Mathlib.Analysis.Normed.Module.FiniteDimension
open Filter Topology
variable (𝕜 : Type*) [NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
{E : Type*} [AddCommGroup E] [UniformSpace E] [T2Space E] [IsUniformAddGroup E]
[Module 𝕜 E] [ContinuousSMul 𝕜 E]
/-- **Main result.** Riesz's theorem for general TVS... | [
{
"kind": "original",
"docs": [
"totallyBounded_iff_subset_finite_iUnion_nhds_zero",
"totallyBounded_iff_subset_finite_iUnion_nhds_one"
]
},
{
"kind": "original",
"docs": [
"TotallyBounded.isVonNBounded"
]
},
{
"kind": "original",
"docs": [
"Bornology.... |
mathlibmpr_075 | 36,490 | CategoryTheory.Adjunction.isCocontinuous_iff_coverPreserving | Let `F ⊣ G : C ⥤ D` be an adjunction equipped with Grothendieck topologies `J` on `C` and `K` on `D`. Then `F` is cocontinuous from `J` to `K` (every `K`-cover of `F U` pulls back to a `J`-cover of `U`) if and only if `G` preserves covers from `K` to `J` (every `K`-cover of `X` is sent by `G` to a `J`-cover of `G X`). | import Mathlib.CategoryTheory.Sites.CoverLifting
import Mathlib.CategoryTheory.Sites.CoverPreserving
namespace CategoryTheory
variable {C D : Type*} [Category C] [Category D]
(J : GrothendieckTopology C) (K : GrothendieckTopology D)
{F : C ⥤ D} {G : D ⥤ C}
/-- **Main result.** For an adjunction `F ⊣ G` between s... | [
{
"kind": "original",
"docs": [
"CategoryTheory.GrothendieckTopology.pullback_stable",
"CategoryTheory.GrothendieckTopology.pullback_stable'"
]
},
{
"kind": "original",
"docs": [
"CategoryTheory.GrothendieckTopology.superset_covering"
]
},
{
"kind": "original",
... |
mathlibmpr_076 | 36,520 | ContinuousLinearMap.orthogonalComplement_iSup_eigenspaces_eq_bot | Let `T : E →L[𝕜] E` be a compact self-adjoint operator on a complete inner product space `E`. Then the orthogonal complement of `⨆ μ, eigenspace T μ` (the closed span of all eigenspaces) is `⊥`, i.e. the eigenspaces of `T` together span a dense subspace of `E`. | import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Analysis.Normed.Operator.Compact
namespace ContinuousLinearMap
open Module.End Submodule
variable {𝕜 E : Type*} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
[CompleteSpace E] {T : E →L[𝕜] E}
/-- **Main result.** The spectral theore... | [
{
"kind": "original",
"docs": [
"IsSelfAdjoint.spectralRadius_eq_nnnorm",
"IsSelfAdjoint.toReal_spectralRadius_eq_norm",
"IsStarNormal.spectralRadius_eq_nnnorm",
"IsometricContinuousFunctionalCalculus.isGreatest_spectrum"
]
},
{
"kind": "original",
"docs": [
"Mo... |
mathlibmpr_077 | 36,946 | PointedCone.DualFG.dual_dual_flip | Let `p : M →ₗ N →ₗ R` be a bilinear pairing between two `R`-modules and let `C : PointedCone R N` be a dually finitely generated cone (i.e. the dual w.r.t. `p` of some finite subset of `M`). Then `C` is recovered from its dual: `dual p (dual p.flip C) = C`. | import Mathlib.Geometry.Convex.Cone.Dual
namespace PointedCone
variable {R M N : Type*}
[CommRing R] [PartialOrder R] [IsOrderedRing R]
[AddCommGroup M] [Module R M]
[AddCommGroup N] [Module R N]
(p : M →ₗ[R] N →ₗ[R] R)
/-- A cone is dually finitely generated (`DualFG`) if it is the dual of
a finite set. -/
... | [
{
"kind": "original",
"docs": [
"PointedCone.dual_dual_flip_dual"
]
}
] |
mathlibmpr_078 | 36,950 | AnalyticOnNhd.sum_divisor_le | Let `f : ℂ → ℂ` be analytic on the closed ball `B(c, R)`, satisfy `f c ≠ 0`, and `‖f z‖ ≤ M` on the sphere `∂B(c, R)` for some `M ≥ 1`. For any `r` with `0 < |r| < |R|`: the sum (with multiplicity) of the orders of zeros of `f` in `B(c, r)` is at most `log(M / ‖f c‖) / log(R / r)`. | import Mathlib.Analysis.Complex.JensenFormula
open Real MeromorphicOn
variable {c : ℂ} {r R M : ℝ} {f : ℂ → ℂ}
/-- **Main result.** Jensen's inequality bounding the number of zeros
in `B(c, r)` by `log(M / ‖f c‖) / log(R / r)`, given `f` analytic on
`B(c, |R|)`, `f c ≠ 0`, and `‖f z‖ ≤ M` on the boundary sphere. -/
... | [
{
"kind": "original",
"docs": [
"MeromorphicOn.circleAverage_log_norm"
]
},
{
"kind": "original",
"docs": [
"Real.circleAverage_mono_on_of_le_circle",
"Real.circleAverage_mono"
]
}
] |
mathlibmpr_079 | 37,296 | ClusterPt.limsSup | Let `α` be a conditionally complete linear order with the order topology and let `f` be a non-trivial filter on `α`. If `f` is bounded and cobounded above, then `f.limsSup` is a cluster point of `f`. | import Mathlib.Topology.Order.LiminfLimsup
open Filter Topology
variable {α : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
/-- **Main result.** Let `f` be a non-trivial filter on a conditionally
complete linear order `α` with the order topology. If `f` is bounded
and cobounded a... | [
{
"kind": "original",
"docs": [
"Filter.frequently_lt_of_lt_limsSup"
]
},
{
"kind": "original",
"docs": [
"Filter.lt_mem_sets_of_limsSup_lt"
]
},
{
"kind": "original",
"docs": [
"clusterPt_iff_frequently",
"Filter.HasBasis.clusterPt_iff_frequently"
... |
mathlibmpr_080 | 37,297 | Matrix.PosSemidef.hadamard | For matrices `A, B : Matrix ι ι 𝕜` over an RCLike-scalar field, if `A` and `B` are positive semidefinite then so is their entrywise (Hadamard) product `A ⊙ B`. | import Mathlib.Analysis.Matrix.Order
open Matrix
open scoped ComplexOrder
open scoped MatrixOrder
theorem Matrix.PosSemidef.hadamard {ι 𝕜 : Type*} [RCLike 𝕜]
{A B : Matrix ι ι 𝕜} (hA : A.PosSemidef) (hB : B.PosSemidef) :
(A ⊙ B).PosSemidef := sorry | [
{
"kind": "original",
"docs": [
"Matrix.PosSemidef.kronecker",
"Matrix.PosDef.kronecker"
]
},
{
"kind": "original",
"docs": [
"Matrix.PosSemidef.submatrix",
"Matrix.posSemidef_submatrix_equiv"
]
},
{
"kind": "original",
"docs": [
"Matrix.posSemid... |
mathlibmpr_081 | 37,612 | Real.exists_measure_rpow_eq_integral_rpowIntegrand₁₂ | For each `p ∈ (1,2)` there exists a positive scalar multiple `μ` of Lebesgue measure on `(0,∞)` such that for every `x ≥ 0`, the function `t ↦ rpowIntegrand₁₂(p,t,x) = t^{p−1}(x/t + t/(t+x) − 1)` is integrable for `μ` and `x^p = ∫₀^∞ t^{p−1}(x/t + t/(t+x) − 1) dμ(t)`. | import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
namespace Real
open Set MeasureTheory
variable {p : ℝ}
/-- The integrand `t ^ (p - 1) * (t⁻¹ * x + t * (t + x)⁻¹ - 1)` whose
integral against a suitable measure... | [
{
"kind": "original",
"docs": [
"Real.rpow_eq_const_mul_integral",
"Real.integral_rpowIntegrand₀₁_eq_rpow_mul_const"
]
}
] |
mathlibmpr_082 | 38,267 | MeasureTheory.IsPredictable.predictablePart_eq | Let `f : ℕ → Ω → E` be predictable with respect to a filtration `ℱ` and integrable at every step. Then its predictable part equals (almost everywhere) the increment process from time `0`: `predictablePart f ℱ μ n =ᵐ[μ] f n − f 0`. | import Mathlib.Probability.Martingale.Centering
open scoped MeasureTheory ProbabilityTheory
namespace MeasureTheory
variable {Ω E : Type*} {m0 : MeasurableSpace Ω} {μ : Measure Ω} [NormedAddCommGroup E]
[NormedSpace ℝ E] [CompleteSpace E] {f : ℕ → Ω → E} {ℱ : Filtration ℕ m0}
/-- **Main result.** A predictable, i... | [
{
"kind": "original",
"docs": [
"Finset.sum_range_sub",
"Finset.sum_Ico_sub",
"List.sum_range_sub"
]
}
] |
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AG6 | null | null | null | null | null |
AG7 | null | null | null | null | null |
End of preview.
RagSolver — índice de recuperación (Mathlib v4.28.0-rc1)
Índice LanceDB (vector + BM25) sobre las declaraciones de Mathlib informalizadas por
LeanSearch v2
(Apache 2.0), embebidas con Qwen3-Embedding-0.6B, más los conjuntos de evaluación del proyecto.
Construido con https://github.com//RagSolver (docs/reproduce.md).
index/lancedb/mathlib.lance: 3903 declaraciones (muestra de evaluación).index/lancedb/mathlib_full.lance: el corpus completo (278160 declaraciones) con índice ANN IVF_SQ (recall@10 0.989 respecto de la búsqueda exacta;RAGSOLVER_MATHLIB_TABLE=mathlib_fullpara usarlo).eval/runs/: resultados por problema × configuración, resúmenes y trazas JSONL por agente de las corridas del informe (eval-v1,eval-v1-full,f5-ablations).corpus/: JSONL normalizado (MathlibChunk, verdocs/interfaces.md).eval/: MathlibQR/MPR, premisas gold de miniF2F (desde pruebas verificadas de DeepSeek-Prover-V2), gold informal, resultados.- Mathlib: Apache 2.0. miniF2F-v2: Apache 2.0. El corpus informal de ejemplo (PDF de olimpiadas, Univ. de Sonora 2002) no se redistribuye: sólo metadatos.
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