chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
8,805 | lean_file_chunk | a6bbe8a4124603db450e1bad3e8b21a64b39604b9dcbfdc5457eb6a17980fd9b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
module
public import Mathlib.Analysis.Complex.AbsMax
public import Mathlib.Analysis.LocallyConvex.WithSeminorms
public import Mathlib.Geometry.Manifold.MFDeriv.Bas... |
4,464 | lean_file_chunk | 600338836a99fb6a24d6aa3c3c70fb4c8ab334878565d20680f24522b2ae11f6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.Immersion
public import Mathlib.Geometry.Manifold.ContMDiff.Defs
public import Mathlib.Geometry.Manifold.Diffeomor... |
23,977 | lean_file_chunk | df0874638369e69222df4725329113c046c4e71e21a21b4f29dc6d71e88cf70c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
public import Mathlib.Geometry.Manifold.LocalSourceTargetProperty
public import Mathlib.Anal... |
14,362 | lean_file_chunk | 20af28f64debb93e24e3515fde2aeb1a63a320eda8e1d893d5c52f4d786fb101 | lean_source_file | pkg_mathlib_source_file_chunks | def complement (h : IsImmersionAt I J n f x) : Type u := by
rw [IsImmersionAt_def] at h
exact Classical.choose h
instance (h : IsImmersionAt I J n f x) : NormedAddCommGroup h.complement := by
rw [IsImmersionAt_def] at h
exact Classical.choose <| Classical.choose_spec h
instance (h : IsImmersionAt I J n f x) :... |
3,243 | lean_file_chunk | f486a371ecb33b85aff0fb9181949bc657c973be0bdc033e2d45fef6811af7bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
module
public import Mathlib.AlgebraicTopology.FundamentalGroupoid.SimplyConnected -- shake: keep (`p_w`)
public import Mathlib.Geometry.Manifold.Diffeomorph
public import Ma... |
19,458 | lean_file_chunk | 9479d80acd8bcf98b0de71b0a877a588b55b7614c11c0d034a0eccdc737a59a3 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Data.EReal.Operations
public import Mathlib.Topology.MetricSpace.Bounded
public import Mathlib.Topology.OpenPartialHomeomorph.Comp... |
23,293 | lean_file_chunk | f715b83410cf738f9d1ced575ebdc9c9f4cb7a7223f65c5f16d1084b706298bc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Charted spaces with a given structure groupoid
-/
@[expose] public section
noncomputable s... |
23,901 | lean_file_chunk | 404d43d698b57bb5d068ddde3b72d6013cd4ed454fefdd71596c9cc6db3b3719 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Geometry.Manifold.Algebra.Structures
public import Mathlib.Geometry.Manifold.BumpFunction
public import Mathlib.Geometry.Manifold.Vect... |
17,220 | lean_file_chunk | 9f96bdcc2f666c31e704de00183a881ea7c6c8d05c6029df1af8b460d31580d9 | lean_source_file | pkg_mathlib_source_file_chunks | with `s` open and `s ⊆ interior t`, there is a smooth function `f : M → [0,1]` which is equal to `s`
in a neighbourhood of `s` and has support contained in `t`. -/
theorem exists_contMDiffMap_one_nhds_of_subset_interior
[T2Space M] [NormalSpace M] [SigmaCompactSpace M]
{s t : Set M} (hs : IsClosed s) (hd : s ⊆ ... |
6,985 | lean_file_chunk | 31e44cfeeb9111ca035e3eaf2870da6874c05a30bd08a59361ae511e9330775b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Patrick Massot
-/
module
public import Mathlib.Geometry.Manifold.Notation
public import Mathlib.Geometry.Manifold.PartitionOfUnity
/-!
# Approximation of continuo... |
23,925 | lean_file_chunk | 825402b5b2849ce036e2c3ffb10414707d0ff2cd0e7da5791d1ad20533543f05 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.HasGroupoid
/-!
# Local properties invariant under a groupoid
We study properties of a tripl... |
11,030 | lean_file_chunk | 7be4e70cc74ffa6458ef67b456bfbd5ab4349f7f070d0af95f00f9673366359c | lean_source_file | pkg_mathlib_source_file_chunks | apply hG.liftPropWithinAt_of_liftPropAt_of_mem_nhds (hG.liftPropAt_of_mem_maximalAtlas hQ he hx)
exact e.open_source.mem_nhds hx
theorem liftPropAt_symm_of_mem_maximalAtlas [HasGroupoid M G] {x : H}
(hG : G.LocalInvariantProp G Q) (hQ : ∀ y, Q id univ y) (he : e ∈ maximalAtlas M G)
(hx : x ∈ e.target) : Lift... |
14,095 | lean_file_chunk | d718ca86f708821263d043b6854e3e1ed5ab28d06d84c735cb96177f33703941 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
module
public import Mathlib.Geometry.RingedSpace.PresheafedSpace
public import Mathlib.Topology.Category.TopCat.Limits.Basic
public import Mathlib.Topology.Sheaves.Limi... |
23,991 | lean_file_chunk | 448631979de8ef4bb6ddd5b1f797eb729c521ed10e7cd0c8237332db67d9b9d8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Topology.Gluing
public import Mathlib.Geometry.RingedSpace.OpenImmersion
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColi... |
4,958 | lean_file_chunk | 0bfc57fcce9bcde82ca23ebf53a586f471bcbf3a9fef4d6256a718b3ff7fdb77 | lean_source_file | pkg_mathlib_source_file_chunks | 6. `f i i` is an isomorphism.
7. `t i i` is the identity.
8. `V i j ×[U i] V i k ⟶ V i j ⟶ V j i` factors through `V j k ×[U j] V j i ⟶ V j i` via some
`t' : V i j ×[U i] V i k ⟶ V j k ×[U j] V j i`.
9. `t' i j k ≫ t' j k i ≫ t' k i j = 𝟙 _`.
We can then glue the spaces `U i` together by identifying `V i j` with `... |
15,418 | lean_file_chunk | d0d2d6e333a34fd1094398b7f20dc1f345746d2e65238ec2b38295bd79235f36 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
module
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
public import Mathlib.Algebra.Category.Ring.Constructions
public import Mathlib.Geometry.RingedSpace.O... |
6,073 | lean_file_chunk | 3134e2c1141eb47ad6ed51ab771e97be2cdf1218a2a0b1469f7719ea1716ad31 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
module
public import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
public import Mathlib.RingTheory.LocalRing.ResidueField.Basic
/-!
# Residue fields of poin... |
3,739 | lean_file_chunk | 1bf82c90e90fb24c188d900469ca0f3872f3dc172c04407e4956e81309adeb99 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies, Patrick Luo, Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Patrick Luo, Eric Rodriguez
-/
module
public import Mathlib.Algebra.Group.Action.Pointwise.Finset
public import Mathlib.Algebra.Group.Subgroup... |
3,145 | lean_file_chunk | 71ee03d394a3214d4f6a6bd090f3daa0933f2755eaf5300a9f318d9b8f0123a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Algebra.Order.BigOperators.Group.Finset
public import Mathlib.GroupTheory.QuotientGroup.Defs
public import Mathlib.Algebra.Group.Pointwise.F... |
4,177 | lean_file_chunk | bfb39f6f7f6a277475cfaad7c8c35a882dda2dadd4ff01ac6b2d56f29c9e77a6 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Geometry.Euclidean.Inversion.Basic
public import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
... |
4,784 | lean_file_chunk | 56ab1edc08f2644732f63a03688ea912f10d9f74c610aa58b418412665276dca | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Geometry.Euclidean.Inversion.Basic
public import Mathlib.Analysis.InnerProductSpace.Calculus
public import Mathlib.Analysis.Calculus.D... |
9,449 | lean_file_chunk | 76bc27d6a1d79e1cf61c4bfe5552a8485979ac051156f47536c45dd84be8fb37 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
module
public import Mathlib.Analysis.InnerProductSpace.Basic
public import Mathlib.Analysis.Normed.Group.AddTorsor
public import Mathlib.Tactic.AdaptationNote
/-... |
12,270 | lean_file_chunk | 6103c041ada87ebca5452a492dd871e471828d7e670276f8e07e29aedf8e500e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Bisector
public import Mathlib.Geometry.Euclidean.Incenter
/-!
# Angles and incenters and excenters.
This file pr... |
23,969 | lean_file_chunk | 9fdf457f95a26d45ed759983ed1765b5f1bc7ff6b7973981ac967af7d2e7afb2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
public import Mathlib.Geometry.Euclidean.Circumcenter
public import Mathlib.Geometry.Euclidean.... |
4,769 | lean_file_chunk | e723069b428cca437b973d50cba8c623321b89a6a8bf62ad15559c4ce4b009cb | lean_source_file | pkg_mathlib_source_file_chunks | convert t'.dist_div_sin_oangle_div_two_eq_circumradius h₁₂ h₁₃ h₂₃ using 3
· rw [← Real.Angle.sin_toReal,
Real.abs_sin_eq_sin_abs_of_abs_le_pi (Real.Angle.abs_toReal_le_pi _),
← angle_eq_abs_oangle_toReal (t'.independent.injective.ne h₁₂)
(t'.independent.injective.ne h₂₃.symm)]
congr
· simp ... |
17,501 | lean_file_chunk | 1e3e4f3746a904fa86ae577a01c02b6677877b338e81f9a324c2593ba8c781f2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Oriented.RightAngle
public import Mathlib.Geometry.Euclidean.Angle.Oriented.Projection
/-!
# Angle bisectors.
Thi... |
11,807 | lean_file_chunk | d8053a2cac18a5f6df5e1c712788fc914261a5175e71ac47363c9584002b7dfd | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Analysis.Convex.Side
public import Mathlib.Analysis.Convex.StrictCombination
public import Mathlib.Geometry.Euclidean.Sphere.Basic
/-!
# Se... |
17,881 | lean_file_chunk | 43124a32780e2cf755b82da24218374255ba12d1bc5e430d376d2cce0b03315a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional
public import Mathlib.Geometry.Euclidean.Projection
public import Mathlib.Geometry.E... |
23,995 | lean_file_chunk | 5f5b5e486c90d6037f35b7ecbc9cc628d752c819777b6ee5419a01bd93342e10 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Analysis.Convex.StrictConvexBetween
public import Mathlib.Analysis.InnerProductSpace.Convex
public import Mathlib.Analysis.Normed.Affine.Con... |
2,264 | lean_file_chunk | 229a00aedcf8a987dd4667f844f7604a322b7ff60382387ce310655ae368ccae | lean_source_file | pkg_mathlib_source_file_chunks | have hp₂' : ‖p₂ -ᵥ s.center‖ = s.radius := by rw [← dist_eq_norm_vsub']; exact mem_sphere'.mp hp₂
have hd : ‖p₂ -ᵥ s.center‖ ^ 2 =
‖p₂ -ᵥ p₁‖ ^ 2 + 2 * ⟪p₂ -ᵥ p₁, p₁ -ᵥ s.center⟫ + ‖p₁ -ᵥ s.center‖ ^ 2 := by
rw [← vsub_add_vsub_cancel p₂ p₁ s.center, norm_add_sq_real]
rw [hp₂', hp₁', ← neg_vsub_eq_vsub_re... |
20,425 | lean_file_chunk | 5b97c4f737a2da5ff527bcd6f46ecce6d034ccbc608b2b32f9816f179607f9c5 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Projection
public import Mathlib.Geometry.Euclidean.Sphere.OrthRadius
/-!
# Tangency for spheres.
This file defines not... |
16,853 | lean_file_chunk | 0651375d343239752fed9dc92f8f40f725b8e0f1e1f3d9710de2ba20d5771d1a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson, Li Jiale
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
public import Mathlib.Geometry.Euclidean.Sphere.Tangent
impo... |
3,379 | lean_file_chunk | e932b77ece4be20e95d8bdff99b27974d87bc8a760c517ca0f2b7e76d70524c7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
module
public import Mathlib.Geometry.Euclidean.Sphere.Power
public import Mathlib.Geometry.Euclidean.Triangle
/-!
# Ptolemy's theorem
This fi... |
16,957 | lean_file_chunk | 88ea682cc9299edf209b534f6abc40a985564ad52e2ee96b8d0310e6646e468c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Weiyi Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Weiyi Wang
-/
module
public import Mathlib.MeasureTheory.Measure.Haar.Unique
public import Mathlib.MeasureTheory.Measure.Hausdorff
public import Mathlib.Analysis.Normed.Lp.MeasurableSpace... |
23,991 | lean_file_chunk | 0e6df982d5f639d247e3e38b64c170452d45334edc7f5128a40914babe26da93 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
module
public import Mathlib.Analysis.InnerProductSpace.TwoDim
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
... |
23,946 | lean_file_chunk | a363fbd9e6d7c944b6eeb253afa19c1893142ec6df19982ca812b04a96f08c91 | lean_source_file | pkg_mathlib_source_file_chunks | nth_rw 1 [← o.oangle_sub_eq_oangle_sub_rev_of_norm_eq h]
rw [eq_sub_iff_add_eq, ← oangle_neg_neg, ← add_assoc]
have hy : y ≠ 0 := by
rintro rfl
rw [norm_zero, norm_eq_zero] at h
exact hn h
have hx : x ≠ 0 := norm_ne_zero_iff.1 (h.symm ▸ norm_ne_zero_iff.2 hy)
convert o.oangle_add_cyc3_neg_right (neg... |
1,316 | lean_file_chunk | bd6215d86eb77cc3ec48832b9ab88543d925417bc0f0169cd352a919755e1de5 | lean_source_file | pkg_mathlib_source_file_chunks | by_cases hs : (o.oangle x y).sign = (o.oangle y z).sign
· rw [o.angle_eq_iff_oangle_eq_of_sign_eq hx hy hy hz hs]
· have hn : o.oangle x y ≠ o.oangle y z := by grind
simp only [hn, iff_false]
intro he
apply hr
by_cases hs' : (o.oangle x y).sign = -(o.oangle y z).sign
· rw [o.angle_eq_iff_oangle_... |
23,939 | lean_file_chunk | 67c713472ed716df516a9a1d74adc77b06e05e3ec004223db4379bb71cd8493e | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in righ... |
20,457 | lean_file_chunk | f8b70403b00ccb4d9d479b352c493263590dfe810352fbe8db6498dbd1230ee8 | lean_source_file | pkg_mathlib_source_file_chunks | /-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side, version subtracting vectors. -/
theorem norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle y (y - x)) = ‖y‖ := by
have hs : (o.oa... |
2,733 | lean_file_chunk | 65e6edfcc8a3f7a60285f75796deb9215c93610e258513d7674fee7cc942fab0 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Projection
/-!
# Oriented angles and ort... |
23,975 | lean_file_chunk | 21dd88e9ee94a7eb8f60b703e71f933e21b112fd8f35dde8a56c1bdd007c7243 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Analysis.Convex.Side
public import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
public import Mathlib.Geometry.Euclidean.Angle.Unorien... |
23,979 | lean_file_chunk | 999eda0b3cffaae774779c973fbb12a5a83e8a3e8380d12a19b3b338b890208c | lean_source_file | pkg_mathlib_source_file_chunks | vsub_sub_vsub_cancel_left p₁ p₃ p₂, ← neg_vsub_eq_vsub_rev p₃ p₂, sub_eq_add_neg,
neg_vsub_eq_vsub_rev p₂ p₁, add_comm, ← @neg_one_smul ℝ]
nth_rw 2 [← one_smul ℝ (p₁ -ᵥ p₂)]
rw [o.oangle_sign_smul_add_smul_right]
simp
/-- Swapping the first and third points in an oriented angle negates the sign of that angle... |
2,851 | lean_file_chunk | 2b4c6032cd966dd9b8a8ddd3cc95ef196bbbe6a1d1633c60acff06cf525e5bb3 | lean_source_file | pkg_mathlib_source_file_chunks | have h₄₂ : p₄ ≠ p₂ := by
rintro rfl
exact hs.right_notMem (right_mem_affineSpan_pair _ _ _)
have ha' : ∡ p₁ p₂ (AffineEquiv.pointReflection ℝ p₂ p₃) =
∡ (AffineEquiv.pointReflection ℝ p₂ p₃) p₂ p₄ := by
rw [oangle_pointReflection_left h₃₂ h₄₂, oangle_pointReflection_right h₁₂ h₃₂]
simpa using ha... |
18,911 | lean_file_chunk | 9729d94adb1fd9397b95a554d61bdb0254fa162685e6b39c91af547907c5494b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
module
public import Mathlib.Analysis.SpecialFunctions.Complex.Circle
public import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by ... |
1,874 | lean_file_chunk | 14cde28cebac68ffa21e3e8b104274aa26360d097dd1984bc966380aa8ba017f | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Vedant Gupta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vedant Gupta, Thomas Browning, Eric Wieser
-/
module
public import Mathlib.Analysis.InnerProductSpace.PiL2
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
public import ... |
15,961 | lean_file_chunk | 8a5eab742d660a77e0e3c28c68165dd79522c9d15dcc83ee25288d875e7a8f14 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
module
public import Mathlib.Analysis.InnerProductSpace.Subspace
public import Mathlib.Analysis.Normed.Module.Normalize
public import Mathlib.Analysis.S... |
23,987 | lean_file_chunk | 853ff2e9d78e0adc0baa4c0be17c4b525adf129d660d594c2e43bbfa3f417108 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangle... |
737 | lean_file_chunk | 15c36f3bd33f4a38221aec7df5e044a8b3fb7dee6188e3ba10cbe8154adf5342 | lean_source_file | pkg_mathlib_source_file_chunks | add_comm, norm_div_sin_angle_add_of_inner_eq_zero h h0]
/-- A side of a right-angled triangle divided by the tangent of the opposite angle equals the
adjacent side. -/
theorem dist_div_tan_angle_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2)
(h0 : p₁ ≠ p₂ ∨ p₃ = p₂) : dist p₁ p₂ / Real.tan (∠ p₂ p₃... |
1,170 | lean_file_chunk | 0e264c6ac7f63d26fb52a5933c0cd7a131991f76756377684b725e1273170053 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
module
public import Mathlib.Analysis.Calculus.Conformal.NormedSpace
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angles and conformal maps
Th... |
13,193 | lean_file_chunk | 48f8e44476b7612b9ef0bd4e8a1c28d683ca35d83e08c7b1f30bd34a71995568 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Ilmārs Cīrulis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ilmārs Cīrulis, Alex Meiburg, Jovan Gerbscheid
-/
module
public import Mathlib.Analysis.Normed.Module.Normalize
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
impor... |
1,426 | lean_file_chunk | b133bad1e8483da31e3fa8f5036728677adaa66cecbe22f96ae89c960172d0db | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
module
public import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
public import Mathlib.Geometry.Euclidean.Projection
/-!
# Angles and orthogonal projection.
Thi... |
23,709 | lean_file_chunk | 9e4c6e06bf0dab6fd39d9b4d30185fc1a3865198931d8ff0840bf7804064dbab | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
module
public import Mathlib.Analysis.Convex.Between
public import Mathlib.Analysis.Normed.Group.AddTorsor
public import Mathlib.Geometry.Euclidean.Angl... |
4,131 | lean_file_chunk | 788dc527efab0138b1e698f974ac05e8a94ac0999b3ab8c67261d96d4fdaf6ba | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2026 Martin Winter. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Winter
-/
module
public import Mathlib.Geometry.Convex.Cone.Dual
public import Mathlib.RingTheory.Finiteness.Basic
/-!
# Duals of finitely generated cones
This file defines th... |
7,501 | lean_file_chunk | 12fb72a82a0b77acc96496a3db6e3da859bce0498d87486b87a469525b9b890a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Bjørn Solheim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bjørn Solheim
-/
module
public import Mathlib.Geometry.Convex.Cone.Dual
public import Mathlib.LinearAlgebra.Dual.Lemmas
public import Mathlib.LinearAlgebra.TensorProduct.Defs
/-!
# Te... |
23,986 | lean_file_chunk | c8c9344762240cc1e38c972a44a7bcaae6e7527bab46c637044f6b506499e789 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
module
public import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In an `R`-module `M`, we define a convex cone as a set `s` such that `a • x... |
3,387 | lean_file_chunk | 12d0ae41fb12fdf82f687a8454f47a3e9bfbe08ec3cb46745981a1e9d85f0728 | lean_source_file | pkg_mathlib_source_file_chunks | /-- The strictly positive cone of an ordered module is always blunt. -/
theorem blunt_strictlyPositive : Blunt (strictlyPositive R M) :=
lt_irrefl 0
end StrictlyPositiveCone
end ConvexCone
/-! ### Cone over a convex set -/
section ConeFromConvex
variable [Field 𝕜] [LinearOrder 𝕜] [IsStrictOrderedRing 𝕜] [Add... |
1,554 | lean_file_chunk | 0b9b5c6e2e0b77c6415ac41e270627fec5600b051b4388cbb5d408fe1f42553d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Bjørn Solheim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bjørn Solheim
-/
module
public import Mathlib.Geometry.Convex.Cone.Pointed
/-!
# Simplicial cones
A **simplicial cone** is a pointed convex cone that equals the conic hull of a finit... |
13,124 | lean_file_chunk | 67b03842c311a14e2d3752250a1d30a6b1afec96561c2704433c3a20afaaa3f7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
module
public import Mathlib.Algebra.Group.Submonoid.Support
public import Mathlib.Algebra.Order.Nonneg.Module
public import Mathlib.Geometry.Convex.Cone.Basic
/-!
#... |
6,169 | lean_file_chunk | 9ad11a6ef03e6b3f6b321e6dc35bd5834059618ce8d87f9b937c2f989f7f0155 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
module
public import Mathlib.Geometry.Convex.Cone.Pointed
/-!
# The algebraic dual of a cone
Given a bilinear pairing `p` between two `R`-modules `M` and `N` and a set... |
23,986 | lean_file_chunk | 3e56e58154fd2b5b5039d9457f36beeb298660f50c736950f37444a324642410 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.Atlas
public import Mathlib.Geometry.Manifold.Riemannian.PathELength
public import Mathlib.Geometry.Mani... |
2,282 | lean_file_chunk | 4dadaebda2f0b24706248f55bc02fda967bed3a408e187feb7134a47138ac12d | lean_source_file | pkg_mathlib_source_file_chunks | rcases setOf_riemannianEDist_lt_subset_nhds I hs with ⟨c, c_pos, hc⟩
exact ⟨c, mod_cast c_pos, hc⟩
variable (M) in
/-- The pseudoemetric space structure associated to a Riemannian metric on a manifold. Designed
so that the topology is defeq to the original one.
This should only be used when constructing data in spe... |
17,161 | lean_file_chunk | 71a2c8549607aeb9f8f904c07bd1c62d4bf423e2531c1bb1370847c2e846ed45 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.AddTorsor.AffineMap
public import Mathlib.Analysis.SpecialFunctions.SmoothTransition
public import Mathlib.Geome... |
10,986 | lean_file_chunk | f6387a0f97175f3b446c8a904fd6a34a01c4a6c520bcec88b196020d2cf42edf | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Winston Yin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Winston Yin
-/
module
public import Mathlib.Geometry.Manifold.IntegralCurve.ExistUnique
/-!
# Uniform time lemma for the global existence of integral curves
## Main results
* `exists_... |
6,124 | lean_file_chunk | 6b4f2f8d4d5fa3ee5e968d898cd1c580cab4815bcdeec470af7e1c0477f3fa1b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Winston Yin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Winston Yin
-/
module
public import Mathlib.Geometry.Manifold.IntegralCurve.Basic
/-!
# Translation and scaling of integral curves
New integral curves may be constructed by translating... |
13,997 | lean_file_chunk | 9483447c640ab9a977c34eb31a38d09c7c99aaae079b8d74b93be6a5c58c7c09 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Winston Yin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Winston Yin
-/
module
public import Mathlib.Analysis.ODE.Gronwall
public import Mathlib.Analysis.ODE.PicardLindelof
public import Mathlib.Geometry.Manifold.IntegralCurve.Transform
public... |
9,077 | lean_file_chunk | f9d77e5298166be0cd4f9811714dbd27f19881a9d5aa6d44f7ae959863e885a4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Winston Yin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Winston Yin
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.Tangent
/-!
# Integral curves of vector fields on a manifold
Let `M` be a manifold and `v : (x : M) → TangentSpace... |
9,724 | lean_file_chunk | a8f4d8e6a44c24a8cb1251cf5ccba346ddb3191477fece96208f67c3ea89a327 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.Atlas
import Mathlib.Geometry.Manifold.Notation
public import Mathlib.Geometry.Manifol... |
19,877 | lean_file_chunk | 4b1b7eaf019245047198fa92f337d2ac94c3fbd86ca9de35a0aeade5eac252f8 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
public import Mathlib.Geometry.Manifold.VectorBundle.Tangent
public ... |
23,911 | lean_file_chunk | 18f79c4257ccd63fb1b65b0a888c77e57735bdc09f09bdde3bfe21a17d54f4a7 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Analysis.Calculus.TangentCone.Prod
public import Mathlib.Geometry.Manifold.MFDeriv.Defs
public import Mathlib.Ge... |
23,945 | lean_file_chunk | 32df87d947ac444f5ce95848b60919fb6a3bf1650fa92f6c7ad790fd686a7d61 | lean_source_file | pkg_mathlib_source_file_chunks | theorem MDifferentiableOn.continuousOn (h : MDifferentiableOn I I' f s) : ContinuousOn f s :=
fun x hx => (h x hx).continuousWithinAt
theorem MDifferentiable.continuous (h : MDifferentiable I I' f) : Continuous f :=
continuous_iff_continuousAt.2 fun x => (h x).continuousAt
/-! ### Deriving continuity from differe... |
16,400 | lean_file_chunk | 0eb9cb1bbad299512cdc51a2c36c93c408ea1990bbd2d1391aa00e6af4fade0a | lean_source_file | pkg_mathlib_source_file_chunks | MDifferentiableWithinAt I I' f₁ s x ↔ MDifferentiableWithinAt I I' f s x :=
differentiableWithinAt_localInvariantProp.liftPropWithinAt_congr_iff_of_eventuallyEq h₁ hx
theorem MDifferentiableWithinAt.congr_of_eventuallyEq (h : MDifferentiableWithinAt I I' f s x)
(h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : f₁ x = f x) : MDiffer... |
4,745 | lean_file_chunk | ae5ef6ee3c6262bbf5ea4d84527263aff9a9ea873d9be475b28e5ba6070cb7d1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.Atlas
public import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
public import... |
18,534 | lean_file_chunk | 957d75ca1c258c7693a33910da2b93720a31645a7829a3c844843d6bcd26866a | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.IsManifold.ExtChartAt
public import Mathlib.Geometry.Manifold.LocalInvariantProperties
/-!
# ... |
23,960 | lean_file_chunk | ec15585a3ac8eeb034c17da9cbcc91f90809f5925ec37f81255946104cd5e9df | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Analysis.Calculus.FDeriv.Mul
public import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
import Mathlib.Geometry.Mani... |
19,673 | lean_file_chunk | 43b065848c071ea007fe192139fee62d7236bc821d553cc44e7597d72d33e372 | lean_source_file | pkg_mathlib_source_file_chunks | refine (mdifferentiableAt_const.mfderiv_prod mdifferentiableAt_id).trans ?_
rw [mfderiv_id, mfderiv_const, ContinuousLinearMap.inr]
theorem tangentMap_prod_right {p : TangentBundle I' M'} {x₀ : M} :
tangentMap I' (I.prod I') (fun y ↦ (x₀, y)) p = ⟨(x₀, p.1), (0, p.2)⟩ := by
simp only [tangentMap, mfderiv_prod_... |
22,119 | lean_file_chunk | d6847ba72557a0cbfd9338fb6f9c96103a016202a82955624ffb7d78e540ea11 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
public import Mathlib.Geometry.Manifold.MFDeriv.SpecificFunctions
/-! #... |
4,983 | lean_file_chunk | 28438c0a7f798222e816d1d0957885cc56bc6d246b543ba88c6cbae50691bd28 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.MFDeriv.Basic
public import Mathlib.Geometry.Manifold.Notation
/-!
### Relations between vect... |
23,922 | lean_file_chunk | 01a2796d747c1b3a35ece3e484a832df2f8dedab3ace60c61d6dde9c6eb6f545 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.VectorField
public import Mathlib.Geometry.Manifold.ContMDiffMFDeriv
public import Mathlib.Geometry.Manifold.MFD... |
23,969 | lean_file_chunk | 91b6e8bfbcdb51fd01ef3bca604cf3d237097d5770335e83e788b6fcacf53ab2 | lean_source_file | pkg_mathlib_source_file_chunks | /- The Lie bracket of vector fields on manifolds is well defined, i.e., it is invariant under
diffeomorphisms. Auxiliary version where one assumes that all relevant sets are contained
in chart domains. -/
private lemma mpullbackWithin_mlieBracketWithin_aux [CompleteSpace E']
{f : M → M'} {V W : Π (x : M'), TangentS... |
9,345 | lean_file_chunk | 205fc400ebbbf3a836cafffb73d3326afcd13cdaf839bfe9a320283e41ac8369 | lean_source_file | pkg_mathlib_source_file_chunks | `[U, [V, W]] = [[U, V], W] + [V, [U, W]]` (also called Jacobi identity). -/
theorem leibniz_identity_mlieBracketWithin_apply
{U V W : Π (x : M), TangentSpace I x} {s : Set M} {x : M}
(hs : UniqueMDiffOn I s) (h's : x ∈ closure (interior s)) (hx : x ∈ s)
(hU : CMDiffAt[s] (minSmoothness 𝕜 2) (T% U) x)
(... |
23,956 | lean_file_chunk | 440eddbbb4d6fbda91351c7c5b63ba0376c6b8de55773362042429cb2626fd9d | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Analysis.Calculus.VectorField
public import Mathlib.Geometry.Manifold.ContMDiffMFDeriv
public import Mathlib.Geometry.Manifold.MFD... |
8,781 | lean_file_chunk | 6795ead367ad6ae3041cca02eb7d8b3b40974719e7e4f087c0641c77c3de9469 | lean_source_file | pkg_mathlib_source_file_chunks | CMDiffAt[s] m (T% (mpullbackWithin I I' f V s)) x₀ := by
subst h
exact ContMDiffWithinAt.mpullbackWithin_vectorField hV hf hf' hx₀ hs hmn hst
/-- The pullback of a `C^m` vector field by a `C^n` function with invertible derivative and
with `m + 1 ≤ n` is `C^m`.
Version within a set at a point, with a set used for t... |
23,953 | lean_file_chunk | 56c20d56b650fe9583eb086fec57e282cd2530d70580daaff617c7ec4a8438e4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Patrick Massot, Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.VectorBundle.Basic
public import Mathlib.Geometry.Manifold.MFDeriv.NormedSpace... |
9,398 | lean_file_chunk | 4cbb91e646b831084578a73ec705cff61308b486afbb13f24e0955fe2f5c5d74 | lean_source_file | pkg_mathlib_source_file_chunks | lemma MDifferentiableAt.sum_section_of_locallyFinite
(ht : LocallyFinite fun i ↦ {x : B | t i x ≠ 0})
(ht' : ∀ i, MDiffAt (T% (t i ·)) x₀) :
MDiffAt (T% (fun x ↦ ∑' i, (t i x))) x₀ := by
simp_rw [← mdifferentiableWithinAt_univ] at ht' ⊢
exact .sum_section_of_locallyFinite ht ht'
/-- The sum of a locall... |
12,023 | lean_file_chunk | cc2bc40b0d62301d70f43c65fc525912051708fb302e6a6484dbace01f6fe6c4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
module
public import Mathlib.Geometry.Manifold.VectorBundle.Hom
public import Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
public import Mathlib.Topo... |
23,925 | lean_file_chunk | 2b2284f4ff7b864b55570e989bb8a224342b00709cc1ef58fd73f8a3363204d4 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.Algebra.Monoid
public import Mathlib.Geometry.Manifold.Notation
public import Mathlib.Geometry.Man... |
3,227 | lean_file_chunk | a194b833c0bd5796a5536c2457aadcfa3c5e2b4a390d696288039c964bb17e8b | lean_source_file | pkg_mathlib_source_file_chunks | /-- A section `s` of `V` is `C^k` on a trivialisation domain `e.baseSet` iff each of its
coefficients `(LinearMap.piApply (b.localFrame_coeff e i) s)` in a local frame near `x` is -/
lemma contMDiffOn_baseSet_iff_localFrame_coeff :
CMDiff[e.baseSet] k (T% s) ↔
∀ i, CMDiff[e.baseSet] k ((LinearMap.piApply (e.l... |
13,403 | lean_file_chunk | e97542c276123c92b198bef48d8db23728b1397ac388aa8fcd39a87eed6a22fc | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2025 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Michael Rothgang, Heather Macbeth
-/
module
public import Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
public import Mathlib.Topology.Algebra.Module.FiniteDi... |
23,990 | lean_file_chunk | ca020bb7f3633848b7519a9fbba391d979e84600e3f61502e3e4ab92c25ee74b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.Atlas
public import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLi... |
8,346 | lean_file_chunk | 06e73f7c10c4f04fccbcfe77d9ad6d776bdbb64463d052276b87c586015c7bbe | lean_source_file | pkg_mathlib_source_file_chunks | refine ⟨contMDiffOn_fst.congr fun x hx ↦ e.proj_symm_apply hx,
contMDiffOn_snd.congr fun x hx ↦ ?_⟩
rw [e.apply_symm_apply hx]
/-- Smoothness of a `C^n` section at `x₀` within a set `a` can be determined
using any trivialisation whose `baseSet` contains `x₀`. -/
theorem contMDiffWithinAt_section {s : ∀ x, E x} (... |
23,985 | lean_file_chunk | abce4d836e1e444be85277210e7c69654af6a235c8a0c11de026cb9829e0e81b | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
module
public import Mathlib.Geometry.Manifold.VectorBundle.Basic
import Mathlib.Geometry.Manifold.Notation
/-! # Tangent bundles
This file de... |
521 | lean_file_chunk | 32a81687c94082471235afdf0970a81f5314b86a4ded754155808b77d79db985 | lean_source_file | pkg_mathlib_source_file_chunks | theorem inTangentCoordinates_eq (f : N → M) (g : N → M') (ϕ : N → E →L[𝕜] E') {x₀ x : N}
(hx : f x ∈ (chartAt H (f x₀)).source) (hy : g x ∈ (chartAt H' (g x₀)).source) :
inTangentCoordinates I I' f g ϕ x₀ x =
(tangentBundleCore I' M').coordChange (achart H' (g x)) (achart H' (g x₀)) (g x) ∘L
ϕ x ... |
2,072 | lean_file_chunk | bb894d8f4e9680321b56bf115f2f690a6ec6d980f3e71587a94e6dda2945a483 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
module
public import Mathlib.Geometry.Manifold.ContMDiffMap
public import Mathlib.Geometry.Manifold.VectorBundle.Basic
/-! # P... |
15,536 | lean_file_chunk | aca44cb8299f23c076428ab7c85414eafc3a76038b4b4522e9de0e801539086c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2023 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Floris van Doorn, Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.Algebra.LieGroup
public import Mathlib.Geometry.Manifold.MFDeriv.Basic
public im... |
23,957 | lean_file_chunk | 9a199f27f570b53c500e27a398b962576f92cf2e03b420abcd0b777d498140eb | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.VectorBundle.Basic
public import Mathlib.Topology.VectorBundle.Hom
public import Mathlib.Geometry.Manifold.VectorB... |
2,203 | lean_file_chunk | 4835c016efeda1634d1c91effb3d3bb5e4fa2dfe4ce78c48130cf7d8910ab850 | lean_source_file | pkg_mathlib_source_file_chunks | (E := fun (x : B) ↦ (E₁ x →L[𝕜] E₂ x →L[𝕜] E₃ x)) (b m) (ψ m)) x)
(hv : MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₁)) (fun m ↦ TotalSpace.mk' F₁ (b m) (v m)) x)
(hw : MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₂)) (fun m ↦ TotalSpace.mk' F₂ (b m) (w m)) x) :
MDifferentiableAt IM (IB.prod 𝓘(𝕜, F₃))
(fun m ↦... |
14,682 | lean_file_chunk | 9147567b2539d83e7dd12b40c2fe48a8ce624c685c762264304a2135baec3fa2 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
/-! # The groupoid of `C^n`, fiberwise-linear maps
This ... |
15,548 | lean_file_chunk | 8e63514206fc6f0f4afb0006740096329ee047c207cf8aebb0a919d766a9aba1 | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.Basic
import Mathlib.Geometry.Manifold.ContMDiff.NormedSpace
/-!
# Smoothness of ch... |
20,094 | lean_file_chunk | 9b19d0aae0600cbaa1b23ee46b288ce9ace4cf603632bd9bfa25aaa488647bfe | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we... |
22,423 | lean_file_chunk | 2a803f336815a4cfcfe19cdd5c8b88cb79fee1b0131f6272b9cfcc114d2b5f1c | lean_source_file | pkg_mathlib_source_file_chunks | /-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Michael Rothgang
-/
module
public import Mathlib.Geometry.Manifold.ContMDiff.Basic
/-!
## Smoothness of standard maps associated to the produc... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.