Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | exact ⟨_, wellFounded_lt.min_mem _ h, fun a' ↦ wf.wf.not_lt_min _ h⟩
lemma exists_covBy_of_wellFoundedGT [wf : WellFoundedGT α] ⦃a : α⦄ (h : ¬ IsMin a) :
∃ a', a' ⋖ a := by
| Mathlib/Order/Cover.lean | 618 | 621 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 885 | 889 | |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... |
@[to_additive existing, deprecated (since := "2025-04-19")]
alias prod_univ_get := prod_univ_getElem
| Mathlib/Algebra/BigOperators/Fin.lean | 90 | 92 |
/-
Copyright (c) 2024 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata, Fabrizio Barroero, Laura Capuano, Nirvana Coppola,
María Inés de Frutos-Fernández, Sam van Gool, Silvain Rideau-Kikuchi, Amos Turchet,
Frances... | -- For any `C > 0`, the limit of `C ^ (1/k)` is 1 as `k → ∞`
private lemma tendsto_const_rpow_inv {C : ℝ} (hC : 0 < C) :
Tendsto (fun k : ℕ ↦ C ^ (k : ℝ)⁻¹) atTop (𝓝 1) :=
((continuous_iff_continuousAt.mpr fun _ ↦ continuousAt_const_rpow hC.ne').tendsto'
0 1 (rpow_zero C)).comp <| tendsto_inv_atTop_zero.comp... | Mathlib/NumberTheory/Ostrowski.lean | 43 | 48 |
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
/-!
# Conditional Probability
This file defines conditional probability and includes basic resu... | @[simp] lemma cond_apply_self (hs₀ : μ s ≠ 0) (hs : μ s ≠ ∞) : μ[s|s] = 1 := by
simpa [cond] using ENNReal.inv_mul_cancel hs₀ hs
theorem cond_inter_self (hms : MeasurableSet s) (t : Set Ω) (μ : Measure Ω) :
μ[s ∩ t|s] = μ[t|s] := by
rw [cond_apply hms, ← Set.inter_assoc, Set.inter_self, ← cond_apply hms]
theo... | Mathlib/Probability/ConditionalProbability.lean | 214 | 227 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... |
variable [AddLeftMono α]
| Mathlib/Algebra/Order/Sub/Defs.lean | 396 | 397 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.... | simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm] using
hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2)
(add_sub_cancel _ _)
/-- If `x ∈ s` and `y ∈ interior s`, then the segment `(x, y]` is included in `interior s`. -/
theorem Convex.add_smul_sub_mem_interior {s : Set E}... | Mathlib/Analysis/Convex/Topology.lean | 213 | 218 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... | · push_cast
exact mul_ne_zero hp hq
end Rat
open Rat
variable [FunLike F α β]
| Mathlib/Data/Rat/Cast/Defs.lean | 197 | 204 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | p : Q($e = $e' * $k)
| Mathlib/Tactic/Ring/Basic.lean | 805 | 806 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Minimal
import Mathlib.Order.Zorn
import Mathlib.Topology.ContinuousOn
/-!
# Irreducibility in topological space... | theorem isIrreducible_iff_sInter :
IsIrreducible s ↔
∀ (U : Finset (Set X)), (∀ u ∈ U, IsOpen u) → (∀ u ∈ U, (s ∩ u).Nonempty) →
(s ∩ ⋂₀ ↑U).Nonempty := by
classical
| Mathlib/Topology/Irreducible.lean | 225 | 229 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | hg.1.stronglyMeasurable_mk (hg.ae_eq hg.1.ae_eq_mk) (hui.ae_eq fun n => (hf n).ae_eq_mk) ?_
have h_ae_forall_eq : ∀ᵐ x ∂μ, ∀ n, f n x = (hf n).mk (f n) x := by
rw [ae_all_iff]
exact fun n => (hf n).ae_eq_mk
filter_upwards [hfg, h_ae_forall_eq, hg.1.ae_eq_mk] with x hx_tendsto hxf_eq hxg_eq
rw [← hxg_e... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 555 | 571 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.MeasureTheory.Measure.Dirac
/-!
# Probability mass functions
This file is about probabil... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 399 | 400 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | rw [sub_smul, ← sub_add, add_right_comm]
simpa using sub_toIcoDiv_zsmul_mem_Ico hp a b
| Mathlib/Algebra/Order/ToIntervalMod.lean | 203 | 205 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... |
/-- A monoid in which multiplication is left-cancellative. -/
@[to_additive]
| Mathlib/Algebra/Group/Defs.lean | 667 | 669 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Sean Leather
-/
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.... | variable {m : Type u → Type u} [Monad m] [LawfulMonad m]
| Mathlib/Control/Fold.lean | 352 | 353 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | (fst φ).1.comp α (by rw [← h, add_comm]) + (snd φ).comp β (zero_add n)
variable (α : Cochain F K m) (β : Cochain G K n) (h : m + 1 = n)
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 297 | 300 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison
-/
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAl... | variable [Module.Free R M]
theorem finrank_eq_zero_of_basis_imp_not_finite
(h : ∀ s : Set M, Basis.{v} (s : Set M) R M → ¬s.Finite) : finrank R M = 0 := by
cases subsingleton_or_nontrivial R
| Mathlib/LinearAlgebra/Dimension/Finite.lean | 485 | 489 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | theorem not_mem_commonNeighbors_left (v w : V) : v ∉ G.commonNeighbors v w := fun h =>
ne_of_adj G h.1 rfl
theorem not_mem_commonNeighbors_right (v w : V) : w ∉ G.commonNeighbors v w := fun h =>
ne_of_adj G h.2 rfl
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 739 | 743 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Chris Hughes, Michael Howes
-/
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Group.Semiconj.Units
/-!
# Conjugacy of group elements
See also `MulAut.conj` a... | r := IsConj
iseqv := ⟨IsConj.refl, IsConj.symm, IsConj.trans⟩
end IsConj
| Mathlib/Algebra/Group/Conj.lean | 110 | 113 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 1,053 | 1,057 | |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
/-!
# Trivial Lie modules and Abelian Lie algebras
The action of a Lie algebra `L` on ... |
theorem lie_abelian_iff_equiv_lie_abelian {R : Type u} {L₁ : Type v} {L₂ : Type w} [CommRing R]
[LieRing L₁] [LieRing L₂] [LieAlgebra R L₁] [LieAlgebra R L₂] (e : L₁ ≃ₗ⁅R⁆ L₂) :
IsLieAbelian L₁ ↔ IsLieAbelian L₂ :=
⟨e.symm.injective.isLieAbelian, e.injective.isLieAbelian⟩
theorem commutative_ring_iff_abelia... | Mathlib/Algebra/Lie/Abelian.lean | 76 | 82 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... |
/-- Expressing uniform convergence on a set using `edist`. -/
theorem tendstoUniformlyOn_iff {ι : Type*} {F : ι → β → α} {f : β → α} {p : Filter ι} {s : Set β} :
TendstoUniformlyOn F f p s ↔ ∀ ε > 0, ∀ᶠ n in p, ∀ x ∈ s, edist (f x) (F n x) < ε := by
refine ⟨fun H ε hε => H _ (edist_mem_uniformity hε), fun H u hu... | Mathlib/Topology/EMetricSpace/Basic.lean | 118 | 124 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | · rw [Set.mem_image] at h; rcases h with ⟨x, h, hx⟩
change x ∈ q.asIdeal at h
convert h
rw [HomogeneousLocalization.ext_iff_val, HomogeneousLocalization.val_mk]
dsimp only [Subtype.coe_mk]; rw [← hx]; rfl)
theorem mem_carrier_iff_of_mem (hm : 0 < m) (q : Spec.T A⁰_ f) (a : A) {n} ... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 285 | 297 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Lee
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.InfiniteSum.Defs
imp... | suffices HasProd f (∏ b' ∈ {b}, f b') by simpa using this
hasProd_prod_of_ne_finset_one <| by simpa [hf]
| Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 132 | 133 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Arithmetic
/-!
# Encodings ... | intro i
simp only [Option.join, map_append, map_map, getElem?_fin, id, Option.bind_eq_some_iff,
getElem?_eq_some_iff, length_append, length_map, length_finRange, exists_eq_right]
refine ⟨lt_of_lt_of_le i.2 le_self_add, ?_⟩
rw [getElem_append_left, getElem_map]
· simp only [getElem_... | Mathlib/ModelTheory/Encoding.lean | 236 | 288 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | (hT' : DominatedFinMeasAdditive μ T' C') (hT'' : DominatedFinMeasAdditive μ T'' C'')
(h_add : ∀ s, MeasurableSet s → μ s < ∞ → T'' s = T s + T' s) (f : α →₁ₛ[μ] E) :
setToL1SCLM α E μ hT'' f = setToL1SCLM α E μ hT f + setToL1SCLM α E μ hT' f :=
setToL1S_add_left' T T' T'' h_add f
theorem setToL1SCLM_smul... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 315 | 356 |
/-
Copyright (c) 2024 Jack McKoen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jack McKoen
-/
import Mathlib.CategoryTheory.Limits.FunctorCategory.Basic
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Types.Limits
impo... | colimit.isoColimitCocone (binaryCoproductColimitCocone F G)
@[simp]
lemma inl_comp_binaryCoproductIso_hom :
Limits.coprod.inl ≫ (binaryCoproductIso F G).hom = coprod.inl := by
| Mathlib/CategoryTheory/Limits/Shapes/FunctorToTypes.lean | 227 | 231 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.Normed.Group.InfiniteSum
/-!
# Tannery's theorem
Tannery's theorem gives a sufficient criterion for th... | /-- **Tannery's theorem**: topological sums commute with termwise limits, when the norms of the
summands are eventually uniformly bounded by a summable function.
(This is the special case of the Lebesgue dominated convergence theorem for the counting measure
on a discrete set. However, we prove it under somewhat weake... | Mathlib/Analysis/Normed/Group/Tannery.lean | 23 | 81 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | end
@[ext]
| Mathlib/CategoryTheory/ComposableArrows.lean | 699 | 701 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | exact mult_comap_le _ _
variable [Algebra k F]
lemma IsUnramified.comap_algHom {w : InfinitePlace F} (h : IsUnramified k w) (f : K →ₐ[k] F) :
IsUnramified k (w.comap (f : K →+* F)) := by
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 814 | 819 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.LinearAlgebra.FreeM... | end LinearMap
section
variable [T2Space E] [T2Space F] [FiniteDimensional 𝕜 E]
namespace LinearEquiv
/-- The continuous linear equivalence induced by a linear equivalence on a finite dimensional
space. -/
| Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 338 | 347 |
/-
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
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 382 | 387 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,647 | 1,658 | |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | y ^ 2 + W'.a₁ * x * y + W'.a₃ * y - (x ^ 3 + W'.a₂ * x ^ 2 + W'.a₄ * x + W'.a₆) = 0 := by
rw [Equation, evalEval_polynomial]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 193 | 195 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | convert hn
change (a : α) = ↑(⟨a', aS⟩ : S)
have := le_one_iff_subsingleton.1 (le_of_eq e)
congr!,
fun ⟨a, e⟩ => by simp [e]⟩
/-! ### Fundamental sequences -/
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 420 | 426 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | δ_v n m hnm _ p q hpq (q - 1) (p + 1) rfl rfl,
δ_v n m hnm _ (p + a) (q + a) (by omega) (q - 1 + a) (p + 1 + a)
(by omega) (by omega),
smul_add, Linear.units_smul_comp, Linear.comp_units_smul, add_right_inj]
rw [smul_comm]
· rw [δ_shape _ _ hnm, δ_shape _ _ hnm, shift_zero, smul_zero]
l... | Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 461 | 468 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
variable [LocallyFiniteOrder α]
| Mathlib/Order/Interval/Finset/Basic.lean | 437 | 438 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.UniformConvergence
/-!
# Algebraic fact... | obtain rfl := hφ.eq_induced; clear hφ
simp only [induced_iInf, UniformOnFun.topologicalSpace_eq, induced_compose]
refine continuousSMul_iInf fun s ↦ continuousSMul_iInf fun hs ↦ ?_
letI : TopologicalSpace H :=
.induced (UniformFun.ofFun ∘ s.restrict ∘ φ) (UniformFun.topologicalSpace s E)
set φ' : H →ₗ[𝕜]... | Mathlib/Topology/Algebra/Module/UniformConvergence.lean | 96 | 109 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | ofReal_zsmul, ofReal_mul, ofReal_ofNat, exp_mul_I_periodic.sub_zsmul_eq]
· rintro ⟨θ, _, rfl⟩
exact ⟨θ, rfl⟩
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 327 | 330 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Minor.Restrict
/-!
# Some constructions of matroids
This file defines some very elementary examples of matroids, namely those with at most o... |
theorem uniqueBaseOn_isBase_iff (hIE : I ⊆ E) : (uniqueBaseOn I E).IsBase B ↔ B = I := by
rw [uniqueBaseOn, isBase_restrict_iff', freeOn_isBasis'_iff, inter_eq_self_of_subset_right hIE]
| Mathlib/Data/Matroid/Constructions.lean | 207 | 209 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | midpoint_vsub_left v₁ v₂
@[simp]
theorem midpoint_sub_right (v₁ v₂ : V) : midpoint R v₁ v₂ - v₂ = (⅟ 2 : R) • (v₁ - v₂) :=
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 140 | 143 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Opposites
import Mathlib.CategoryTheory.Groupoid
/-!
# Facts about epimorphisms and monomorphisms.
The definitions of `Epi` an... | theorem IsSplitEpi.id {X Y : C} (f : X ⟶ Y) [hf : IsSplitEpi f] : section_ f ≫ f = 𝟙 Y :=
hf.exists_splitEpi.some.id
| Mathlib/CategoryTheory/EpiMono.lean | 122 | 123 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | γ.leftShift_v a n' hn' p (p + n') rfl (p + a) (by omega),
(γ.comp γ' h).leftShift_v a t' (by omega) p q hpq (p + a) (by omega),
smul_smul, Linear.units_smul_comp, assoc, Int.negOnePow_add, ← mul_assoc, ← h',
comp_v _ _ h (p + a) (p + n') q (by omega) (by omega)]
congr 2
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 381 | 385 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | subalgebra.
When `L` is Noetherian, it follows from Engel's theorem that the converse holds. See
`LieAlgebra.zeroRootSubalgebra_eq_iff_is_cartan` -/
theorem is_cartan_of_zeroRootSubalgebra_eq (h : zeroRootSubalgebra R L H = H) :
H.IsCartanSubalgebra :=
{ nilpotent := inferInstance
self_normalizing := by rw [... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 218 | 231 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Basic
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Data.Int.Basic
import Mathlib.Lean.Meta.CongrTheorems
import Mathli... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 533 | 537 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
/-!
# Higher differentiability over `ℝ` or `ℂ`
-/
... | simp [← NNReal.coe_le_coe, hC x]
end Real
| Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 144 | 148 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Ker... |
@[reassoc (attr := simp)]
lemma inr_opProdIsoCoprod_inv : coprod.inr ≫ (opProdIsoCoprod A B).inv = prod.snd.op := by
rw [Iso.comp_inv_eq, snd_opProdIsoCoprod_hom]
| Mathlib/CategoryTheory/Limits/Opposites.lean | 783 | 786 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,379 | 1,381 | |
/-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
/-!
# Prime ideals
## Main definitions
Throughout this file, `P` is at least a preorder, but some sections require mo... | Mathlib/Order/PrimeIdeal.lean | 231 | 235 | |
/-
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
-/
import Mathlib.Analysis.Calculus.BumpFunction.Basic
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.Lebesgu... | protected theorem integrable_normed : Integrable (f.normed μ) μ :=
f.integrable.div_const _
| Mathlib/Analysis/Calculus/BumpFunction/Normed.lean | 53 | 54 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
/-!
# Equicontinuity of a family of functions
Let `X` be a topological space and `α` a `UniformS... |
/-- A family `𝓕 : ι → X → α` is equicontinuous at `x₀` within `S` iff the function
`swap 𝓕 : X → ι → α` is continuous at `x₀` within `S`
*when `ι → α` is equipped with the topology of uniform convergence*. This is very useful for
| Mathlib/Topology/UniformSpace/Equicontinuity.lean | 488 | 491 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 1,626 | 1,633 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,510 | 1,520 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 187 | 190 | |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | theorem intervalIntegral_tendsto_integral_Iic (b : ℝ) (hfi : IntegrableOn f (Iic b) μ)
(ha : Tendsto a l atBot) :
Tendsto (fun i => ∫ x in a i..b, f x ∂μ) l (𝓝 <| ∫ x in Iic b, f x ∂μ) := by
let φ i := Ioi (a i)
have hφ : AECover (μ.restrict <| Iic b) l φ := aecover_Ioi ha
refine (hφ.integral_tendsto_of_... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 614 | 623 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.List.MinMax
import Mathlib.Data.Nat.Order.Lemmas
import Mathlib.Logic.Encodable.Basic
/-!
# Denumerable type... | theorem lt_succ_iff_le {x y : s} : x < succ y ↔ x ≤ y :=
⟨fun h => le_of_not_gt fun h' => not_le_of_gt h (succ_le_of_lt h'), fun h =>
lt_of_le_of_lt h (lt_succ_self _)⟩
/-- Returns the `n`-th element of a set, according to the usual ordering of `ℕ`. -/
def ofNat (s : Set ℕ) [DecidablePred (· ∈ s)] [Infinite s] :... | Mathlib/Logic/Denumerable.lean | 222 | 230 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Normed.Module.Completion
/-!
# Liouville's ... | have : HasDerivAt (e ∘ f) (e (deriv f c)) c :=
e.hasFDerivAt.comp_hasDerivAt c
(hd.differentiableAt isOpen_ball <| mem_ball_self hR).hasDerivAt
calc
‖deriv f c‖ = ‖deriv (e ∘ f) c‖ := by
rw [this.deriv]
exact (UniformSpace.Completion.norm_coe _).symm
_ ≤ C / R :=
norm_deriv_le_au... | Mathlib/Analysis/Complex/Liouville.lean | 71 | 84 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Basis.Submodule
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.Invarian... | (i : LinearIndependent R v) (w : Set M) [Fintype w] (s : range v ≤ span R w) :
Fintype.card ι ≤ Fintype.card w := by
-- We construct an injective linear map `(ι → R) →ₗ[R] (w → R)`,
-- by thinking of `f : ι → R` as a linear combination of the finite family `v`,
-- and expressing that (using the axiom of c... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 177 | 191 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Option.Defs
import Mathlib.Control.Functor
import Batteries.Data.List.Basic
import Mathlib.Control.Basic
/-!
# Traversable type class
Type classes for... | section Preserves
variable (η : ApplicativeTransformation F G)
@[functor_norm]
| Mathlib/Control/Traversable/Basic.lean | 121 | 125 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Data.List.Iterate
import Mathlib.GroupTheory.Perm.Cycle.Basic
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.Tactic.Group
/-!
# ... | (fun _ _ ↦ forall₂_imp (fun _ _ h hx hy ↦ (h hx hy).commute))
lemma disjoint_ofSubtype_noncommPiCoprod (u : Perm (Function.fixedPoints f))
(v : (c : { x // x ∈ f.cycleFactorsFinset }) → (Subgroup.zpowers (c : Perm α))) :
Disjoint (ofSubtype u) ((Subgroup.noncommPiCoprod f.pairwise_commute_of_mem_zpowers) v... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 606 | 614 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | simp only [idxOf_get] at x_eq_y; exact x_eq_y, fun h => by subst h; rfl⟩
@[deprecated (since := "2025-01-30")] alias indexOf_inj := idxOf_inj
| Mathlib/Data/List/Basic.lean | 677 | 680 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 930 | 931 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | let m := minFac (k + 2)
have : (k + 2) / m < (k + 2) := factors_lemma
rw [primeFactorsList]
refine List.Chain.cons ((le_minFac.2 h).resolve_left (by simp)) (primeFactorsList_chain ?_)
exact fun p pp d => minFac_le_of_dvd pp.two_le (d.trans <| div_dvd_of_dvd <| minFac_dvd _)
theorem primeF... | Mathlib/Data/Nat/Factors.lean | 93 | 104 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
/-!
# Pi
This file contains lemmas which establish bounds on `Real.pi`.
Notably, t... | 4756/3363, 14965/8099, 21183/10799, 49188/24713, 2-53/22000, 2-71/117869, 2-47/312092,
2-17/451533, 2-4/424971]
| Mathlib/Data/Real/Pi/Bounds.lean | 175 | 176 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,446 | 1,448 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topo... | classical exact ge_of_tendsto' hg fun s => sum_nonneg fun i _hi => (norm_nonneg _).trans (h i)
/-- If `∑' i, ‖f i‖` is summable, then `‖∑' i, f i‖ ≤ (∑' i, ‖f i‖)`. Note that we do not assume
| Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 119 | 121 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... |
theorem vecMul_surjective_iff_isUnit {A : Matrix m m R} :
Function.Surjective A.vecMul ↔ IsUnit A := by
rw [vecMul_surjective_iff_exists_left_inverse, exists_left_inverse_iff_isUnit]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 335 | 338 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | @[to_additive]
theorem finprod_mem_eq_prod_of_subset (f : α → M) {s : Set α} {t : Finset α}
| Mathlib/Algebra/BigOperators/Finprod.lean | 428 | 429 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | (l ++ l').getD n d = l.getD n d := by
rw [getD_eq_getElem _ _ (Nat.lt_of_lt_of_le h (length_append ▸ Nat.le_add_right _ _)),
getElem_append_left h, getD_eq_getElem]
| Mathlib/Data/List/GetD.lean | 73 | 76 |
/-
Copyright (c) 2021 Ivan Sadofschi Costa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ivan Sadofschi Costa
-/
import Mathlib.Data.Finsupp.Single
/-!
# `cons` and `tail` for maps `Fin n →₀ M`
We interpret maps `Fin n →₀ M` as `n`-tuples of elements of `M`,
We def... | intro i hi
suffices i = 0 ∨ ∃ a, ¬s a = 0 ∧ a.succ = i by simpa
apply (Fin.eq_zero_or_eq_succ i).imp id (Exists.imp _)
rintro i rfl
| Mathlib/Data/Finsupp/Fin.lean | 89 | 92 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Homology
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Preadditive.Opposite
/-!
# Homolo... | @[simp]
lemma opcyclesMap'_neg :
opcyclesMap' (-φ) h₁ h₂ = -opcyclesMap' φ h₁ h₂ := by
have γ : RightHomologyMapData φ h₁ h₂ := default
simp only [γ.opcyclesMap'_eq, γ.neg.opcyclesMap'_eq, RightHomologyMapData.neg_φQ]
| Mathlib/Algebra/Homology/ShortComplex/Preadditive.lean | 211 | 215 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.AssociatedPrime.Basic
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-20")
| Mathlib/RingTheory/Ideal/AssociatedPrime.lean | 83 | 103 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... |
theorem sum_take_map_length_splitWrtComposition (l : List α) (c : Composition l.length) (i : ℕ) :
(((l.splitWrtComposition c).map length).take i).sum = c.sizeUpTo i := by
congr
| Mathlib/Combinatorics/Enumerative/Composition.lean | 746 | 749 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Hom.BoundedLattice
/-!
# Complete lattice homomorphisms
This file defines frame homomorphisms and complete la... | theorem coe_comp (f : CompleteLatticeHom β γ) (g : CompleteLatticeHom α β) : ⇑(f.comp g) = f ∘ g :=
rfl
| Mathlib/Order/Hom/CompleteLattice.lean | 639 | 641 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... | section Group
variable (G : ι → Type*) [∀ i, Group (G i)]
| Mathlib/GroupTheory/CoprodI.lean | 220 | 222 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | `P a`. -/
@[simps]
def piecewise (f : Subtype P →₀ M) (g : {a // ¬ P a} →₀ M) : α →₀ M where
toFun a := if h : P a then f ⟨a, h⟩ else g ⟨a, h⟩
support := (f.support.map (.subtype _)).disjUnion (g.support.map (.subtype _)) <| by
simp_rw [Finset.disjoint_left, mem_map, forall_exists_index, Embedding.coe_subtype,
... | Mathlib/Data/Finsupp/Basic.lean | 1,256 | 1,265 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | coe_injective <| by
push_cast
exact image2_inter_right hf
theorem image₂_inter_subset_left [DecidableEq α] :
| Mathlib/Data/Finset/NAry.lean | 179 | 183 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 1,265 | 1,266 | |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.BigOperators.Fin
/-!
# Lemmas for tuples `Fin m → α`
This file contains alternative definitions of common opera... | def sum [Add α] [Zero α] : ∀ {m} (_ : Fin m → α), α
| 0, _ => 0
| 1, v => v 0
| _ + 2, v => sum (fun i => v (Fin.castSucc i)) + v (Fin.last _)
| Mathlib/Data/Fin/Tuple/Reflection.lean | 130 | 134 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | HasLineDerivWithinAt 𝕜 f f' s x v := by
apply HasDerivWithinAt.mono_of_mem_nhdsWithin h
apply ContinuousWithinAt.preimage_mem_nhdsWithin'' _ hst (by simp)
apply Continuous.continuousWithinAt; fun_prop
@[deprecated (since := "2024-10-31")]
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 242 | 247 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,656 | 1,666 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.Normed.Module.Ray
import Mathlib.Analysis.NormedSpace.Pointwise
/-!
# Strictly conv... | exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
variable [NormedSpace ℝ E]
/-- A real normed vector space is strictly convex provided that the unit ball is strictly convex. -/
theorem StrictConvexSpace.of_strictConvex_unitClosedBall [LinearMap.CompatibleSMul E E 𝕜 ℝ]
| Mathlib/Analysis/Convex/StrictConvexSpace.lean | 76 | 81 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | -- classical
theorem iInter_eq_empty_iff {f : ι → Set α} : ⋂ i, f i = ∅ ↔ ∀ x, ∃ i, x ∉ f i := by
| Mathlib/Data/Set/Lattice.lean | 909 | 910 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.Shift.ShiftSequence
import Mathlib.CategoryTheory.Triangulated.Functor
import Mathlib.CategoryT... |
variable (T)
variable [HasZeroObject C] [Preadditive C] [∀ (n : ℤ), (CategoryTheory.shiftFunctor C n).Additive]
[Pretriangulated C] [Abelian A] [F.IsHomological]
variable [F.ShiftSequence ℤ] (T T' : Triangle C) (hT : T ∈ distTriang C)
| Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean | 178 | 182 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Logic.Encodable.Pi
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.MeasurableSpace.... | apply boundedBy_caratheodory
intro t
simp_rw [piPremeasure]
refine Finset.prod_add_prod_le' (Finset.mem_univ i) ?_ ?_ ?_
· simp [image_inter_preimage, image_diff_preimage, measure_inter_add_diff _ hs, le_refl]
| Mathlib/MeasureTheory/Constructions/Pi.lean | 197 | 201 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.Algebra.Polynomial.Eval.SMul
import Mathli... | le_trans (natDegree_sub_le _ _) (by simp)
rw [← isNilpotent_reflect_iff aux, reflect_sub, ← reverse, M.reverse_charpoly]
simpa [p, hp]
end reverse
end Matrix
| Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 366 | 374 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
/-!
# Multiplicative... | [CharP R' p] : χ ^ p = χ := by
ext x
rw [pow_apply_coe]
rcases hχ x with (hx | hx | hx) <;> rw [hx]
| Mathlib/NumberTheory/MulChar/Basic.lean | 479 | 482 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.NormedSpace.Extr
import Mathlib.Data.Complex... | set e : F →L[ℂ] F̂ := UniformSpace.Completion.toComplL
have he : ∀ x, ‖e x‖ = ‖x‖ := UniformSpace.Completion.norm_coe
replace hz : IsMaxOn (norm ∘ e ∘ f) (closedBall z (dist w z)) z := by
simpa only [IsMaxOn, Function.comp_def, he] using hz
simpa only [he, Function.comp_def]
using norm_max_aux₁ (e.diffe... | Mathlib/Analysis/Complex/AbsMax.lean | 144 | 151 |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... |
@[simp] theorem simple_mul_simple_pow' (i i' : B) : (s i' * s i) ^ M i i' = 1 :=
M.symmetric i' i ▸ cs.simple_mul_simple_pow i' i
/-- The simple reflections of `W` generate `W` as a group. -/
| Mathlib/GroupTheory/Coxeter/Basic.lean | 230 | 234 |
/-
Copyright (c) 2022 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.L1Space.AEEqFun
import Mathlib.MeasureTheory.Function.LpSpace.Complete
import Mathlib.MeasureThe... | of their images is a subset of `{0}`).
-/
@[elab_as_elim]
theorem MemLp.induction [_i : Fact (1 ≤ p)] (hp_ne_top : p ≠ ∞) (motive : (α → E) → Prop)
(indicator : ∀ (c : E) ⦃s⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun _ => c))
(add : ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → Me... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 839 | 891 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,701 | 1,711 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Analysis.SpecialFunctions.... | refine (Complex.tendsto_norm_tan_of_cos_eq_zero hx).comp ?_
refine Tendsto.inf Complex.continuous_ofReal.continuousAt ?_
exact tendsto_principal_principal.2 fun y => mt Complex.ofReal_inj.1
theorem tendsto_abs_tan_atTop (k : ℤ) :
Tendsto (fun x => abs (tan x)) (𝓝[≠] ((2 * k + 1) * π / 2)) atTop :=
tendsto... | Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean | 34 | 40 |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | -- -- squeezed from `ext, simp [sphere_ext_iff, stereographic'_symm_apply, real_inner_comm]`
simp only [PartialHomeomorph.trans_toPartialEquiv, PartialHomeomorph.symm_toPartialEquiv,
| Mathlib/Geometry/Manifold/Instances/Sphere.lean | 390 | 391 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | QuaternionAlgebra R (-1) (0) (-1)
| Mathlib/Algebra/Quaternion.lean | 731 | 731 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad
-/
import Mathlib.Logic.Basic
import Mathlib.Logic.Function.Defs
import Mathlib.Order.Defs.LinearOrder
/-!
# Booleans
This file proves various... | theorem coe_sort_false : (false : Prop) = False := by simp
theorem coe_sort_true : (true : Prop) = True := by simp
| Mathlib/Data/Bool/Basic.lean | 68 | 70 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... | Mathlib/Topology/Basic.lean | 1,249 | 1,252 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 223 | 225 | |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,259 | 1,260 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | induction hx using iSup_induction' with
| mem i x hx =>
refine
(Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => ?_).trans
(orthogonalProjection_eq_self_iff.mpr hx)
rw [orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Submodule.coe_zero]
exact hV.isOrtho hij.... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,250 | 1,258 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
-- TODO: Rename `List.reverse_perm` to `List.reverse_perm_self`
@[simp] lemma reverse_perm' : l₁.reverse ~ l₂ ↔ l₁ ~ l₂ where
mp := l₁.reverse_perm.symm.trans
mpr := l₁.reverse_perm.trans
| Mathlib/Data/List/Basic.lean | 284 | 288 |
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